这半个月的 16 篇 Agent Memory 论文方向出奇一致：GraphRAG 式的静态记忆图被集体宣判不够用，记忆要变成会巩固、会演化的子系统。

我把这批论文按四条趋势主线做了梳理，分篇速览和链接在文末。

最近半个月的 arXiv 和 ACL 上，Graph-based Agent Memory 方向密集冒出了一批论文。把它们放在一起看，能清晰看到这个领域正在发生的一次转向：记忆从“检索中间件“变成“会演化的系统“。

四条趋势主线

1. 从静态图 → 自演化记忆引擎

去年 GraphRAG 系的工作把记忆图当作静态检索中间件：建好图，查它。今年的论文几乎都在打破这一点。SAGE 直接把目标定为 Self-Evolving——记忆图要在使用中更新结构、恢复证据链，而不是建完就不动。EverMemOS 更进一步，把记忆做成“自组织操作系统“，长期交互里记忆自己整理自己。

2. 从单一语义相似度 → 多信号路由

旧方法（RAG 和第一代 GraphRAG）靠语义相似度把查询映射到记忆片段，被论文们称为 System-1 检索：快，但分不清时间、因果、实体关系。新一批工作把多个信号拆开再融合——MAGMA 用多图分离时间/因果/实体信息；Mnemis 用层次图做双路由，显式加入 System-2 式推理；LiCoMemory 试图用轻量结构保持认知性，避免图越建越重。

3. 认知科学回归：情景-语义-巩固

一批论文直接回到人类记忆的认知模型。Episodic-Semantic、Synapse 用情景记忆（具体事件）+ 语义记忆（抽象知识）双通道；RecMem 和 TiMem 引入“记忆巩固“（consolidation）——像睡眠一样把短期交互压成长期结构，而不是每条交互都原样存下来。

4. 成本意识：记忆系统不能比模型还贵

RecMem 的出发点很直白：现有记忆系统对每条进来的交互都调用一次 LLM 做抽取，长期运行贵得离谱。它用循环结构替代逐条 LLM 调用。LiCoMemory、HiMem 也都在讲“轻量“和“可扩展“——记忆自身的 token 成本开始被当作一等公民。

分论文速览

一句话总结

这一批论文的共同动作，是把记忆从“一个查相似度的数据库“升级成“一个会巩固、会演化、分情景和语义、按成本设计的子系统“。对做 agent 工程的人来说，可抄的作业是：别再用一个向量库当记忆了——时间、因果、实体分开存，短期交互定期巩固成长期结构，记忆维护别逐条烧 LLM。

原文链接： • SAGE: https://arxiv.org/abs/2605.12061 • MAGMA: https://arxiv.org/abs/2601.03236 • Mnemis: https://aclanthology.org/2026.acl-long.1096/… • EverMemOS: https://aclanthology.org/2026.acl-long.2125/… • RecMem: https://aclanthology.org/2026.findings-acl.1619/… • Taxonomy: https://arxiv.org/abs/2602.05665 #AgentMemory #GraphRAG #LLMAgents

SAGE: A Self-Evolving Agentic Graph-Memory Engine for Structure-Aware Associative Memory

Source: https://arxiv.org/html/2605.12061 Juntong Wang1,2Haoyue Zhao3Guanghui Pan3Yanbo Wang1,2 Xiyuan Wang1,2Qiyan Deng3Muhan Zhang1 1Institute for Artificial Intelligence, Peking University 2School of Intelligence Science and Technology, Peking University 3School of Computer Science and Technology, Beijing Institute of Technology [email protected],[email protected],[email protected], [email protected],[email protected],[email protected],[email protected]

Abstract

Long-term memory is becoming a central bottleneck for language agents. Existing RAG and GraphRAG systems largely treat memory graphs as static retrieval middleware, which limits their ability to recover complete evidence chains from partial cues, exploit reusable graph-structural roles, and improve the memory itself through downstream feedback. We introduceSAGE, aSelf-evolvingAgenticGraph-memoryEngine that models graph memory as a dynamic long-term memory substrate.SAGEcouples two roles: a memory writer that incrementally constructs structured graph memory from interaction histories, and a Graph Foundation Model-based memory reader to perform retrieval and provide feedback to the memory writer. We provide rigorous theoretical analyses supporting the effectiveness of carefully designed architectural components and the framework. Across multi-hop QA, open-domain retrieval, domain-specific review QA, and long-term agent-memory benchmarks, SAGE improves evidence recovery, answer grounding, and retrieval efficiency: after two self-evolution rounds, it achieves the best average rank on multi-hop QA; in zero-shot open-domain transfer, it reaches 82.5/91.6 Recall@2/5 on NQ. Further results on LongMemEval and HaluMem show that training and reader–writer feedback improve multiple long-term memory and hallucination-diagnostic metrics, suggesting that self-evolving, structure-aware graph memory is a promising foundation for robust long-horizon language agents. Our code is availablehere.

1Introduction

As large language models evolve from single-turn question-answering systems into general-purpose agents for multi-turn dialogue, personalized assistance, multi-agent collaboration, and open-environment exploration, the system bottleneck is shifting from whether a model can answer within the current context to whether it can accumulate, organize, invoke, and update memory over longer time scales. Memory is a core system capability that determines whether Agents can achieve long-term consistency, personalized adaptation, cross-turn reasoning, and self-improvement.Memory is to Agents what parameters are to foundation models(Parket al.,2023; Zhonget al.,2024; Packeret al.,2023; Wuet al.,2025a; Yanget al.,2026). Recent memory benchmarks have made this bottleneck explicit, evaluating agents on ultra-long conversational consistency, multi-session reasoning, temporal reasoning, knowledge updating, selective forgetting, abstention, and hallucination control(Maharanaet al.,2024;Di Wuet al.,; Huet al.,2025; Chenet al.,2025a; Liet al.,2026b).

In engineering practice, RAG has become the dominant non-parametric interface for extending language models with external memory, alleviating the static nature of parametric knowledge and the limited size of context windows(Lewiset al.,2020). Yet standard RAG usually retrieves independent text chunks, whereas long-term agent memory often requires recovering evidence distributed across entities, events, aliases, temporal constraints, and multi-hop dependencies. GraphRAG takes an important step by organizing documents, entities, relations, and summaries as graphs, making cross-document dependencies and reasoning paths more explicit(Edgeet al.,2024; Gutiérrezet al.,2024). However, for long-horizon agents, graph structure should not merely serve as an external retrieval index. In this work, we studyagent graph memoryas a coupled write–read–update problem. Given interaction histories or external documents, a memory writer should construct anevolving graphwhose nodes and edges encode entities, episodes, documents, aliases, temporal constraints, and cross-fragment relations. Given a query, a memory reader should not simply expand from a few matched entities; it should return acompact, verifiable evidence chain. The retrieval outcome should furtherprovide feedback about what the graph lacks. In other words, the graph is not only built before retrieval and searched afterward; it is theworking substrate through which memory is written, read, corrected, and self-improved. Around this goal, we identify the following three core challenges.

Figure 1:Overview of the three core challenges in agent graph memory, illustrated with a concrete example. Given the query,“Alice mentioned a work in last week’s lab meeting that seemed to be inspired by the Cornu Ammonis. Among works in the same field as that work, are there any that can also help with agent memory? Give one example,”the memory reader must address associative and selective reading by expanding sparse partial cues into the correct evidence chain while avoiding noisy distractors. It must then exploit structural information in the memory graph, such as aliases, bridges, and hubs, to traverse from Cornu Ammonis to SAGE. Self-evolving memory highlights the closed loop between writing and reading.Challenge I: Agent memory requires global associative reading from fragmented cues.The first challenge is not merely to retrieve text that is semantically similar to the query, but to reconstruct a complete reasoning chain from sparse, fragmented, and sometimes indirect cues. In long-term agent memory, a query may mention only an episodic clue, an alias, or a distant conceptual hint, while the answer depends on intermediate entities that are not explicitly named. Standard vector retrieval tends to return locally similar snippets, and many graph-based retrieval methods start propagation from a small set of query-matched anchor entities. However, if these anchors only cover a local subgraph, the necessary bridge nodes may lie outside the activated region, leaving the evidence chain disconnected even after graph propagation. Thus, agent memory reading should not commit too early to a small set of partial cues(Trivediet al.,2023; Gutiérrezet al.,2024).

Challenge II: Agent memory requires learned structural use rather than fixed structural expansion.The second challenge is that graph structure should not be used only as a fixed index after graph construction. Many GraphRAG-style systems exploit structure through pre-built communities, paths, graph indexes, or heuristic expansion rules, but once the graph is constructed, the role of structure is largely fixed: a hub remains broadly expanded, a bridge may be missed if it is not reached by the initial anchors, and noisy shortcuts may be treated similarly to useful evidence edges. This is insufficient for agent memory, where the graph itself is continuously updated by new interactions and where the same topological pattern may have different meanings across domains. A structure-aware reader should therefore learn how structural roles affect retrieval(Edgeet al.,2024; Gutiérrezet al.,2024; Liuet al.,2025; Luoet al.,2025).

The example in Figure1illustrates both challenges. Given the query, the explicit cues are onlyAlice,lab meeting,Cornu Ammonis, andagent memory. A retrieval system that only anchors on the most query-matched nodes may retrieve the meeting note or the biological cue, but still fail to connect them tohippocampus,HippoRAG,GraphRAG, and finallySAGE. This is the associative-reading challenge: the system must piece together a long chain from scattered cues. At the same time, the correct path depends on structural roles:HippoRAGis a bridge,GraphRAGandRAGare hubs that must be controlled rather than blindly expanded, and the edge fromGraphRAGtoSAGEis critical for reaching the final answer. This is the structural-information challenge: the reader must use graph topology in a learned and selective way, not simply propagate uniformly over a fixed graph.

Challenge III: Existing methods mostly optimize retrieval trajectories, but rarely optimize the self-evolution of the memory system itself.Existing RAG and GraphRAG systems often assume that the external memory graph or knowledge base is already available, so the main problem becomes how to retrieve from it. For long-term agents, however, writing is itself part of the memory problem. Conversely, retrieval failures provide useful signals about what the memory graph lacks. For example, if the reader repeatedly needs to traverse fromCornu Ammonistohippocampus-style GraphRAGand then to the GraphRAG literature, the memory system should gradually add or strengthen useful structural links, such as a more direct edge fromhippocampus-style GraphRAGtoGraphRAG. Thus, a true agent memory system should not only optimize retrieval trajectories; it should optimize the memory graph itself through a closed loop in which better reading exposes writing deficiencies, and better writing makes future reading more accurate, selective, and efficient(Chenet al.,2025a).

To address these challenges, we proposeSAGE, aSelf-evolvingAgenticGraph-memoryEngine. Unlike GraphRAG systems that mainly use graphs as retrieval middleware,SAGEtreats the graph as a dynamic long-term memory object. It couples two mutually reinforcing components: a memory writer incrementally constructs and revises graph memory; a Graph Foundation Model-based memory reader to perform retrieval and provide feedback to the memory writer.SAGEdirectly targets the three challenges above: it recovers long reasoning chains from fragmented cues, learns how to use structural roles rather than propagate uniformly, and continuously improves the graph memory itself for future queries.

2Related Work

Retrieval-Augmented Generation and GraphRAG.

Retrieval-Augmented Generation (RAG) provides a non-parametric interface for language models by retrieving external evidence before generation(Lewiset al.,2020). Many variants further improve retrieval timing, reasoning interaction, adaptive policies, and hierarchical organization(Jianget al.,2023; Trivediet al.,2023; Asaiet al.,2023; Jeonget al.,2024; Sarthiet al.,2024). GraphRAG enables structured retrieval over cross-document dependencies and multi-hop evidence paths(Edgeet al.,2024; Heet al.,2024; Guoet al.,2024; Liet al.,2024; Wang and Han,2025; Xuet al.,2025a; Zhaoet al.,2025; Zhanget al.,2025c; Gutiérrezet al.,2024,2025; Luoet al.,2025). Another line of work improves retrieval by optimizing retrieval trajectories, including interleaved retrieval and reasoning, self-reflective retrieval, adaptive retrieval, multi-agent RAG, and reinforcement-learning-based query rewriting(Trivediet al.,2023; Asaiet al.,2023; Jeonget al.,2024; Chenet al.,2025b; Chaet al.,2025; Tsanget al.,2025).

Agent Memory.

Agent memory studies how LLM-based agents store, update, retrieve, and use past experiences(Parket al.,2023; Zhonget al.,2024; Packeret al.,2023; Chhikaraet al.,2025; Xuet al.,2025b; Rasmussenet al.,2025; Kanget al.,2025; Zhanget al.,2025a; Wuet al.,2025b; Zhanget al.,2025b; Huanget al.,2025; Yueet al.,2026). Recent surveys also highlight the importance of human-inspired and graph-based memory mechanisms for LLM agents(Wuet al.,2025a; Yanget al.,2026). Meanwhile, memory benchmarks evaluate long-term consistency, event reasoning, multi-session reasoning, temporal reasoning, knowledge updating, selective forgetting, abstention, and hallucination control(Maharanaet al.,2024;Di Wuet al.,; Huet al.,2025; Chenet al.,2025a; Liet al.,2026b).

Graph Foundation Models.

Graph Foundation Models (GFMs) aim to learn transferable graph representations through large-scale pretraining, allowing models to reuse structural priors and semantic patterns across graphs, tasks, and domains(Liuet al.,2025). Representative early works include GCC, GPT-GNN, and GraphCL, which learn transferable graph representations through cross-network contrast, generative graph pretraining, and graph augmentation based contrastive learning(Qiuet al.,2020; Huet al.,2020; Youet al.,2020; Yuet al.,2025).

3Preliminary

Given a knowledge-intensive memory samplex=(q,𝒟,𝒟+,y)x=(q,\mathcal{D},\mathcal{D}^{+},y), whereqqdenotes the query,𝒟={di}i=1N\mathcal{D}=\{d_{i}\}_{i=1}^{N}denotes the set of candidate historical memory fragments,𝒟+⊆𝒟\mathcal{D}^{+}\subseteq\mathcal{D}denotes the gold evidence set that supports the answer, andyydenotes the ground-truth answer. The writer is viewed as a structured policy model: at stephh, given stateshs_{h}, the policy samples a writing actionah∼πθ(⋅∣sh)a_{h}\sim\pi_{\theta}(\cdot\mid s_{h})and updates the partial graph as𝒢h+1=𝒢h⊕ah\mathcal{G}_{h+1}=\mathcal{G}_{h}\oplus a_{h}. The memory readerℛϕ\mathcal{R}_{\phi}performs query-conditioned propagation over the graph, obtains entity relevance scores𝐬E=fϕ​(q,𝒢)∈ℝ|𝒱E|\mathbf{s}_{E}=f_{\phi}(q,\mathcal{G})\in\mathbb{R}^{|\mathcal{V}_{E}|}, and then projects them into memory-fragment scores. The reader finally outputs(𝒟^k,𝒢^q,Πq)=ℛϕ​(q,𝒢,𝐌)(\widehat{\mathcal{D}}_{k},\widehat{\mathcal{G}}_{q},\Pi_{q})=\mathcal{R}_{\phi}(q,\mathcal{G},\mathbf{M}), where𝒟^k=TopKd∈𝒟​(𝐬D​(d))\widehat{\mathcal{D}}_{k}=\mathrm{TopK}_{d\in\mathcal{D}}(\mathbf{s}_{D}(d)),𝒢^q\widehat{\mathcal{G}}_{q}is the query-activated subgraph, andΠq\Pi_{q}denotes optional relational paths. The generation model then produces the answery^=LLM​(q,𝒟^k,Πq)\widehat{y}=\mathrm{LLM}(q,\widehat{\mathcal{D}}_{k},\Pi_{q}).

4Method

Figure 2:Overall pipeline of the proposedSAGE. The memory writer incrementally constructs and updates graph memory from observations through state-conditioned writing actions, and receives rewards from downstream memory use. The resulting retrieval feedback closes the loop between writing and reading, enabling graph memory to improve over time.At a high level, our method builds a self-evolving graph memory pipeline (Figure2). The memory writer𝒲θ\mathcal{W}_{\theta}first transforms the query and candidate historical memory fragments into a heterogeneous graph memory𝒢\mathcal{G}. The memory readerℛϕ\mathcal{R}_{\phi}then performs query-conditioned activation over𝒢\mathcal{G}: it softly locates query-relevant entities, propagates evidence signals through relational structures, and projects the activated entity-level information back to memory fragments.

4.1Memory Writer: Graph Memory Writing via Reading Feedback

Policy-based writing.

The writer is modeled as a sequential decision-making policy. At steptt, the state is defined asst=(q,𝒟,𝒢t−1,𝒟t−1proc)s_{t}=\big(q,\mathcal{D},\mathcal{G}_{t-1},\mathcal{D}^{\mathrm{proc}}_{t-1}\big), where𝒢t−1\mathcal{G}_{t-1}is the partially written graph, and𝒟t−1proc\mathcal{D}^{\mathrm{proc}}_{t-1}denotes the set of processed documents. The actionata_{t}contains entity-relation triples(u,r,v)(u,r,v)together with their source anchors(u,source,d)(u,\texttt{source},d). Detailed implementation information is provided in AppendixO.

Reader-aware Writing Reward.

The writer’s reward stems from the task utility of its written graph after being accessed by the memory reader. Given the current graph𝒢\mathcal{G}, the frozen reader returns the evidencePk​(q,𝒢)P_{k}(q,\mathcal{G}). Inspired by(Tsanget al.,2025), we employ two complementary types of rewards. The first category measures whether the graph is sufficient as a knowledge carrier to support the derivation of the answer:rded​(q,y,𝒢)=𝕀​[Judge⁡(q,y∣Pk​(q,𝒢))=Yes]r_{\mathrm{ded}}(q,y,\mathcal{G})=\mathbb{I}\left[\operatorname{Judge}\big(q,y\mid P_{k}(q,\mathcal{G})\big)=\texttt{Yes}\right]. The second category measures whether the graph can serve as a knowledge index to recover the supporting text:rrec​(q,𝒟+,𝒢)=|Pk​(q,𝒢)∩𝒟+||𝒟+|,rpre​(q,𝒟+,𝒢)=|Pk​(q,𝒢)∩𝒟+||Pk​(q,𝒢)|r_{\mathrm{rec}}(q,\mathcal{D}^{+},\mathcal{G})=\frac{|P_{k}(q,\mathcal{G})\cap\mathcal{D}^{+}|}{|\mathcal{D}^{+}|},r_{\mathrm{pre}}(q,\mathcal{D}^{+},\mathcal{G})=\frac{|P_{k}(q,\mathcal{G})\cap\mathcal{D}^{+}|}{|P_{k}(q,\mathcal{G})|}. Whererrecr_{\mathrm{rec}}encourages the coverage of necessary evidence, whilerprer_{\mathrm{pre}}penalizes the expansion of irrelevant evidence. To align with end-to-end question answering, we also use an answer-level auxiliary rewardrans​(q,y,𝒢)=maxy′∈𝒴​(y)⁡F1​(y^,y′),y^=LLM⁡(q,Pk​(q,𝒢))r_{\mathrm{ans}}(q,y,\mathcal{G})=\max_{y^{\prime}\in\mathcal{Y}(y)}\mathrm{F1}\big(\hat{y},y^{\prime}\big),\hat{y}=\operatorname{LLM}\big(q,P_{k}(q,\mathcal{G})\big), where𝒴​(y)\mathcal{Y}(y)is the set of answer aliases. In practice, we adopt a hybrid task rewardrtask=α​rrec+β​rpre+γ​rdedα+β+γr_{\mathrm{task}}=\frac{\alpha r_{\mathrm{rec}}+\beta r_{\mathrm{pre}}+\gamma r_{\mathrm{ded}}}{\alpha+\beta+\gamma}.

Furthermore, to prevent the policy from inflating the graph size by stacking duplicate triples, we define a repetition rate:ρrep​(𝒢)=|𝒯​(𝒢)|−|uniq⁡(𝒯​(𝒢))||𝒯​(𝒢)|\rho_{\mathrm{rep}}(\mathcal{G})=\dfrac{|\mathcal{T}(\mathcal{G})|-|\operatorname{uniq}(\mathcal{T}(\mathcal{G}))|}{|\mathcal{T}(\mathcal{G})|}and derive the trajectory returnR​(τ)=rtask​(τ)−λrep​ρrep​(𝒢τ)+λfmt​∑t=1|τ|rtfmtR(\tau)=r_{\mathrm{task}}(\tau)-\lambda_{\mathrm{rep}}\rho_{\mathrm{rep}}(\mathcal{G}_{\tau})+\lambda_{\mathrm{fmt}}\sum_{t=1}^{|\tau|}r^{\mathrm{fmt}}_{t}. This directly addresses the issue revealed by works such as HaluMem: errors in memory systems often do not emerge only at the answering stage, but are already written during the extraction and updating phases(Chenet al.,2025a). We employ standard clipped GRPO to update the writer.

4.2Memory Reader: Memory Retrieval Based on Graph Foundation Model

The memory reader must operate stably over graph memory that is continuously updated by the writer. Dense retrievers mainly learn query–document semantic matching and thus struggle to exploit entity roles, bridge paths, and cross-community dependencies, while conventional GNN retrievers are often tied to fixed graph distributions and generalize poorly across domains, users, and evolution stages. We therefore adopt a Graph Foundation Model (GFM) as the memory reader, whose multi-graph pre-training enables transferable structural priors and lightweight calibration on new graphs(Luoet al.,2025; Zhanget al.,2025c). Formally, the memory reader outputs an entity distribution, a document distribution, and an optional retrieval subgraphfϕ​(q,𝒢,𝒟)=(pϕ​(e∣q,𝒢),pϕ​(d∣q,𝒢,𝒟),𝒢q)f_{\phi}(q,\mathcal{G},\mathcal{D})=\big(p_{\phi}(e\mid q,\mathcal{G}),p_{\phi}(d\mid q,\mathcal{G},\mathcal{D}),\mathcal{G}_{q}\big). Wherepϕ​(e∣q,𝒢)p_{\phi}(e\mid q,\mathcal{G})represents the entity memory activated by the query,pϕ​(d∣q,𝒢,𝒟)p_{\phi}(d\mid q,\mathcal{G},\mathcal{D})denotes the final retrieved textual evidence, and𝒢q\mathcal{G}_{q}provides an interpretable retrieval path. To obtain a compact and query-aligned activated subgraph, we further introduce a lightweight query-conditioned subgraph selector; implementation details are provided in AppendixI.

Cognition-inspired Structured Query Planning.

When humans extract long-term memories, the brain often automatically generates multi-dimensional retrieval cues to anchor the target based on only a vague final intention. Inspired by this, we no longer treat the natural language query as a single retrieval command. Instead, we introduce a planning function𝒫ω\mathcal{P}_{\omega}to simulate the cue reconstruction process of the human brain before awakening memory, decomposing the initial query into a set of rich associative probes:𝒫ω​(q)=(ℰexp,𝒜,𝒞rel,𝒞hard,τ,{(q~m,αm,tm)}m=1M)\mathcal{P}_{\omega}(q)=\Big(\mathcal{E}_{\mathrm{exp}},\mathcal{A},\mathcal{C}_{\mathrm{rel}},\mathcal{C}_{\mathrm{hard}},\tau,\{(\tilde{q}_{m},\alpha_{m},t_{m})\}_{m=1}^{M}\Big). Detailed definitions of the notation, additional information, and the concrete prompt templates and output schema are provided in AppendixK. This multi-path concurrent awakening method effectively overcomes the “tip-of-the-tongue phenomenon” (i.e., difficulties in alias alignment or missing bridging entities) and naturally stitches together forgotten implicit relationships(Trivediet al.,2023; Asaiet al.,2023; Wuet al.,2025b; Zhanget al.,2025b).

Soft Addressing and Pre-activation of Memory Fragments.

Cognitive neuroscience reveals that human memory retrieval involves not only the extraction of perfectly matching information but also the instinctive awakening of peripherally related memories throughSemantic Priming. And to address the first challenge, we treat the calculation of the query-conditioned entry scorese​(q)s_{e}(q)as a comprehensive assessment of the stimulus intensity across differentMemory Engrams:

se​(q)=\displaystyle s_{e}(q)=λ1​Exact⁡(e,ℰexp)+λ2​Alias⁡(e,𝒜)+λ3​maxm≤M⁡cos⁡(Emb⁡(desc⁡(e)),Emb⁡(q~m))\displaystyle\lambda_{1}\,\operatorname{Exact}(e,\mathcal{E}_{\mathrm{exp}})+\lambda_{2}\,\operatorname{Alias}(e,\mathcal{A})+\lambda_{3}\max_{m\leq M}\cos\big(\operatorname{Emb}(\operatorname{desc}(e)),\operatorname{Emb}(\tilde{q}_{m})\big)+λ4​Type⁡(e,τ)+λ5​Cons⁡(e,𝒞hard)+λ6​∑ξ∈NER⁡(q)EL⁡(e∣ξ).\displaystyle+\lambda_{4}\,\operatorname{Type}(e,\tau)+\lambda_{5}\,\operatorname{Cons}(e,\mathcal{C}_{\mathrm{hard}})+\lambda_{6}\sum_{\xi\in\operatorname{NER}(q)}\operatorname{EL}(e\mid\xi).(1)Subsequently, the system employs a Softmax function with a temperature coefficientT0T_{0}to simulate the brain’s limited Attention Allocation mechanism during retrieval. This normalizes the multi-dimensional stimulus signals to form the initial activation distribution of the memory atlasp0​(e∣q)=exp⁡(se​(q)/T0)∑v∈𝒱Eexp⁡(sv​(q)/T0)p_{0}(e\mid q)=\frac{\exp(s_{e}(q)/T_{0})}{\sum_{v\in\mathcal{V}_{E}}\exp(s_{v}(q)/T_{0})}. Based on this distribution, we define the initial state of the memory nodes as𝐡e(0)=(p0​(e∣q))η​Wq​Emb⁡(q)+Wx​𝐱e\mathbf{h}_{e}^{(0)}=\big(p_{0}(e\mid q)\big)^{\eta}W_{q}\operatorname{Emb}(q)+W_{x}\mathbf{x}_{e}. In this process,𝐱e\mathbf{x}_{e}acts as the solidifiedlong-term memory(static representation of entities) in the brain, while the query vector adjusted by the cognitive recall degreep0​(e∣q)p_{0}(e\mid q)represents the currentworking memory(task context).

Synapse-inspired Structurally Conditioned Associative Propagation.

To address the second challenge while avoiding indiscriminate diffusion, we introduce edge-level vector structural gating in the GFM. The node-level structural features, edge-pair structural features, and graph-level summary are defined as:

ϕ​(v)\displaystyle\phi(v)=[log⁡(1+dv),cv,κv,d¯𝒩​(v)],\displaystyle=\big[\log(1+d_{v}),c_{v},\kappa_{v},\bar{d}_{\mathcal{N}(v)}\big],(2)ψ​(u,v)\displaystyle\psi(u,v)=[|du−dv|,|𝒩​(u)∩𝒩​(v)|,Jaccard⁡(𝒩​(u),𝒩​(v))],\displaystyle=\big[|d_{u}-d_{v}|,|\mathcal{N}(u)\cap\mathcal{N}(v)|,\operatorname{Jaccard}(\mathcal{N}(u),\mathcal{N}(v))\big],(3)𝐫𝒢\displaystyle\mathbf{r}_{\mathcal{G}}=[meanv∈𝒱E⁡ϕ​(v);stdv∈𝒱E⁡ϕ​(v);dens⁡(𝒢)].\displaystyle=\big[\operatorname{mean}_{v\in\mathcal{V}_{E}}\phi(v);\operatorname{std}_{v\in\mathcal{V}_{E}}\phi(v);\operatorname{dens}(\mathcal{G})\big].(4)Detailed definitions and normalization procedures are provided in AppendixL. The edge structural context for thell-th layer is𝐳u​v(l)=[En(l)​(ϕ​(u));En(l)​(ϕ​(v));Ep(l)​(ψ​(u,v));Eg(l)​(𝐫𝒢)]\mathbf{z}_{uv}^{(l)}=\big[E_{n}^{(l)}(\phi(u));E_{n}^{(l)}(\phi(v));E_{p}^{(l)}(\psi(u,v));E_{g}^{(l)}(\mathbf{r}_{\mathcal{G}})\big], which generates the vector gating𝐠u​v(l)=𝟏+δ​tanh⁡(MLPg(l)⁡(𝐳u​v(l)))\mathbf{g}_{uv}^{(l)}=\mathbf{1}+\delta\tanh\big(\operatorname{MLP}_{g}^{(l)}(\mathbf{z}_{uv}^{(l)})\big). Letηu​v\eta_{uv}be the normalized adjacency weight with self-loops; the message and node updates are𝐦u→v(l)=ηu​v​𝐠u​v(l)⊙Wm(l)​𝐡u(l−1),𝐡v(l)=LayerNorm⁡(𝐡v(l−1)+PReLU⁡(𝐛(l)+∑u∈𝒩​(v)𝐦u→v(l)))\mathbf{m}_{u\to v}^{(l)}=\eta_{uv}\,\mathbf{g}_{uv}^{(l)}\odot W_{m}^{(l)}\mathbf{h}_{u}^{(l-1)},\mathbf{h}_{v}^{(l)}=\operatorname{LayerNorm}\left(\mathbf{h}_{v}^{(l-1)}+\operatorname{PReLU}\left(\mathbf{b}^{(l)}+\sum_{u\in\mathcal{N}(v)}\mathbf{m}_{u\to v}^{(l)}\right)\right). Unlike traditional heuristic path expansion, PPR walks, or community summarization(Edgeet al.,2024; Guoet al.,2024; Wang and Han,2025), the system here can actively performInhibitionof non-specific generalized memories (suppressing hub edges), keenly capture and preservelong-distance associationsacross different cognitive clusters (lateral thinking/bridge edge preservation), and undergoHabituation(weakening redundant edges) toward highly repetitive local information, much like the human brain.

Traditional query-dependent GNNs or PPR-style expansion can perform multi-hop propagation along graph structures. But the key issue is not simply to expand the propagation range, but to preserve the advantage of query-relevant evidence signal over distractor noise under a limited top-kkbudget. Proposition4.2(i) summarizes this signal–budget view: soft addressing improves the initial evidence activation, structural gating preserves bridge/evidence paths while suppressing noisy neighborhoods, and controlled entity-to-document projection converts the entity-level advantage into more efficient document-level retrieval. Complete definitions, assumptions, and proofs are provided in AppendixB.

Target Graph Calibration and Cross-graph Structural Priors.

Human memory, on one hand, reorganizes cues based on the current context, while on the other, it retains relatively stable structured recall habits. In our self-evolving graph memory, each𝒢\mathcal{G}generated by the writer per round alters the local topology and noise distribution; therefore, the reader cannot rely solely on propagation patterns from a fixed graph.

Since the writer continuously changes the memory graph, the reader must simultaneously adapt to the current target graph and preserve cross-graph structural priors. This is precisely why we introduce the context–schema decomposition. As summarized in Proposition4.2(ii), the schema channel provides a transferable structural prior, while the context channel corrects the target-graph residual induced by the current writer, current domain, entity granularity, and local noise. The complete theoretical motivation is provided in AppendixCand AppendixD.

First, a feature prompt vector𝐩f\mathbf{p}_{f}is used for a lightweight calibration of the query-activated input𝐡~e(0)=𝐩f⊙𝐡e(0)\tilde{\mathbf{h}}_{e}^{(0)}=\mathbf{p}_{f}\odot\mathbf{h}_{e}^{(0)}. The contextual calibration channel performs gated propagation on the current graph𝒢\mathcal{G}:𝐇ctx=Fgate​(𝐇~(0),𝒢;Θgate)\mathbf{H}_{\mathrm{ctx}}=F_{\mathrm{gate}}(\tilde{\mathbf{H}}^{(0)},\mathcal{G};\Theta_{\mathrm{gate}}). Where𝐇ctx\mathbf{H}_{\mathrm{ctx}}captures the immediate structural state within the current memory graph. Simultaneously, the schema prior channel maintains a set of cross-graph structural prompt bases{𝐏j(l)}j=1K\{\mathbf{P}_{j}^{(l)}\}_{j=1}^{K}, which are used to encode stable reading habits formed during multi-graph training:ωj(l)=softmaxj⁡(𝐚(l)/Tp),𝐏schema(l)=∑j=1Kωj(l)​𝐏j(l)\omega_{j}^{(l)}=\operatorname{softmax}_{j}(\mathbf{a}^{(l)}/T_{p}),\mathbf{P}_{\mathrm{schema}}^{(l)}=\sum_{j=1}^{K}\omega_{j}^{(l)}\mathbf{P}_{j}^{(l)}. Propagation is executed based on these schema prompts to obtain:𝐇sch=Fprompt​(𝐇~(0),𝒢;{𝐏schema(l)}l=1L)\mathbf{H}_{\mathrm{sch}}=F_{\mathrm{prompt}}\big(\tilde{\mathbf{H}}^{(0)},\mathcal{G};\{\mathbf{P}_{\mathrm{schema}}^{(l)}\}_{l=1}^{L}\big). The final entity representation is jointly determined by the current context and the long-term schema:𝐇​(q,𝒢)=𝐇ctx+βsch​𝐇sch\mathbf{H}(q,\mathcal{G})=\mathbf{H}_{\mathrm{ctx}}+\beta_{\mathrm{sch}}\mathbf{H}_{\mathrm{sch}}. Here,𝐇ctx\mathbf{H}_{\mathrm{ctx}}is analogous to a context-dependent immediate recall state, responsible for adapting to the specific graph structure generated by the current writer;𝐇sch\mathbf{H}_{\mathrm{sch}}is akin to a memory schema formed across experiences, retaining the ability to recognize stable patterns such as bridge nodes, community boundaries, core–periphery structures, and noise short-circuits.

Reader Training.

Reader training aims to learn cross-graph transferable retrieval biases through a two-stage procedure. First, we perform structural contrastive pre-training on multiple augmented graph views. Then, in the supervised fine-tuning stage, we align these transferable capabilities with question-driven evidence retrieval by training the reader to identify and rank supporting entities for each query using weighted classification and multi-positive ranking objectives. Implementation details are provided in AppendicesMandN.

Writer–Reader Self-evolution.

To address the third challenge, we propose a self-evolution framework. Each of our self-evolution iterations consists of two phases. First, we fix the reader and train the writer using its retrieval results as rewards. Subsequently, we use the updated writer to generate new graphs and continue training the reader. The overall procedure is detailed in Algorithm1.

Proposition 1. Theoretical consequences of SAGE(i)Signal–budget efficiency.Soft addressing, structural gating, and controlled entity-to-document projection jointly improve evidence signal over distractor noise, thereby reducing the top-kkbudget needed for evidence coverage.[⊳\rhdSNR proof][⊳\rhdBudget proof](ii)Context–schema decomposition.The reader combines transferable structural priors with target-graph calibration, so adaptation only needs to correct graph-specific residuals.[⊳\rhdProof](iii)Evolution stability.Under bounded graph drift, consecutive writer updates induce bounded document-score changes.[⊳\rhdProof]

From a theoretical perspective, this process can be interpreted as approximate coordinate improvement over a joint memory utility: the writer update improves the readability of the graph memory, while the reader update reduces writer-induced graph distribution shift and reward bias. We provide the full coordinate-improvement result, the surrogate reward bias bound, and the analysis of single-sided update bottlenecks in AppendixF. In addition, Proposition4.2(iii) shows that although each writer update changes the graph structure in self-evolving memory, the reader output does not oscillate arbitrarily with graph evolution. We provide detailed training, inference, memory, and selector-regularizer complexity analyses in AppendixJ.

Table 1:Open-domain retrieval results onNQandPopQA. We report passage/document-level Recall (%) at top-2 and top-5 when comparable numbers are available in original papers or later works that reproduce/cite these methods. Best available results are inboldand runner-ups areunderlined.Only rows marked with0-shotare our zero-shot transfer results; baseline rows are not marked as zero-shot.Zero-shot setting applies only toSAGEonNQandPopQA.DatasetNQPopQAMethodR@2D{}_{\textsf{D}}R@5D{}_{\textsf{D}}R@2D{}_{\textsf{D}}R@5D{}_{\textsf{D}}BM25(⊳\rhdSIGIR’94) 28.2 †\dagger56.1 †\dagger24.0 †\dagger35.7 †\daggerContriever(⊳\rhdTMLR’22) 29.1 †\dagger54.6 †\dagger27.0 †\dagger43.2 †\daggerGTR(⊳\rhdEMNLP’22) 35.0 †\dagger63.4 †\dagger40.1 †\dagger49.4 †\daggerColBERTv2(⊳\rhdNAACL’22) 36.8 ⋆\star64.3 ⋆\star––RAPTOR(⊳\rhdICLR’24) 40.3 †\dagger68.3 †\dagger40.2 †\dagger48.7 †\daggerProposition(⊳\rhdEMNLP’24) 33.1 ⋆\star62.2 ⋆\star––HippoRAG(⊳\rhdNeurIPS’24) 21.3 †\dagger44.4 †\dagger40.0 †\dagger53.8†\daggerHippoRAG2(⊳\rhdICML’25) 45.6†\dagger78.0†\dagger43.9†\dagger51.7 †\daggerPropRAG(⊳\rhdEMNLP’25) –77.9 ‡\ddagger–56.2‡\ddaggerSAGE(ours)0-shot82.50-shot91.60-shot41.50-shot52.3 0-shot

†\daggerValues are from the reproduced passage Recall@2/5 evaluation in⊳\rhd.‡\ddaggerValues are from the Recall@5 table in⊳\rhd; Recall@2 is not reported there.⋆\starValues are from the reproduced single-step retrieval table in⊳\rhd; PopQA is not reported there.

5Experiments

This section presents an experimental evaluation centered around four research questions (RQs).RQ1: whetherSAGEcan bring consistent benefits in tasks such as multi-hop QA and open-domain transfer;

RQ2: whetherSAGEis an agent memory system capable of handling long-term conversation history, knowledge updates, and memory hallucination;

RQ3: whether the writer–reader closed loop truly yields self-evolution benefits;

RQ4: further analysis of where and how the performance gains come from specific designs.

Datasets.

We evaluateSAGEon five complementary scenarios. The first category consists of general QA benchmarks and three multi-hop QA benchmarks, including NQ, PopQA, HotpotQA, 2WikiMultiHopQA, and MuSiQue, used to examine whether the system can recover bridge entities across documents and combine evidence and reasoning paths. The second category focuses on a practical e-commerce application scenario, using a Review-Based Question Answering Task: AmazonQA, to assess its value in real e-commerce applications with real noisy reviews. The third category comprises long-term agent memory datasets, including LongMemEval and HaluMem, used to test information extraction from long interaction histories, multi-session reasoning, temporal reasoning, knowledge updating, abstention, and operation-level hallucination. Table13summarizes the details of each dataset. Further details on baselines and metrics can be found in AppendixR.

Figure 3:Visualization of the retrieved results.

5.1End-to-End Effectiveness

Multi-hop Question Answering

Table 2:Performance of representative memory systems onLongMemEval. We report accuracy (%) on six task categories: single-session user (SS-U), single-session assistant (SS-A), merged single-session recall (SSR), single-session preference (SS-P), knowledge update (KU), temporal reasoning (TR), and multi-session reasoning (MS). SSR is computed as the weighted average of SS-U and SS-A when both are available; if a source reports only merged single-session recall, SS-U/SS-A are left blank. Best results are inboldand runner-ups areunderlined. The darker the cell, the better. Results are grouped by reporting protocol and should not be treated as a single strict leaderboard.Only rows marked with0-shotare our zero-shot transfer results; baseline rows and trained variants are not marked as zero-shot.Zero-shot setting applies only toOursrows marked with0-shotonLongMemEval.DatasetLongMemEval-S / LongMemEvalMethodSS-USS-ASSRSS-PKUTRMSOverallUnified protocol in TiMem (GPT-4o-mini, LLJ accuracy)MemoryBank(⊳\rhd) 50.09.832.10.021.80.00.011.5A-MEM(⊳\rhd) 82.987.584.939.372.836.140.355.4Mem0(⊳\rhd) 94.351.875.450.078.749.266.265.0MemoryOS(⊳\rhd) 81.178.279.851.356.153.444.858.1MemOS(⊳\rhd) 93.767.982.250.776.765.158.868.7TiMem(⊳\rhd) 95.782.189.763.386.268.470.876.9MemOS evaluation suite (short-answer prompt)MIRIX(⊳\rhd) 72.863.668.853.352.625.630.143.5Mem0(⊳\rhd) 82.926.857.990.066.772.263.166.4Zep(⊳\rhd) 92.975.084.953.374.454.147.463.2memobase111memobase:https://github.com/memodb-io/memobase.92.823.261.980.089.775.966.972.4Supermemory222Supermemory:https://github.com/supermemoryai/supermemory.85.758.973.890.055.144.452.658.4MemU333MemU:https://github.com/NevaMind-AI/memU.67.119.646.076.741.017.342.138.4Our methodOurs(0-shot)0-shot60.381.468.016.723.513.89.428.4Ours(trained)73.380.076.023.127.822.812.534.3Ours+1 round79.480.579.615.826.222.810.734.1

Table3reports the main results on general QA benchmarks and three multi-hop QA benchmarks. Table9reports the results of retrieval performance on multi-hop QA benchmarks. It is worth mentioning that even when we directly test on NQ and PopQA using a model trained only on MuSiQue, HotpotQA, and 2WikiMultiHopQA, we still achieve very strong performance, especially on NQ; see Table1for the detailed results.

Table 3:Results of multi-hop question answering (QA) performance. We report Exact Match (EM) and F1 score, both reported as percentages (%). Best results are inboldand runner-ups areunderlined. The darker the cell, the better.DatasetHotpotQAMuSiQue2WikiMultiHopQAAvg. RankMethodEMF1EMF1EMF1BM25(⊳\rhdarXiv’24) 40.053.219.523.646.957.915.5Contriever(⊳\rhdTMLR’22) 34.951.216.321.624.333.920.5GTR(⊳\rhdEMNLP’22) 33.851.915.125.233.742.519.7ColBERTv2(⊳\rhdNAACL’22) 43.457.715.526.433.443.317.3RAPTOR(⊳\rhdICLR’24) 48.259.217.628.930.642.015.8GraphRAG(⊳\rhdarXiv’24) 35.354.613.429.528.346.918.0G-Retriever(⊳\rhdNeurIPS’24) 33.250.318.025.942.345.618.2LightRAG(⊳\rhdarXiv’24) 36.848.318.127.545.149.516.7HippoRAG(⊳\rhdNeurIPS’24) 41.855.019.229.846.659.513.8HippoRAG2(⊳\rhdICML’25) 57.369.633.940.575.478.24.7SubgraphRAG(⊳\rhdICLR’25) 52.665.833.039.271.076.27.5PropRAG(⊳\rhdEMNLP’25) 57.270.138.941.373.079.93.8GFM-RAG(⊳\rhdNeurIPS’25) 51.666.930.240.469.877.77.8FLARE(⊳\rhdEMNLP’23) 48.760.616.228.446.765.412.7Adaptive-RAG(⊳\rhdNAACL’24) 45.559.613.825.648.962.814.3ColBERTv2+IRCoT(⊳\rhdACL’23) 45.558.419.130.535.445.114.4HippoRAG+IRCoT(⊳\rhdACL’23) 45.759.221.933.347.762.711.3GFM-RAG+IRCoT(⊳\rhdACL’23) 56.071.836.649.272.580.83.7SAGE(ours)51.372.728.047.364.575.37.5ours +1 round60.473.432.252.371.779.54.2ours +2 round56.180.834.253.174.080.42.5ours +IRCoT52.375.035.252.672.282.93.3

Domain-specific Memory

Table 4:Retrieval efficiency comparison. We report retrieval time in seconds onHotpotQA,MuSiQue, and2Wiki. For Time, lower is better. Best results are inboldand runner-ups areunderlined. The darker the cell, the better.DatasetHotpotQAMuSiQue2WikiMethodTime↓\downarrowTime↓\downarrowTime↓\downarrowSingle-step retrieval methodsColBERTv20.0350.0300.029HippoRAG0.2550.2510.158LightRAG0.8611.1090.911GraphRAG(MS) 2.7593.0371.204GFM-RAG0.1070.1240.060Iterative retrieval methodsIRCoT+ColBERTv21.1461.1522.095IRCoT+HippoRAG3.1623.1043.441SAGE0.0320.0340.019

Table10reports the results on AmazonQA.SAGEconsistently outperforms the neural baseline R-Net across all metrics, indicating strong cross-task generalization. After training on AmazonQA, Ours achieves substantial gains. Overall, training and interaction rounds steadily enhance performance, while the zero-shot results demonstrate promising transfer ability.

5.2Long-term Agent Memory Evaluation

The LongMemEval results are shown in Table2. The HaluMem results are shown in Table11.SAGEis compared against highly specialized long-term memory systems, making this a challenging evaluation setting. AlthoughSAGEdoes not yet surpass the strongest system-level baselines. Notably,SAGE+1 round already outperforms Memobase on several metrics, suggesting that it is competitive despite being less system-engineered. The remaining gap mainly lies in memory updating and high-coverage extraction, indicating clear potential for further gains with stronger memory management and update mechanisms.

5.3Further Analysis

As shown inTable˜4,SAGEdemonstrates a strong speed advantage. It achieves the fastest retrieval time, indicating strong potential for practical and large-scale deployment. To further analyze the interpretability ofSAGE, we visualize the retrieved subgraph for a representative case, as shown in Figure. A detailed case study can be found inP.1. The detailed ablation study design, analysis, and results for the Memory Writer and Reader can be found in AppendixHand AppendixG, respectively.

6Conclusion

We presented SAGE, a self-evolving agentic graph-memory engine that treats memory as a dynamic substrate for writing, reading, and continual improvement. Experiments show that SAGE improves evidence recovery, grounding, and retrieval efficiency, suggesting that self-evolving graph memory is a promising foundation for long-horizon language agents.

References

• A. Asai, Z. Wu, Y. Wang, A. Sil, and H. Hajishirzi (2023)Self-rag: learning to retrieve, generate, and critique through self-reflection.InThe Twelfth International Conference on Learning Representations,Cited by:§2,§4.2.
• S. Cha, D. Kim, T. Hahn, M. Kim, Y. Han, and B. Jeon (2025)Annotation-free reinforcement learning query rewriting via verifiable search reward.arXiv preprint arXiv:2507.23242.Cited by:§2.
• D. Chen, S. Niu, K. Li, P. Liu, X. Zheng, B. Tang, X. Li, F. Xiong, and Z. Li (2025a)Halumem: evaluating hallucinations in memory systems of agents.arXiv preprint arXiv:2511.03506.Cited by:NeurIPS Paper Checklist,§1,§1,§2,§4.1.
• T. Chen, H. Wang, S. Chen, W. Yu, K. Ma, X. Zhao, H. Zhang, and D. Yu (2024)Dense x retrieval: what retrieval granularity should we use?.InProceedings of the 2024 Conference on Empirical Methods in Natural Language Processing,pp. 15159–15177.Cited by:2nd item,NeurIPS Paper Checklist.
• Y. Chen, L. Yan, W. Sun, X. Ma, Y. Zhang, S. Wang, D. Yin, Y. Yang, and J. Mao (2025b)Improving retrieval-augmented generation through multi-agent reinforcement learning.arXiv preprint arXiv:2501.15228.Cited by:§2.
• P. Chhikara, D. Khant, S. Aryan, T. Singh, and D. Yadav (2025)Mem0: building production-ready ai agents with scalable long-term memory.arXiv preprint arXiv:2504.19413.Cited by:NeurIPS Paper Checklist,§2.
• [7]H. W. Di Wu, W. Yu, Y. Zhang, K. Chang, and D. Y. LongmemevalBenchmarking chat assistants on long-term interactive memory, 2024.URL https://arxiv. org/abs/2410.108132,pp. 14.Cited by:NeurIPS Paper Checklist,§1,§2.
• D. Edge, H. Trinh, N. Cheng, J. Bradley, A. Chao, A. Mody, S. Truitt, D. Metropolitansky, R. O. Ness, and J. Larson (2024)From local to global: a graph rag approach to query-focused summarization.arXiv preprint arXiv:2404.16130.Cited by:3rd item,NeurIPS Paper Checklist,§1,§1,§2,§4.2.
• Z. Guo, L. Xia, Y. Yu, T. Ao, and C. Huang (2024)Lightrag: simple and fast retrieval-augmented generation.arXiv preprint arXiv:2410.057792(3).Cited by:3rd item,NeurIPS Paper Checklist,§2,§4.2.
• M. Gupta, N. Kulkarni, R. Chanda, A. Rayasam, and Z. C. Lipton (2019)Amazonqa: a review-based question answering task.arXiv preprint arXiv:1908.04364.Cited by:NeurIPS Paper Checklist.
• B. J. Gutiérrez, Y. Shu, Y. Gu, M. Yasunaga, and Y. Su (2024)Hipporag: neurobiologically inspired long-term memory for large language models.Advances in neural information processing systems37,pp. 59532–59569.Cited by:3rd item,NeurIPS Paper Checklist,§1,§1,§1,§2.
• B. J. Gutiérrez, Y. Shu, W. Qi, S. Zhou, and Y. Su (2025)From rag to memory: non-parametric continual learning for large language models.arXiv preprint arXiv:2502.14802.Cited by:NeurIPS Paper Checklist,§2.
• X. He, Y. Tian, Y. Sun, N. V. Chawla, T. Laurent, Y. LeCun, X. Bresson, and B. Hooi (2024)G-retriever: retrieval-augmented generation for textual graph understanding and question answering.Advances in Neural Information Processing Systems37,pp. 132876–132907.Cited by:3rd item,NeurIPS Paper Checklist,§2.
• Y. Hu, Y. Wang, and J. McAuley (2025)Evaluating memory in llm agents via incremental multi-turn interactions.arXiv preprint arXiv:2507.05257.Cited by:§1,§2.
• Z. Hu, Y. Dong, K. Wang, K. Chang, and Y. Sun (2020)Gpt-gnn: generative pre-training of graph neural networks.InProceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining,pp. 1857–1867.Cited by:§2.
• Z. Huang, Z. Tian, Q. Guo, F. Zhang, Y. Zhou, D. Jiang, Z. Xie, and X. Zhou (2025)Licomemory: lightweight and cognitive agentic memory for efficient long-term reasoning.arXiv preprint arXiv:2511.01448.Cited by:§2.
• A. Hurst, A. Lerer, A. P. Goucher, A. Perelman, A. Ramesh, A. Clark, A. Ostrow, A. Welihinda, A. Hayes, A. Radford,et al.(2024)GPT-4o system card.arXiv preprint arXiv:2410.21276.Cited by:1st item.
• G. Izacard, M. Caron, L. Hosseini, S. Riedel, P. Bojanowski, A. Joulin, and E. Grave (2021)Unsupervised dense information retrieval with contrastive learning.arXiv preprint arXiv:2112.09118.Cited by:2nd item,NeurIPS Paper Checklist.
• S. Jeong, J. Baek, S. Cho, S. J. Hwang, and J. C. Park (2024)Adaptive-rag: learning to adapt retrieval-augmented large language models through question complexity.InProceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers),pp. 7036–7050.Cited by:4th item,NeurIPS Paper Checklist,§2.
• Z. Jiang, F. F. Xu, L. Gao, Z. Sun, Q. Liu, J. Dwivedi-Yu, Y. Yang, J. Callan, and G. Neubig (2023)Active retrieval augmented generation.InProceedings of the 2023 conference on empirical methods in natural language processing,pp. 7969–7992.Cited by:4th item,NeurIPS Paper Checklist,§2.
• J. Kang, M. Ji, Z. Zhao, and T. Bai (2025)Memory os of ai agent.InProceedings of the 2025 Conference on Empirical Methods in Natural Language Processing,pp. 25972–25981.Cited by:NeurIPS Paper Checklist,§2.
• P. Lewis, E. Perez, A. Piktus, F. Petroni, V. Karpukhin, N. Goyal, H. Küttler, M. Lewis, W. Yih, T. Rocktäschel,et al.(2020)Retrieval-augmented generation for knowledge-intensive nlp tasks.Advances in neural information processing systems33,pp. 9459–9474.Cited by:§1,§2.
• K. Li, X. Yu, Z. Ni, Y. Zeng, Y. Xu, Z. Zhang, X. Li, J. Sang, X. Duan, X. Wang,et al.(2026a)TiMem: temporal-hierarchical memory consolidation for long-horizon conversational agents.arXiv preprint arXiv:2601.02845.Cited by:NeurIPS Paper Checklist.
• M. Li, S. Miao, and P. Li (2024)Simple is effective: the roles of graphs and large language models in knowledge-graph-based retrieval-augmented generation.arXiv preprint arXiv:2410.20724.Cited by:3rd item,NeurIPS Paper Checklist,§2.
• Y. Li, W. Guo, L. Zhang, R. Xu, M. Huang, H. Liu, L. Xu, Y. Xu, and J. Liu (2026b)Locomo-plus: beyond-factual cognitive memory evaluation framework for llm agents.arXiv preprint arXiv:2602.10715.Cited by:§1,§2.
• Z. Li, S. Song, H. Wang, S. Niu, D. Chen, J. Yang, C. Xi, H. Lai, J. Zhao, Y. Wang,et al.(2025)Memos: an operating system for memory-augmented generation (mag) in large language models.arXiv preprint arXiv:2505.22101.Cited by:NeurIPS Paper Checklist.
• J. Liu, C. Yang, Z. Lu, J. Chen, Y. Li, M. Zhang, T. Bai, Y. Fang, L. Sun, P. S. Yu,et al.(2025)Graph foundation models: concepts, opportunities and challenges.IEEE Transactions on Pattern Analysis and Machine Intelligence.Cited by:§1,§2.
• L. Luo, Z. Zhao, G. Haffari, D. Phung, C. Gong, and S. Pan (2025)GFM-rag: graph foundation model for retrieval augmented generation.arXiv preprint arXiv:2502.01113.Cited by:3rd item,NeurIPS Paper Checklist,§1,§2,§4.2.
• A. Maharana, D. Lee, S. Tulyakov, M. Bansal, F. Barbieri, and Y. Fang (2024)Evaluating very long-term conversational memory of llm agents.InProceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers),pp. 13851–13870.Cited by:§1,§2.
• J. Ni, C. Qu, J. Lu, Z. Dai, G. H. Abrego, J. Ma, V. Zhao, Y. Luan, K. Hall, M. Chang,et al.(2022)Large dual encoders are generalizable retrievers.InProceedings of the 2022 Conference on Empirical Methods in Natural Language Processing,pp. 9844–9855.Cited by:2nd item,NeurIPS Paper Checklist.
• C. Packer, V. Fang, S. Patil, K. Lin, S. Wooders, and J. Gonzalez (2023)MemGPT: towards llms as operating systems..Cited by:§1,§2.
• J. S. Park, J. O’Brien, C. J. Cai, M. R. Morris, P. Liang, and M. S. Bernstein (2023)Generative agents: interactive simulacra of human behavior.InProceedings of the 36th annual acm symposium on user interface software and technology,pp. 1–22.Cited by:§1,§2.
• J. Qiu, Q. Chen, Y. Dong, J. Zhang, H. Yang, M. Ding, K. Wang, and J. Tang (2020)Gcc: graph contrastive coding for graph neural network pre-training.InProceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining,pp. 1150–1160.Cited by:§2.
• P. Rasmussen, P. Paliychuk, T. Beauvais, J. Ryan, and D. Chalef (2025)Zep: a temporal knowledge graph architecture for agent memory.arXiv preprint arXiv:2501.13956.Cited by:NeurIPS Paper Checklist,§2.
• S. E. Robertson and S. Walker (1994)Some simple effective approximations to the 2-poisson model for probabilistic weighted retrieval.InSIGIR’94: Proceedings of the Seventeenth Annual International ACM-SIGIR Conference on Research and Development in Information Retrieval, organised by Dublin City University,pp. 232–241.Cited by:2nd item,NeurIPS Paper Checklist.
• K. Santhanam, O. Khattab, J. Saad-Falcon, C. Potts, and M. Zaharia (2022)Colbertv2: effective and efficient retrieval via lightweight late interaction.InProceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies,pp. 3715–3734.Cited by:2nd item,NeurIPS Paper Checklist.
• P. Sarthi, S. Abdullah, A. Tuli, S. Khanna, A. Goldie, and C. D. Manning (2024)Raptor: recursive abstractive processing for tree-organized retrieval.InThe Twelfth International Conference on Learning Representations,Cited by:2nd item,NeurIPS Paper Checklist,§2.
• H. Trivedi, N. Balasubramanian, T. Khot, and A. Sabharwal (2022)MuSiQue: multihop questions via single-hop question composition.Transactions of the Association for Computational Linguistics10,pp. 539–554.Cited by:NeurIPS Paper Checklist.
• H. Trivedi, N. Balasubramanian, T. Khot, and A. Sabharwal (2023)Interleaving retrieval with chain-of-thought reasoning for knowledge-intensive multi-step questions.InProceedings of the 61st annual meeting of the association for computational linguistics (volume 1: long papers),pp. 10014–10037.Cited by:4th item,Appendix R,NeurIPS Paper Checklist,§1,§2,§4.2.
• H. T. Tsang, J. Bai, H. Huang, Q. Xiao, T. Zheng, B. Xu, S. Liu, and Y. Song (2025)Autograph-r1: end-to-end reinforcement learning for knowledge graph construction.arXiv preprint arXiv:2510.15339.Cited by:§2,§4.1.
• J. Wang and J. Han (2025)Proprag: guiding retrieval with beam search over proposition paths.InProceedings of the 2025 Conference on Empirical Methods in Natural Language Processing,pp. 6223–6238.Cited by:3rd item,NeurIPS Paper Checklist,§2,§4.2.
• Y. Wang and X. Chen (2025)Mirix: multi-agent memory system for llm-based agents.arXiv preprint arXiv:2507.07957.Cited by:NeurIPS Paper Checklist.
• Y. Wu, S. Liang, C. Zhang, Y. Wang, Y. Zhang, H. Guo, R. Tang, and Y. Liu (2025a)From human memory to ai memory: a survey on memory mechanisms in the era of llms.arXiv preprint arXiv:2504.15965.Cited by:§1,§2.
• Y. Wu, Y. Zhang, S. Liang, and Y. Liu (2025b)Sgmem: sentence graph memory for long-term conversational agents.arXiv preprint arXiv:2509.21212.Cited by:§2,§4.2.
• T. Xu, H. Zheng, C. Li, H. Chen, Y. Liu, R. Chen, and L. Sun (2025a)NodeRAG: structuring graph-based rag with heterogeneous nodes.arXiv preprint arXiv:2504.11544.Cited by:§2.
• W. Xu, Z. Liang, K. Mei, H. Gao, J. Tan, and Y. Zhang (2025b)A-mem: agentic memory for llm agents.arXiv preprint arXiv:2502.12110.Cited by:NeurIPS Paper Checklist,§2.
• C. Yang, C. Zhou, Y. Xiao, S. Dong, L. Zhuang, Y. Zhang, Z. Wang, Z. Hong, Z. Yuan, Z. Xiang,et al.(2026)Graph-based agent memory: taxonomy, techniques, and applications.arXiv preprint arXiv:2602.05665.Cited by:§1,§2.
• Z. Yang, P. Qi, S. Zhang, Y. Bengio, W. Cohen, R. Salakhutdinov, and C. D. Manning (2018)HotpotQA: a dataset for diverse, explainable multi-hop question answering.InProceedings of the 2018 Conference on Empirical Methods in Natural Language Processing,pp. 2369–2380.Cited by:NeurIPS Paper Checklist.
• Y. You, T. Chen, Y. Sui, T. Chen, Z. Wang, and Y. Shen (2020)Graph contrastive learning with augmentations.Advances in neural information processing systems33,pp. 5812–5823.Cited by:§2.
• X. Yu, Z. Gong, C. Zhou, Y. Fang, and H. Zhang (2025)Samgpt: text-free graph foundation model for multi-domain pre-training and cross-domain adaptation.InProceedings of the ACM on Web Conference 2025,pp. 1142–1153.Cited by:§2.
• J. Yue, C. Hu, J. Sheng, Z. Zhou, W. Zhang, T. Liu, L. Guo, and Y. Deng (2026)HyperMem: hypergraph memory for long-term conversations.arXiv preprint arXiv:2604.08256.Cited by:§2.
• G. Zhang, M. Fu, G. Wan, M. Yu, K. Wang, and S. Yan (2025a)G-memory: tracing hierarchical memory for multi-agent systems.arXiv preprint arXiv:2506.07398.Cited by:§2.
• K. Zhang, X. Zhang, E. Ahmed, H. Jiang, C. Kumar, K. Sun, Z. Lin, S. Sharma, S. Oraby, A. Colak,et al.(2025b)Assomem: scalable memory qa with multi-signal associative retrieval.arXiv preprint arXiv:2510.10397.Cited by:§2,§4.2.
• Q. Zhang, S. Chen, Y. Bei, Z. Yuan, H. Zhou, Z. Hong, H. Chen, Y. Xiao, C. Zhou, J. Dong,et al.(2025c)A survey of graph retrieval-augmented generation for customized large language models.arXiv preprint arXiv:2501.13958.Cited by:§2,§4.2.
• Y. Zhao, J. Zhu, Y. Guo, K. He, and X. Li (2025)Eˆ 2graphrag: streamlining graph-based rag for high efficiency and effectiveness.arXiv preprint arXiv:2505.24226.Cited by:§2.
• W. Zhong, L. Guo, Q. Gao, H. Ye, and Y. Wang (2024)Memorybank: enhancing large language models with long-term memory.InProceedings of the AAAI conference on artificial intelligence,Vol.38,pp. 19724–19731.Cited by:NeurIPS Paper Checklist,§1,§2.

Input:Training set

𝒟train\mathcal{D}_{\mathrm{train}}, writer

πθ0\pi_{\theta_{0}}, GFM reader

fϕ0f_{\phi_{0}}, self-evolution iterations

TT Output:Trained writer

πθT\pi_{\theta_{T}}and reader

fϕTf_{\phi_{T}} 1fort=0,…,T−1t=0,\dots,T-1do

//Writer update: fixed GFM reader as reward environment

2for*each samplex=(q,𝒟,𝒟+,y)∈𝒟trainx=(q,\mathcal{D},\mathcal{D}^{+},y)\in\mathcal{D}_{\mathrm{train}}*do

3Sample

GGgraph construction trajectories

{τi}i=1G\{\tau_{i}\}_{i=1}^{G}from

πθt\pi_{\theta_{t}};

4fori=1,…,Gi=1,\dots,Gdo

5Obtain graph

𝒢i\mathcal{G}_{i}and retrieve

Pk​(q,𝒢i)P_{k}(q,\mathcal{G}_{i})using

fϕtf_{\phi_{t}};

6Calculate return

RiR_{i};

7

8end for

9Update writer

πθt\pi_{\theta_{t}};

10

11end for

//Reader update: improved graphs as memory substrate

12Construct a set of graph memories{𝒢x}\{\mathcal{G}_{x}\}for the training corpus usingπθt+1\pi_{\theta_{t+1}};

13Update GFM reader

fϕtf_{\phi_{t}}on

{𝒢x}\{\mathcal{G}_{x}\};

14

15end for

Algorithm 1Writer–Reader Self-evolution Training forSAGE## Appendix AAdditional Analysis of the Memory Writer

This appendix provides a detailed analysis of the memory writer experiments in the main text.

A.1Reward Design and Writer Behavior

In Table5, different RL rewards induce different writer behaviors. GFM-pretrained-only achieves Precision/Recall/Deducible of0.838/0.818/0.5100.838/0.818/0.510, while GFM-finetuned achieves0.824/0.813/0.5120.824/0.813/0.512, indicating that relying solely on supervised finetuning cannot stably improve the utility of graph memory for a frozen reader. This result is also consistent with our setup: the goal of the memory writer is not to reproduce a static graph format.

RL-Recall improves Precision and Recall to0.889/0.8350.889/0.835, but Deducible drops to0.5020.502. This shows that rewarding only supporting context coverage encourages the writer to store more locally relevant evidence, but does not necessarily lead to a complete multi-hop reasoning chain. RL-F1 further raises Recall to0.8810.881, but Deducible is only0.4970.497, again indicating a gap between retrieval matching quality and answer deducibility: the reader hitting the supporting contexts does not guarantee that these contexts are organized in a way sufficient to support answer reasoning. In contrast, RL-Deduce achieves0.861/0.892/0.5170.861/0.892/0.517, showing that using answer deducibility directly as feedback can encourage the writer to focus more on bridging entities, cross-document relations, and answer-relevant causal or attribute paths.

RL-Hybrid achieves Precision and Recall of0.902\mathbf{0.902}and0.917\mathbf{0.917}, respectively, representing improvements of+0.064+0.064and+0.099+0.099over pretrained-only, while Deducible reaches0.5220.522. This indicates that hybrid rewards can mitigate the bias of a single reward: they both avoid the introduction of too much weakly relevant evidence caused by a pure recall reward and prevent a pure deducibility reward from overfavoring short paths or local answer clues. Hybrid + frozen answer API achieves the highest Deducible, at0.526\mathbf{0.526}, but Precision and Recall drop to0.832/0.8740.832/0.874. This suggests that stronger answer-side feedback can further improve reasoning usability, but it may also make the writer more conservative, writing only evidence directly related to the final answer and thereby sacrificing some supporting context coverage.

A.2Cross-domain Transfer

Table6shows that the base writer trained on HotpotQA/MuSiQue has a certain degree of transferability to new domains, but training on the target domain remains very important. On GRBench, Base→\rightarrowGRBench achieves Precision/Recall/Deducible of0.575/0.609/0.4110.575/0.609/0.411, while GRBench train→\rightarrowval improves to0.794/0.833/0.5960.794/0.833/0.596. This improvement indicates that the writing strategy learned for multi-hop QA can transfer to structured product or domain graph memory tasks, but the entity types, attribute relations, and evidence granularity in the target domain still need to be re-adapted.

On HaluMem and LongMemEval, cross-domain differences are even more pronounced. The Base→\rightarrowHaluMem results are0.230/0.448/0.2990.230/0.448/0.299, which improve to0.312/0.708/0.4380.312/0.708/0.438after training on the target domain; the Base→\rightarrowLongMemEval results are0.232/0.376/0.4750.232/0.376/0.475, which improve to0.377/0.439/0.5310.377/0.439/0.531after training on the target domain. These results indicate that memory writing in agent memory tasks requires not only extracting explicit facts, but also maintaining user preferences, temporal order, state updates, and long-term consistency. In traditional multi-hop QA, supporting contexts often form a relatively static set of evidence centered around a single question, whereas information in long-term memory tasks changes over time and involves personalization, conflict updates, and context dependence. Therefore, although reader-aware RL feedback can provide transferable writing principles, interaction feedback from the target domain remains crucial for achieving stable performance.

A.3Writing Protocol and Interaction Budget

Table7shows that the writing protocol significantly changes the trade-off among Precision, Recall, and Deducible. The results for Tight=True are0.836/0.806/0.5150.836/0.806/0.515; the results for Tight=False are0.845/0.851/0.5060.845/0.851/0.506. After relaxing the protocol, Recall improves noticeably, indicating that the writer can write more potentially relevant evidence; however, Deducible declines, suggesting that the additional evidence also contains more noise, redundant facts, or weakly related local information. Although this content may increase the coverage of supporting context, it can dilute the reasoning chain that truly supports the answer.

Iterative writing further highlights the role of the interaction budget. For Iterative, 12 turns, tight, Precision/Recall/Deducible are0.852/0.829/0.5160.852/0.829/0.516; after increasing to 20 turns, Recall reaches the highest value of0.881\mathbf{0.881}, indicating that multi-turn reader feedback helps the writer complete cross-document bridging paths. For Iterative, 24 turns, loose, Precision and Deducible reach0.863\mathbf{0.863}and0.531\mathbf{0.531}, respectively, but Recall falls back to0.8260.826. This shows that more rounds of interaction are not simply “the longer, the better”: the benefit comes from the writer revising the graph structure based on reader feedback, whereas when the protocol is too loose or the writing space becomes too large, the additional content may alter the reader’s ranking, causing some gold supporting contexts to be pushed out of the top results.

A.4Reader-side Sensitivity

Figure 4:Freeze the sensitivity analysis on the reader side. The impact of the initial-entity weight on reward and Deducible is the most stable; the top-kkand ranker variants exhibit a non-monotonic budget–noise trade-off.Figure4analyzes the impact of the frozen reader setting on writer training results. First, the top-kkbudget sweep shows a non-monotonic trend: atk=5k=5, reward and Deducible are0.623/0.5280.623/0.528, the better setting in this group; atk=40k=40, reward remains at0.6220.622, but Deducible drops to0.5180.518; atk=60k=60, reward further declines to0.5910.591. This indicates that expanding the retrieval budget does not necessarily lead to better reader feedback. Although a larger top-kkimproves potential coverage, it also introduces more weakly related or redundant evidence, diluting the reasoning chain that truly supports the answer.

The ranker variants also reflect a similar coverage–noise trade-off. topk20 achieves the highest reward and Deducible, at0.630/0.5440.630/0.544, respectively; idf-only and raw both have a reward of0.6260.626, but their Deducible scores are0.5380.538and0.5190.519, respectively; idf-topk60 declines to0.595/0.5060.595/0.506. This shows that the reader ranker cannot rely solely on entity overlap or on expanding the candidate set, but instead must strike a balance among entity matching, semantic relevance, and contextual compactness. For the writer, an overly weak ranker makes it difficult for effective graph structure to be read out, while an overly broad candidate space amplifies the negative impact of noisy writes.

Initial-entity weight is the most stable factor among the three groups of reader-side settings. When initial-entity weight is enabled, reward reaches0.6390.639and Deducible is0.5250.525; when it is disabled, reward drops significantly to0.5470.547and Deducible falls to0.5050.505. Even when the budget is increased after disabling it, the rewards for off@10 and off@40 recover only to0.6130.613and0.6150.615. This indicates that the initial entity anchor is crucial in multi-hop graph retrieval: it helps the reader enter the correct local subgraph from the question entity and expand along the bridging relations written by the writer to the evidence supporting the answer. Without this anchor, simply increasing the retrieval budget cannot fully compensate for the deviation in graph traversal direction.

A.5Training Stability and Regularization

Figure 5:Training regularization and scaling analysis. A larger rollout group size and a moderate warmup ratio show a slight advantage in this batch of results, but the gains are smaller than the effects of reward design, the reader-side initial-entity weight, and the writing protocol.Figure5shows the effects of training regularization and rollout settings on the writer. First, the repetition penalty affects both reward and Deducible, but the trend is not monotonic. Without any penalty, the result is0.585/0.5100.585/0.510; with a penalty of0.100.10, it improves to0.610/0.5200.610/0.520; with a penalty of0.500.50, it reaches the best result in this group at0.619/0.5220.619/0.522; after further increasing it to1.001.00, it drops to0.597/0.5000.597/0.500. This indicates that a moderate penalty on repeated triples can suppress redundant edges and cyclic expressions, but an overly strong penalty may limit the writer’s necessary restatement of key facts. Especially in multi-hop reasoning, the same bridging entity often needs to appear in multiple relational paths, so repetition is not always meaningless noise.

Rollout filtering brings consistent but limited gains. When filtering is disabled, reward/Deducible is0.585/0.5100.585/0.510; after applyingthr_80,thr_90, andthr_95, reward increases to0.6210.621,0.6170.617, and0.6250.625, respectively, while Deducible remains stable at0.5160.516–0.5180.518. This suggests that filtering out low-quality rollouts can reduce the interference of noisy trajectories with policy updates, allowing the writer to learn effective writing strategies more stably. However, the differences between thresholds are small, indicating that the main role of filtering is to remove obviously negative samples rather than determine the final performance ceiling.

Rollout group size and warmup ratio further affect training stability. As group size increases fromn=1n=1ton=10n=10, reward/Deducible rises from0.610/0.5110.610/0.511to0.620/0.5310.620/0.531, indicating that a larger group size can provide more reliable relative preference estimates and help RL distinguish more accurately between effective and ineffective writing. The optimal warmup ratio appears at0.200.20, where Deducible reaches0.5290.529; too little warmup may lead to unstable early policy updates, while too much warmup may delay the effect of the RL signal. Overall, these regularization and training scale settings can improve stability, but their gains are smaller than those from reward design, reader-side initial-entity weight, and the interaction protocol itself. This shows that the core improvement of the memory writer comes from reader-aware RL feedback: it forces the graph constructor to learn to preserve bridging entities, cross-document relations, and evidence chains that support answer derivation, while also reducing repetitive structures and irrelevant local facts.

Appendix BSignal-to-Noise Ratio and Retrieval Budget of Structurally Gated Propagation

This section analyzes the structural capability of the GFM memory reader inSAGEfrom the perspective of signal propagation and retrieval budget. Unlike graph-isomorphism expressivity analyses centered onkk-WL, we focus on the following question: on noisy graph memories dynamically written by the memory writer, how do soft addressing, structurally gated propagation, context–schema dual-channel calibration, and entity-to-document projection jointly improve the ratio of query-relevant evidence signal to distractor noise, thereby reducing the top-kkretrieval budget required to achieve a given level of evidence coverage?

B.1Review of the SAGE-GFM Reader Formalization

Given a samplex=(q,D,D+,y)x=(q,D,D^{+},y), whereqqdenotes the query,D={di}i=1ND=\{d_{i}\}_{i=1}^{N}denotes the candidate memory fragments, andD+⊆DD^{+}\subseteq Ddenotes the gold evidence set supporting the answeryy, the memory writer constructs a heterogeneous graph

G=Wθ​(q,D)=(VE∪VD,EE​E∪EE​D),G=W_{\theta}(q,D)=(V_{E}\cup V_{D},E_{EE}\cup E_{ED}),(5)whereVEV_{E}is the set of entity nodes,VDV_{D}is the set of memory-fragment nodes,EE​EE_{EE}denotes entity–entity relation edges, andEE​DE_{ED}denotes entity–text-fragment anchoring edges. The GFM memory reader outputs an entity distribution, a document distribution, and an optional activated subgraph:

fϕ​(q,G,D)=(pϕ​(e∣q,G),pϕ​(d∣q,G,D),Gq).f_{\phi}(q,G,D)=\big(p_{\phi}(e\mid q,G),\,p_{\phi}(d\mid q,G,D),\,G_{q}\big).(6) The reader first uses query planning and soft addressing to generate query-conditioned initial activation for entities. Letse​(q)s_{e}(q)denote the entry score of entityee, which integrates multiple cues such as explicit entities, aliases, pseudo-query similarity, answer type, hard constraints, and entity linking. The initial activation distribution is then given by

p0​(e∣q)=exp⁡(se​(q)/T0)∑v∈VEexp⁡(sv​(q)/T0).p_{0}(e\mid q)=\frac{\exp(s_{e}(q)/T_{0})}{\sum_{v\in V_{E}}\exp(s_{v}(q)/T_{0})}.(7)The initial entity representation is written as

he(0)=(p0​(e∣q))η​Wq​Emb⁡(q)+Wx​xe,0≤η≤1.h_{e}^{(0)}=\big(p_{0}(e\mid q)\big)^{\eta}W_{q}\operatorname{Emb}(q)+W_{x}x_{e},\qquad 0\leq\eta\leq 1.(8)The reader then constructs structural gates using node-level structural features, edge-pair structural features, and graph-level structural summaries. Specifically, let

φ​(v)\displaystyle\varphi(v)=(log⁡(1+dv),cv,κv,d¯𝒩​(v)),\displaystyle=\big(\log(1+d_{v}),\,c_{v},\,\kappa_{v},\,\bar{d}_{\mathcal{N}(v)}\big),(9)ψ​(u,v)\displaystyle\psi(u,v)=(|du−dv|,|𝒩​(u)∩𝒩​(v)|,Jaccard⁡(𝒩​(u),𝒩​(v))),\displaystyle=\big(|d_{u}-d_{v}|,\,|\mathcal{N}(u)\cap\mathcal{N}(v)|,\,\operatorname{Jaccard}(\mathcal{N}(u),\mathcal{N}(v))\big),(10)rG\displaystyle r_{G}=(meanv∈VE⁡φ​(v);stdv∈VE⁡φ​(v);dens⁡(G)).\displaystyle=\big(\operatorname{mean}_{v\in V_{E}}\varphi(v);\,\operatorname{std}_{v\in V_{E}}\varphi(v);\,\operatorname{dens}(G)\big).(11)The edge structural context at layerllis

zu​v(l)=(En(l)​(φ​(u));En(l)​(φ​(v));Ep(l)​(ψ​(u,v));Eg(l)​(rG)),z_{uv}^{(l)}=\big(E_{n}^{(l)}(\varphi(u));\,E_{n}^{(l)}(\varphi(v));\,E_{p}^{(l)}(\psi(u,v));\,E_{g}^{(l)}(r_{G})\big),(12)which generates the vector-valued gate

gu​v(l)=1+δ​tanh⁡(MLPg(l)⁡(zu​v(l))).g_{uv}^{(l)}=1+\delta\tanh\big(\operatorname{MLP}_{g}^{(l)}(z_{uv}^{(l)})\big).(13)Letηu​v≥0\eta_{uv}\geq 0denote the normalized adjacency weight with self-loops. The message and node update are

mu→v(l)\displaystyle m_{u\to v}^{(l)}=ηu​v​gu​v(l)⊙Wm(l)​hu(l−1),\displaystyle=\eta_{uv}g_{uv}^{(l)}\odot W_{m}^{(l)}h_{u}^{(l-1)},(14)hv(l)\displaystyle h_{v}^{(l)}=LayerNorm⁡(hv(l−1)+PReLU⁡(b(l)+∑u∈𝒩​(v)mu→v(l))).\displaystyle=\operatorname{LayerNorm}\left(h_{v}^{(l-1)}+\operatorname{PReLU}\left(b^{(l)}+\sum_{u\in\mathcal{N}(v)}m_{u\to v}^{(l)}\right)\right).(15)In addition, the reader combines a contextual calibration channel on the current graph with a cross-graph schema prior channel:

H​(q,G)=Hctx+βsch​Hsch.H(q,G)=H_{\mathrm{ctx}}+\beta_{\mathrm{sch}}H_{\mathrm{sch}}.(16)

B.2Recoverable Evidence Region and Effective Signal-to-Noise Ratio

Definition B.1(Recoverable Evidence Region).

Fix a queryqqand the current memory graphGG. LetRq⊆VER_{q}\subseteq V_{E}denote the recoverable evidence region under queryqq, namely the set of entities jointly determined by the current graph structure, anchoring edges, and reader-reachable paths. This set contains nodes that support the answer, connect supporting documents, or serve as bridge entities. If the entity anchor set of documentddis denoted byA​(d)⊆VEA(d)\subseteq V_{E}, then the anchor coverage of the current graph over the gold evidence is defined as

ρA=|{d∈D+:A​(d)∩Rq≠∅}||D+|.\rho_{A}=\frac{\left|\{d\in D^{+}:A(d)\cap R_{q}\neq\varnothing\}\right|}{|D^{+}|}.(17)

Definition B.2(Query-Relevant Scalar Activation).

Letrqr_{q}be the direction induced by the query representation or the final scoring head. The nonnegative query-relevant activation of nodevvat layerllis defined as

av(l)=[⟨rq,hv(l)⟩]+,a_{v}^{(l)}=\big[\langle r_{q},h_{v}^{(l)}\rangle\big]_{+},(18)where[t]+=max⁡{t,0}[t]_{+}=\max\{t,0\}. The evidence signal mass, noise mass, and effective signal-to-noise ratio at layerllare respectively defined as

Sl\displaystyle S_{l}=∑v∈Rqav(l),\displaystyle=\sum_{v\in R_{q}}a_{v}^{(l)},(19)Nl\displaystyle N_{l}=∑v∈VE∖Rqav(l),\displaystyle=\sum_{v\in V_{E}\setminus R_{q}}a_{v}^{(l)},(20)SNRl\displaystyle\operatorname{SNR}_{l}=SlNl,\displaystyle=\frac{S_{l}}{N_{l}},(21)with the convention thatSNRl=+∞\operatorname{SNR}_{l}=+\inftywhenNl=0N_{l}=0.

B.3Aggregate Propagation Assumptions and Structural Gating Coefficients

Idealized analyses often assume that every evidence edge has a uniform lower gate boundg+g_{+}and every noisy edge has a uniform upper gate boundg−g_{-}. However, in graph memories dynamically constructed by an LLM writer, edges may be missing, erroneous, or repeated; some evidence edges may be underestimated, while some distractor edges may receive high gates. We therefore adopt an aggregate propagation assumption.

Assumption B.4(Query-Relevant Effective Propagation Operator).

Fix a queryqq, the current graph memoryGG, and the reader representation at layerll. For each layerl∈{1,…,L}l\in\{1,\ldots,L\}, there exists a nonnegative matrixTl∈ℝ≥0|VE|×|VE|T_{l}\in\mathbb{R}_{\geq 0}^{|V_{E}|\times|V_{E}|}and a nonnegative perturbation vectorϵl∈ℝ≥0|VE|\epsilon_{l}\in\mathbb{R}_{\geq 0}^{|V_{E}|}such that the query-relevant activation vector at layerll,

a(l)=(av(l))v∈VE,a^{(l)}=\big(a_{v}^{(l)}\big)_{v\in V_{E}},is controlled by the previous-layer activationa(l−1)a^{(l-1)}in the following coordinate-wise sense:

a(l)⪯Tl​a(l−1)+ϵl.a^{(l)}\preceq T_{l}a^{(l-1)}+\epsilon_{l}.(22)Here,⪯\preceqdenotes coordinate-wise inequality. The operatorTlT_{l}denotes the effective propagation operator induced by thell-th layer on the query-relevant scalar channel. It absorbs the combined effects of normalized adjacency weightsηu​v\eta_{uv}, structural gatesgu​v(l)g_{uv}^{(l)}, message projectionWm(l)W_{m}^{(l)}, context–schema representation composition, and final scoring-channel projection into a single nonnegative propagation kernel. In other words,Tl​(u,v)T_{l}(u,v)is the effective nonnegative contribution strength of the query-relevant activation of nodevvat the previous layer to nodeuuat layerll.

The perturbation termϵl\epsilon_{l}absorbs residual effects that are difficult to exactly characterize by nonnegative linear propagation, including LayerNorm, PReLU, residual connections, vector-direction rotation, scoring-channel mismatch, and finite-parameter approximation error.

Furthermore, we only require the propagation process to preserve effective signal in the evidence region in an aggregate sense. We do not require the structural gate to perfectly distinguish every evidence edge from every noisy edge. The operatorTlT_{l}may allow some evidence edges to be underestimated and some distractor edges to be overestimated; the aggregate propagation coefficients defined below only characterize the overall effect of these local errors on the evidence and noise regions.

Let

R¯q=VE∖Rq.\bar{R}_{q}=V_{E}\setminus R_{q}.PartitionTlT_{l}according to the node setsRqR_{q}andR¯q\bar{R}_{q}:

Tl=(TR​R(l)TR​R¯(l)TR¯​R(l)TR¯​R¯(l)).T_{l}=\begin{pmatrix}T_{RR}^{(l)}&T_{R\bar{R}}^{(l)}\\ T_{\bar{R}R}^{(l)}&T_{\bar{R}\bar{R}}^{(l)}\end{pmatrix}.(23)Here,TR​R(l)T_{RR}^{(l)}denotes effective propagation within the evidence region,TR¯​R¯(l)T_{\bar{R}\bar{R}}^{(l)}denotes effective propagation within the noise region,TR¯​R(l)T_{\bar{R}R}^{(l)}denotes leakage propagation from the evidence region to the noise region, andTR​R¯(l)T_{R\bar{R}}^{(l)}denotes propagation from the noise region to the evidence region.

Definition B.5(Aggregate Propagation Coefficients).

Given the effective propagation operatorTlT_{l}at layerlland its block decomposition in Eq. (23), define the three aggregate propagation coefficientsAlA_{l},BlB_{l}, andClC_{l}as follows.

AlA_{l}is the evidence-retention coefficient. It characterizes the minimum fraction of total mass that remains inside the evidence regionRqR_{q}after any nonnegative evidence signalxxpropagates one layer withinRqR_{q}. Formally,AlA_{l}is any nonnegative constant satisfying

Al≤infx∈ℝ≥0|Rq|,1⊤​x>0𝟏⊤​TR​R(l)​x𝟏⊤​x.A_{l}\leq\inf_{x\in\mathbb{R}_{\geq 0}^{|R_{q}|},\ \mathbf{1}^{\top}x>0}\frac{\mathbf{1}^{\top}T_{RR}^{(l)}x}{\mathbf{1}^{\top}x}.(24)Equivalently, for any nonnegative evidence signalx∈ℝ≥0|Rq|x\in\mathbb{R}_{\geq 0}^{|R_{q}|}with𝟏⊤​x>0\mathbf{1}^{\top}x>0,

𝟏⊤​TR​R(l)​x≥Al​1⊤​x.\mathbf{1}^{\top}T_{RR}^{(l)}x\geq A_{l}\,\mathbf{1}^{\top}x. BlB_{l}is the noise self-propagation coefficient. It characterizes the maximum extent to which any nonnegative noise signalyycan be retained or expanded after one layer of propagation inside the noise regionR¯q\bar{R}_{q}. Formally,BlB_{l}is any nonnegative constant satisfying

Bl≥supy∈ℝ≥0|R¯q|,1⊤​y>0𝟏⊤​TR¯​R¯(l)​y𝟏⊤​y.B_{l}\geq\sup_{y\in\mathbb{R}_{\geq 0}^{|\bar{R}_{q}|},\ \mathbf{1}^{\top}y>0}\frac{\mathbf{1}^{\top}T_{\bar{R}\bar{R}}^{(l)}y}{\mathbf{1}^{\top}y}.(25)Equivalently, for any nonnegative noise signaly∈ℝ≥0|R¯q|y\in\mathbb{R}_{\geq 0}^{|\bar{R}_{q}|}with𝟏⊤​y>0\mathbf{1}^{\top}y>0,

𝟏⊤​TR¯​R¯(l)​y≤Bl​1⊤​y.\mathbf{1}^{\top}T_{\bar{R}\bar{R}}^{(l)}y\leq B_{l}\,\mathbf{1}^{\top}y. ClC_{l}is the evidence-to-noise leakage coefficient. It characterizes the maximum fraction of an arbitrary nonnegative signal in the evidence region that can leak into the non-evidence regionR¯q\bar{R}_{q}after one layer of propagation. Formally,ClC_{l}is any nonnegative constant satisfying

Cl≥supx∈ℝ≥0|Rq|,1⊤​x>0𝟏⊤​TR¯​R(l)​x𝟏⊤​x.C_{l}\geq\sup_{x\in\mathbb{R}_{\geq 0}^{|R_{q}|},\ \mathbf{1}^{\top}x>0}\frac{\mathbf{1}^{\top}T_{\bar{R}R}^{(l)}x}{\mathbf{1}^{\top}x}.(26)Equivalently, for any nonnegative evidence signalx∈ℝ≥0|Rq|x\in\mathbb{R}_{\geq 0}^{|R_{q}|}with𝟏⊤​x>0\mathbf{1}^{\top}x>0,

𝟏⊤​TR¯​R(l)​x≤Cl​1⊤​x.\mathbf{1}^{\top}T_{\bar{R}R}^{(l)}x\leq C_{l}\,\mathbf{1}^{\top}x. Finally, let

ξl=𝟏⊤​ϵl,R¯\xi_{l}=\mathbf{1}^{\top}\epsilon_{l,\bar{R}}(27)denote the total perturbation mass injected into the noise regionR¯q\bar{R}_{q}at layerllby nonlinearities, normalization, representation-direction shifts, and approximation errors. Here,ϵl,R¯\epsilon_{l,\bar{R}}denotes the restriction of the perturbation vectorϵl\epsilon_{l}toR¯q\bar{R}_{q}.

Lemma B.6(Aggregate Propagation Recurrence).

Under AssumptionB.4and DefinitionB.5, ifSl−1>0S_{l-1}>0andNl−1≥0N_{l-1}\geq 0, then layerllsatisfies

Sl\displaystyle S_{l}≥Al​Sl−1,\displaystyle\geq A_{l}S_{l-1},(28)Nl\displaystyle N_{l}≤Bl​Nl−1+Cl​Sl−1+ξl.\displaystyle\leq B_{l}N_{l-1}+C_{l}S_{l-1}+\xi_{l}.(29)

Proof.

LetaR(l−1)a_{R}^{(l-1)}andaR¯(l−1)a_{\bar{R}}^{(l-1)}be the restrictions ofa(l−1)a^{(l-1)}toRqR_{q}andR¯q\bar{R}_{q}, respectively. By the definition of the evidence-retention coefficient in Eq. (24), the total mass retained within the evidence region throughRq→RqR_{q}\to R_{q}propagation is at least

𝟏⊤​TR​R(l)​aR(l−1)≥Al​𝟏⊤​aR(l−1)=Al​Sl−1,\mathbf{1}^{\top}T_{RR}^{(l)}a_{R}^{(l-1)}\geq A_{l}\mathbf{1}^{\top}a_{R}^{(l-1)}=A_{l}S_{l-1},(30)and thusSl≥Al​Sl−1S_{l}\geq A_{l}S_{l-1}.

On the other hand, the layer-llmass in the noise region can be upper-bounded by three terms: noise self-propagation, evidence leakage, and perturbation:

Nl≤𝟏⊤​TR¯​R¯(l)​aR¯(l−1)+𝟏⊤​TR¯​R(l)​aR(l−1)+𝟏⊤​ϵl,R¯.N_{l}\leq\mathbf{1}^{\top}T_{\bar{R}\bar{R}}^{(l)}a_{\bar{R}}^{(l-1)}+\mathbf{1}^{\top}T_{\bar{R}R}^{(l)}a_{R}^{(l-1)}+\mathbf{1}^{\top}\epsilon_{l,\bar{R}}.(31)Using Eqs. (25), (26), and (27), we obtain

Nl≤Bl​Nl−1+Cl​Sl−1+ξl.N_{l}\leq B_{l}N_{l-1}+C_{l}S_{l-1}+\xi_{l}.(32)This proves the lemma. ∎

B.4Realistic Aggregate Signal-to-Noise Ratio Bound

Theorem B.7(Realistic Aggregate SNR Bound).

Assume that for alll=1,…,Ll=1,\ldots,L, there existAl>0A_{l}>0,Bl≥0B_{l}\geq 0,Cl≥0C_{l}\geq 0, andξl≥0\xi_{l}\geq 0such that the recurrences in Eqs.(28)–(29)hold. Let

Ql=SNRl−1=NlSl.Q_{l}=\operatorname{SNR}_{l}^{-1}=\frac{N_{l}}{S_{l}}.(33)Then

QL≤(∏l=1LBlAl)​Q0+∑i=1L(CiAi+ξiAi​Si−1)​∏t=i+1LBtAt.Q_{L}\leq\left(\prod_{l=1}^{L}\frac{B_{l}}{A_{l}}\right)Q_{0}+\sum_{i=1}^{L}\left(\frac{C_{i}}{A_{i}}+\frac{\xi_{i}}{A_{i}S_{i-1}}\right)\prod_{t=i+1}^{L}\frac{B_{t}}{A_{t}}.(34)Equivalently, if the right-hand side is finite, then

SNRL≥[(∏l=1LBlAl)​SNR0−1+∑i=1L(CiAi+ξiAi​Si−1)​∏t=i+1LBtAt]−1.\operatorname{SNR}_{L}\geq\left[\left(\prod_{l=1}^{L}\frac{B_{l}}{A_{l}}\right)\operatorname{SNR}_{0}^{-1}+\sum_{i=1}^{L}\left(\frac{C_{i}}{A_{i}}+\frac{\xi_{i}}{A_{i}S_{i-1}}\right)\prod_{t=i+1}^{L}\frac{B_{t}}{A_{t}}\right]^{-1}.(35)The empty product is defined as11.

Proof.

By LemmaB.6, for anyll,

Sl≥Al​Sl−1,Nl≤Bl​Nl−1+Cl​Sl−1+ξl.S_{l}\geq A_{l}S_{l-1},\qquad N_{l}\leq B_{l}N_{l-1}+C_{l}S_{l-1}+\xi_{l}.(36)Therefore,

Ql=NlSl\displaystyle Q_{l}=\frac{N_{l}}{S_{l}}≤Bl​Nl−1+Cl​Sl−1+ξlAl​Sl−1\displaystyle\leq\frac{B_{l}N_{l-1}+C_{l}S_{l-1}+\xi_{l}}{A_{l}S_{l-1}}(37)=BlAl​Ql−1+ClAl+ξlAl​Sl−1.\displaystyle=\frac{B_{l}}{A_{l}}Q_{l-1}+\frac{C_{l}}{A_{l}}+\frac{\xi_{l}}{A_{l}S_{l-1}}.(38)Let

rl=BlAl,dl=ClAl+ξlAl​Sl−1.r_{l}=\frac{B_{l}}{A_{l}},\qquad d_{l}=\frac{C_{l}}{A_{l}}+\frac{\xi_{l}}{A_{l}S_{l-1}}.(39)Then

Ql≤rl​Ql−1+dl.Q_{l}\leq r_{l}Q_{l-1}+d_{l}.(40)Expanding this first-order nonhomogeneous recurrence yields

QL≤(∏l=1Lrl)​Q0+∑i=1Ldi​∏t=i+1Lrt.Q_{L}\leq\left(\prod_{l=1}^{L}r_{l}\right)Q_{0}+\sum_{i=1}^{L}d_{i}\prod_{t=i+1}^{L}r_{t}.(41)Substituting backrlr_{l}anddld_{l}proves Eq. (34). SinceSNRL=QL−1\operatorname{SNR}_{L}=Q_{L}^{-1}, Eq. (35) follows. ∎

Corollary B.8(Layer-Homogeneous Case).

IfAl=A>0A_{l}=A>0,Bl=B≥0B_{l}=B\geq 0,Cl=C≥0C_{l}=C\geq 0, andξl=0\xi_{l}=0, then

SNRL≥[(BA)L​SNR0−1+CA​∑i=0L−1(BA)i]−1.\operatorname{SNR}_{L}\geq\left[\left(\frac{B}{A}\right)^{L}\operatorname{SNR}_{0}^{-1}+\frac{C}{A}\sum_{i=0}^{L-1}\left(\frac{B}{A}\right)^{i}\right]^{-1}.(42)If furtherC=0C=0, then

SNRL≥(AB)L​SNR0.\operatorname{SNR}_{L}\geq\left(\frac{A}{B}\right)^{L}\operatorname{SNR}_{0}.(43)

Proof.

SubstitutingAl=AA_{l}=A,Bl=BB_{l}=B,Cl=CC_{l}=C, andξl=0\xi_{l}=0into TheoremB.7and simplifying the resulting geometric series gives the result. ∎

Corollary B.9(Ideal Edge-Wise Gating as a Special Case).

Suppose there exist constantsg+>g−>0g_{+}>g_{-}>0,α+>0\alpha_{+}>0,α−>0\alpha_{-}>0,g0≥0g_{0}\geq 0, andλleak≥0\lambda_{\mathrm{leak}}\geq 0such that the effective retention inside the evidence region isA=g+​α+A=g_{+}\alpha_{+}, the self-propagation inside the noise region isB=g−​α−B=g_{-}\alpha_{-}, the evidence-to-noise leakage isC=g0​λleakC=g_{0}\lambda_{\mathrm{leak}}, andξl=0\xi_{l}=0. Then TheoremB.7reduces to

SNRL≥[(g−​α−g+​α+)L​SNR0−1+g0​λleakg+​α+​∑i=0L−1(g−​α−g+​α+)i]−1.\operatorname{SNR}_{L}\geq\left[\left(\frac{g_{-}\alpha_{-}}{g_{+}\alpha_{+}}\right)^{L}\operatorname{SNR}_{0}^{-1}+\frac{g_{0}\lambda_{\mathrm{leak}}}{g_{+}\alpha_{+}}\sum_{i=0}^{L-1}\left(\frac{g_{-}\alpha_{-}}{g_{+}\alpha_{+}}\right)^{i}\right]^{-1}.(44)Ifλleak=0\lambda_{\mathrm{leak}}=0, then

SNRL≥(g+​α+g−​α−)L​SNR0.\operatorname{SNR}_{L}\geq\left(\frac{g_{+}\alpha_{+}}{g_{-}\alpha_{-}}\right)^{L}\operatorname{SNR}_{0}.(45)

B.5Document Retrieval Budget

The final retrieval targets of the SAGE-GFM reader are memory fragments or documents. Therefore, we need to convert the entity-level SNR bound into a document-level top-kkbudget bound. Let the final document score beSD​(d)S_{D}(d), and let the top-kkretrieval result be

Pk​(q,G)=Top​-​kd∈D​SD​(d).P_{k}(q,G)=\mathrm{Top}\text{-}k_{d\in D}S_{D}(d).(46)

Definition B.10(ρ\rho-Coverage Retrieval Budget).

Given0<ρ≤ρA0<\rho\leq\rho_{A}, let

mρ=⌈ρ​|D+|⌉.m_{\rho}=\lceil\rho|D^{+}|\rceil.(47)The minimum top-kkbudget required to achieveρ\rho-level gold evidence coverage is defined as

ℬρ​(q,G)=min⁡{k:|Pk​(q,G)∩D+|≥mρ}.\mathcal{B}_{\rho}(q,G)=\min\left\{k:\ |P_{k}(q,G)\cap D^{+}|\geq m_{\rho}\right\}.(48)Letτρ+\tau_{\rho}^{+}denote themρm_{\rho}-th largest score among gold evidence documents, namely the gold score threshold required to achieveρ\rho-coverage.

Lemma B.11(Quantile Retrieval Budget Bound).

Let the total score mass of distractor documents be

ML−=∑d∈D∖D+SD​(d).M_{L}^{-}=\sum_{d\in D\setminus D^{+}}S_{D}(d).(49)Ifτρ+>0\tau_{\rho}^{+}>0, then

ℬρ​(q,G)≤mρ+ML−τρ+.\mathcal{B}_{\rho}(q,G)\leq m_{\rho}+\frac{M_{L}^{-}}{\tau_{\rho}^{+}}.(50)

Proof.

Define the set of distractor documents whose scores are not lower thanτρ+\tau_{\rho}^{+}as

𝒩ρ={d∈D∖D+:SD​(d)≥τρ+}.\mathcal{N}_{\rho}=\{d\in D\setminus D^{+}:S_{D}(d)\geq\tau_{\rho}^{+}\}.(51)For anyd∈𝒩ρd\in\mathcal{N}_{\rho}, we haveSD​(d)≥τρ+S_{D}(d)\geq\tau_{\rho}^{+}, and hence

|𝒩ρ|​τρ+≤∑d∈𝒩ρSD​(d)≤ML−.|\mathcal{N}_{\rho}|\tau_{\rho}^{+}\leq\sum_{d\in\mathcal{N}_{\rho}}S_{D}(d)\leq M_{L}^{-}.(52)Thus|𝒩ρ|≤ML−/τρ+|\mathcal{N}_{\rho}|\leq M_{L}^{-}/\tau_{\rho}^{+}. To ensure that the top-kkresults contain at leastmρm_{\rho}gold evidence documents, it suffices to include thesemρm_{\rho}gold documents and all distractors whose scores are not lower than the thresholdτρ+\tau_{\rho}^{+}. Therefore,

ℬρ​(q,G)≤mρ+|𝒩ρ|≤mρ+ML−τρ+.\mathcal{B}_{\rho}(q,G)\leq m_{\rho}+|\mathcal{N}_{\rho}|\leq m_{\rho}+\frac{M_{L}^{-}}{\tau_{\rho}^{+}}.(53)This proves the lemma. ∎

To use entity-level SNR for document-level retrieval, we need to control the noise expansion introduced by entity-to-document projection.

Assumption B.12(Projection Noise and Gold Score Concentration).

There exist constantsKA≥0K_{A}\geq 0,ζA≥0\zeta_{A}\geq 0, andcρ∈(0,1]c_{\rho}\in(0,1]such that the final document scores satisfy

ML−\displaystyle M_{L}^{-}≤KA​NL+ζA,\displaystyle\leq K_{A}N_{L}+\zeta_{A},(54)τρ+\displaystyle\tau_{\rho}^{+}≥cρmρ​SL.\displaystyle\geq\frac{c_{\rho}}{m_{\rho}}S_{L}.(55)Here,KAK_{A}is the noise expansion factor of entity-to-document projection,ζA\zeta_{A}denotes the projection residual caused by incorrect anchors, missing anchors, or additional text-similarity terms, andcρc_{\rho}measures whether the evidence signal is effectively distributed over at leastmρm_{\rho}gold documents.

Theorem B.13(Realistic Signal–Noise–Budget Bound).

Under the conditions of TheoremB.7, further assume that AssumptionB.12holds. Then

ℬρ​(q,G)≤mρ+mρ​KAcρ​SNRL−1+mρ​ζAcρ​SL.\mathcal{B}_{\rho}(q,G)\leq m_{\rho}+\frac{m_{\rho}K_{A}}{c_{\rho}}\operatorname{SNR}_{L}^{-1}+\frac{m_{\rho}\zeta_{A}}{c_{\rho}S_{L}}.(56)Substituting TheoremB.7further yields the explicit upper bound

ℬρ​(q,G)≤\displaystyle\mathcal{B}_{\rho}(q,G)\leqmρ+mρ​KAcρ[(∏l=1LBlAl)SNR0−1\displaystyle\ m_{\rho}+\frac{m_{\rho}K_{A}}{c_{\rho}}\Bigg[\left(\prod_{l=1}^{L}\frac{B_{l}}{A_{l}}\right)\operatorname{SNR}_{0}^{-1}(57)+∑i=1L(CiAi+ξiAi​Si−1)∏t=i+1LBtAt]+mρ​ζAcρ​SL.\displaystyle\quad+\sum_{i=1}^{L}\left(\frac{C_{i}}{A_{i}}+\frac{\xi_{i}}{A_{i}S_{i-1}}\right)\prod_{t=i+1}^{L}\frac{B_{t}}{A_{t}}\Bigg]+\frac{m_{\rho}\zeta_{A}}{c_{\rho}S_{L}}.

Proof.

By LemmaB.11,

ℬρ​(q,G)≤mρ+ML−τρ+.\mathcal{B}_{\rho}(q,G)\leq m_{\rho}+\frac{M_{L}^{-}}{\tau_{\rho}^{+}}.(58)By AssumptionB.12,

ML−τρ+≤KA​NL+ζA(cρ/mρ)​SL=mρ​KAcρ​NLSL+mρ​ζAcρ​SL.\frac{M_{L}^{-}}{\tau_{\rho}^{+}}\leq\frac{K_{A}N_{L}+\zeta_{A}}{(c_{\rho}/m_{\rho})S_{L}}=\frac{m_{\rho}K_{A}}{c_{\rho}}\frac{N_{L}}{S_{L}}+\frac{m_{\rho}\zeta_{A}}{c_{\rho}S_{L}}.(59)SinceNL/SL=SNRL−1N_{L}/S_{L}=\operatorname{SNR}_{L}^{-1}, Eq. (56) follows. Substituting the upper bound onSNRL−1\operatorname{SNR}_{L}^{-1}from TheoremB.7into Eq. (56) gives Eq. (57). ∎

Corollary B.14(Full Evidence Recovery Budget).

Ifρ=1\rho=1andρA=1\rho_{A}=1, thenmρ=|D+|m_{\rho}=|D^{+}|, andℬρ​(q,G)\mathcal{B}_{\rho}(q,G)reduces to the full evidence recovery budget. In this case, TheoremB.13provides an upper bound on the top-kkbudget required to recover all gold evidence.

B.6Interpretation of the Theoretical Bound for the SAGE Design

TheoremB.7and TheoremB.13unify four reader design factors under the same retrieval-budget upper bound. To avoid relying only on intuitive discussion, we provide several direct monotonicity propositions.

Proposition 2(Monotonicity of the Budget Bound).

Fixmρm_{\rho},KAK_{A},cρc_{\rho},ζA\zeta_{A},SLS_{L}, andSNR0\operatorname{SNR}_{0}, and define

ΓL=(∏l=1LBlAl)​SNR0−1+∑i=1L(CiAi+ξiAi​Si−1)​∏t=i+1LBtAt.\Gamma_{L}=\left(\prod_{l=1}^{L}\frac{B_{l}}{A_{l}}\right)\operatorname{SNR}_{0}^{-1}+\sum_{i=1}^{L}\left(\frac{C_{i}}{A_{i}}+\frac{\xi_{i}}{A_{i}S_{i-1}}\right)\prod_{t=i+1}^{L}\frac{B_{t}}{A_{t}}.(60)Then the budget upper bound

Uρ=mρ+mρ​KAcρ​ΓL+mρ​ζAcρ​SLU_{\rho}=m_{\rho}+\frac{m_{\rho}K_{A}}{c_{\rho}}\Gamma_{L}+\frac{m_{\rho}\zeta_{A}}{c_{\rho}S_{L}}(61)is monotonically nondecreasing inΓL\Gamma_{L},KAK_{A}, andζA\zeta_{A}, and monotonically nonincreasing incρc_{\rho}andSLS_{L}. If all other terms in the products are fixed, decreasing any ofBl/AlB_{l}/A_{l},Cl/AlC_{l}/A_{l}, orξl/(Al​Sl−1)\xi_{l}/(A_{l}S_{l-1})cannot increaseUρU_{\rho}.

Proof.

The partial derivatives ofUρU_{\rho}with respect toΓL\Gamma_{L},KAK_{A}, andζA\zeta_{A}are nonnegative, while the partial derivatives with respect tocρc_{\rho}andSLS_{L}are nonpositive. Moreover,ΓL\Gamma_{L}is a nonnegative linear or multiplicative combination ofBl/AlB_{l}/A_{l},Cl/AlC_{l}/A_{l}, andξl/(Al​Sl−1)\xi_{l}/(A_{l}S_{l-1}). When the other terms are fixed, decreasing any such nonnegative term cannot increaseΓL\Gamma_{L}. The proposition follows. ∎

Proposition 3(Effect of Soft Addressing).

If soft addressing increases the initial evidence signalS0S_{0}and decreases the initial noise massN0N_{0}, thereby increasingSNR0\operatorname{SNR}_{0}, then, with all other coefficients fixed, the budget upper bound in TheoremB.13does not increase. In particular, explicit entities, aliases, pseudo-queries, type constraints, hard constraints, and entity-linking signals in query planning improve the final budget bound whenever they increaseS0/N0S_{0}/N_{0}in an aggregate sense.

Proposition 4(Aggregate Advantage of Structural Gating).

Compared with a reader without structural gating, suppose the structurally gated reader satisfies

BlgateAlgate≤BlplainAlplain,ClgateAlgate≤ClplainAlplain,ξlgateAlgate​Sl−1gate≤ξlplainAlplain​Sl−1plain.\frac{B_{l}^{\mathrm{gate}}}{A_{l}^{\mathrm{gate}}}\leq\frac{B_{l}^{\mathrm{plain}}}{A_{l}^{\mathrm{plain}}},\qquad\frac{C_{l}^{\mathrm{gate}}}{A_{l}^{\mathrm{gate}}}\leq\frac{C_{l}^{\mathrm{plain}}}{A_{l}^{\mathrm{plain}}},\qquad\frac{\xi_{l}^{\mathrm{gate}}}{A_{l}^{\mathrm{gate}}S_{l-1}^{\mathrm{gate}}}\leq\frac{\xi_{l}^{\mathrm{plain}}}{A_{l}^{\mathrm{plain}}S_{l-1}^{\mathrm{plain}}}.(62)Then the budget upper bound of the structurally gated reader is no larger than that of the ungated reader.

Proposition 5(Stability Interpretation of the Context–Schema Dual Channel).

Letδl\delta_{l}denote the failure probability of the aggregate recurrences in Eqs. (28)–(29) at layerll. If the schema prior channel reduces the variance of cross-graph structural-role estimation and the context calibration channel reduces the current-graph adaptation error, so thatδl\delta_{l}decreases toδl′\delta_{l}^{\prime}withδl′≤δl\delta_{l}^{\prime}\leq\delta_{l}, then the probability lower bound under which TheoremB.7and TheoremB.13simultaneously hold improves from1−∑l=1Lδl1-\sum_{l=1}^{L}\delta_{l}to1−∑l=1Lδl′1-\sum_{l=1}^{L}\delta_{l}^{\prime}.

The core quantities derived above are

SNRL−1≤(∏l=1LBlAl)​SNR0−1+∑i=1L(CiAi+ξiAi​Si−1)​∏t=i+1LBtAt,\operatorname{SNR}_{L}^{-1}\leq\left(\prod_{l=1}^{L}\frac{B_{l}}{A_{l}}\right)\operatorname{SNR}_{0}^{-1}+\sum_{i=1}^{L}\left(\frac{C_{i}}{A_{i}}+\frac{\xi_{i}}{A_{i}S_{i-1}}\right)\prod_{t=i+1}^{L}\frac{B_{t}}{A_{t}},(63)and

ℬρ​(q,G)≤mρ+mρ​KAcρ​SNRL−1+mρ​ζAcρ​SL.\mathcal{B}_{\rho}(q,G)\leq m_{\rho}+\frac{m_{\rho}K_{A}}{c_{\rho}}\operatorname{SNR}_{L}^{-1}+\frac{m_{\rho}\zeta_{A}}{c_{\rho}S_{L}}.(64)Equation (63) shows that soft addressing reduces the amount of noise that subsequent propagation must overcome by improving the initialSNR0\operatorname{SNR}_{0}; structural gating improves aggregate evidence retention and noise suppression by increasingAlA_{l}and decreasingBlB_{l}andClC_{l}; the context–schema dual channel makes these aggregate inequalities more stable on dynamic graph memories by reducing cross-graph structural-role estimation error; and entity-to-document projection converts entity-level SNR into document-level budget efficiency by decreasingKAK_{A}andζA\zeta_{A}and increasingcρc_{\rho}. Equation (64) further shows that the advantage of SAGE-GFM does not rely on perfect edge-wise classification or zero-leakage assumptions. As long as evidence-retention dominance is achieved in an aggregate or high-probability sense, i.e.,Bl/AlB_{l}/A_{l}andCl/AlC_{l}/A_{l}are sufficiently small, the reader can improve query-relevant SNR and reduce the top-kkretrieval budget required to achieve a given level of evidence coverage.

Appendix CTarget Graph Calibration and Cross-graph Structural Priors

C.1Structural Role Decomposition Assumption

Definition C.1(Structural role mapping).

Given a graphGG, a mapping

ρG:V​(G)→ℛ\rho_{G}:V(G)\to\mathcal{R}(65)is called a structural role mapping, whereℛ\mathcal{R}is the structural role space.ρG​(v)\rho_{G}(v)can be jointly determined byφG​(v)\varphi_{G}(v), the structural statistics of edges incident tovv, local community-boundary statistics, and other graph-structure summaries. Typical structural roles include hub, bridge, community core, boundary node, and noisy shortcut.

Definition C.2(Target graph reading risk).

Fix a target graphGG. Let𝒟G\mathcal{D}_{G}be the query–node sampling distribution on the target graph, and letfG⋆​(q,v)f_{G}^{\star}(q,v)be the ideal evidence relevance function. For any measurable functionff, define the squared risk as

ℛG​(f)=𝔼(q,v)∼𝒟G​[(f​(q,v,G)−fG⋆​(q,v))2].\mathcal{R}_{G}(f)=\mathbb{E}_{(q,v)\sim\mathcal{D}_{G}}\left[\left(f(q,v,G)-f_{G}^{\star}(q,v)\right)^{2}\right].(66)

Assumption C.3(Context–schema decomposability).

For every target graphGG, the ideal reading function can be decomposed as

fG⋆​(q,v)=fsch⋆​(q,ρG​(v))+fctx,G⋆​(q,v),f_{G}^{\star}(q,v)=f_{\mathrm{sch}}^{\star}(q,\rho_{G}(v))+f_{\mathrm{ctx},G}^{\star}(q,v),(67)wherefsch⋆f_{\mathrm{sch}}^{\star}denotes the cross-graph shared structural reading rule, andfctx,G⋆f_{\mathrm{ctx},G}^{\star}denotes the target-graph residual induced by the current writer, current domain, current entity naming, local noise, and writing style.

Equation (67) corresponds exactly to the structural design ofSAGE:HschH_{\mathrm{sch}}is used to approximatefsch⋆f_{\mathrm{sch}}^{\star}, andHctxH_{\mathrm{ctx}}is used to approximatefctx,G⋆f_{\mathrm{ctx},G}^{\star}. The next subsection gives the risk meaning of this decomposition.

C.2Approximation Risk of Context–schema Decomposition

Theorem C.4(Context–schema decomposition reduces target-graph approximation risk).

Suppose Assumption 2.3 holds. Letℋsch\mathcal{H}_{\mathrm{sch}}be the schema function class, letℋctx,G\mathcal{H}_{\mathrm{ctx},G}be the target-graph context function class, and define the sum class

ℋsch+ℋctx,G={fs+fc:fs∈ℋsch,fc∈ℋctx,G}.\mathcal{H}_{\mathrm{sch}}+\mathcal{H}_{\mathrm{ctx},G}=\{f_{s}+f_{c}:f_{s}\in\mathcal{H}_{\mathrm{sch}},f_{c}\in\mathcal{H}_{\mathrm{ctx},G}\}.(68)If there existϵsch,ϵctx≥0\epsilon_{\mathrm{sch}},\epsilon_{\mathrm{ctx}}\geq 0such that

inffs∈ℋsch𝔼​[(fs​(q,ρG​(v))−fsch⋆​(q,ρG​(v)))2]≤ϵsch,\inf_{f_{s}\in\mathcal{H}_{\mathrm{sch}}}\mathbb{E}\left[(f_{s}(q,\rho_{G}(v))-f_{\mathrm{sch}}^{\star}(q,\rho_{G}(v)))^{2}\right]\leq\epsilon_{\mathrm{sch}},(69)inffc∈ℋctx,G𝔼​[(fc​(q,v,G)−fctx,G⋆​(q,v))2]≤ϵctx,\inf_{f_{c}\in\mathcal{H}_{\mathrm{ctx},G}}\mathbb{E}\left[(f_{c}(q,v,G)-f_{\mathrm{ctx},G}^{\star}(q,v))^{2}\right]\leq\epsilon_{\mathrm{ctx}},(70)where both expectations are over(q,v)∼𝒟G(q,v)\sim\mathcal{D}_{G}, then

inff∈ℋsch+ℋctx,GℛG​(f)≤2​ϵsch+2​ϵctx.\inf_{f\in\mathcal{H}_{\mathrm{sch}}+\mathcal{H}_{\mathrm{ctx},G}}\mathcal{R}_{G}(f)\leq 2\epsilon_{\mathrm{sch}}+2\epsilon_{\mathrm{ctx}}.(71)

Proof.

Take anyα>0\alpha>0. By the two approximation error conditions, there existf^s∈ℋsch\hat{f}_{s}\in\mathcal{H}_{\mathrm{sch}}andf^c∈ℋctx,G\hat{f}_{c}\in\mathcal{H}_{\mathrm{ctx},G}such that

𝔼​[(f^s−fsch⋆)2]≤ϵsch+α,𝔼​[(f^c−fctx,G⋆)2]≤ϵctx+α.\mathbb{E}[(\hat{f}_{s}-f_{\mathrm{sch}}^{\star})^{2}]\leq\epsilon_{\mathrm{sch}}+\alpha,\qquad\mathbb{E}[(\hat{f}_{c}-f_{\mathrm{ctx},G}^{\star})^{2}]\leq\epsilon_{\mathrm{ctx}}+\alpha.(72)Letf^=f^s+f^c\hat{f}=\hat{f}_{s}+\hat{f}_{c}. By the decomposition in Eq. (67), we have

f^−fG⋆=(f^s−fsch⋆)+(f^c−fctx,G⋆).\hat{f}-f_{G}^{\star}=(\hat{f}_{s}-f_{\mathrm{sch}}^{\star})+(\hat{f}_{c}-f_{\mathrm{ctx},G}^{\star}).(73)Using(a+b)2≤2​a2+2​b2(a+b)^{2}\leq 2a^{2}+2b^{2}, we obtain

ℛG​(f^)\displaystyle\mathcal{R}_{G}(\hat{f})=𝔼​[(f^−fG⋆)2]\displaystyle=\mathbb{E}[(\hat{f}-f_{G}^{\star})^{2}]≤2​𝔼​[(f^s−fsch⋆)2]+2​𝔼​[(f^c−fctx,G⋆)2]\displaystyle\leq 2\mathbb{E}[(\hat{f}_{s}-f_{\mathrm{sch}}^{\star})^{2}]+2\mathbb{E}[(\hat{f}_{c}-f_{\mathrm{ctx},G}^{\star})^{2}]≤2​ϵsch+2​ϵctx+4​α.\displaystyle\leq 2\epsilon_{\mathrm{sch}}+2\epsilon_{\mathrm{ctx}}+4\alpha.(74)Sinceα>0\alpha>0is arbitrary, taking the infimum yields the conclusion. ∎

Proposition 6(Residual bias of schema-only models).

Further assume thatL2​(𝒟G)L_{2}(\mathcal{D}_{G})is a Hilbert space,ℋsch\mathcal{H}_{\mathrm{sch}}is a closed linear subspace of it, andfsch⋆∈ℋschf_{\mathrm{sch}}^{\star}\in\mathcal{H}_{\mathrm{sch}}. If only a schema-only modelfs∈ℋschf_{s}\in\mathcal{H}_{\mathrm{sch}}is used, then

inffs∈ℋschℛG​(fs)=distL2​(𝒟G)2⁡(fctx,G⋆,ℋsch).\inf_{f_{s}\in\mathcal{H}_{\mathrm{sch}}}\mathcal{R}_{G}(f_{s})=\operatorname{dist}_{L_{2}(\mathcal{D}_{G})}^{2}(f_{\mathrm{ctx},G}^{\star},\mathcal{H}_{\mathrm{sch}}).(75)Therefore, as long as the target-graph residualfctx,G⋆f_{\mathrm{ctx},G}^{\star}does not belong toℋsch\mathcal{H}_{\mathrm{sch}}, a schema-only reader has an irreducible target-graph bias.

Proof.

Byfsch⋆∈ℋschf_{\mathrm{sch}}^{\star}\in\mathcal{H}_{\mathrm{sch}}and the linearity ofℋsch\mathcal{H}_{\mathrm{sch}}, anyfs∈ℋschf_{s}\in\mathcal{H}_{\mathrm{sch}}can be written asfs=fsch⋆+gf_{s}=f_{\mathrm{sch}}^{\star}+g, whereg∈ℋschg\in\mathcal{H}_{\mathrm{sch}}. Thus,

ℛG​(fs)\displaystyle\mathcal{R}_{G}(f_{s})=‖fs−fsch⋆−fctx,G⋆‖L2​(𝒟G)2\displaystyle=\left\lVert f_{s}-f_{\mathrm{sch}}^{\star}-f_{\mathrm{ctx},G}^{\star}\right\rVert_{L_{2}(\mathcal{D}_{G})}^{2}=‖g−fctx,G⋆‖L2​(𝒟G)2.\displaystyle=\left\lVert g-f_{\mathrm{ctx},G}^{\star}\right\rVert_{L_{2}(\mathcal{D}_{G})}^{2}.(76)Taking the infimum overfs∈ℋschf_{s}\in\mathcal{H}_{\mathrm{sch}}is equivalent to taking the infimum overg∈ℋschg\in\mathcal{H}_{\mathrm{sch}}, and Eq. (75) follows from the definition of distance. ∎

C.3Sample Complexity Advantage of Schema Prior

Lemma C.6(Uniform convergence for bounded loss classes).

Letℱ\mathcal{F}be a function class, and let the lossℓf​(z)∈[0,1]\ell_{f}(z)\in[0,1]. Givennnindependent samplesS={zi}i=1nS=\{z_{i}\}_{i=1}^{n}, define the true riskR​(f)=𝔼​[ℓf​(z)]R(f)=\mathbb{E}[\ell_{f}(z)]and the empirical riskR^S​(f)=n−1​∑i=1nℓf​(zi)\widehat{R}_{S}(f)=n^{-1}\sum_{i=1}^{n}\ell_{f}(z_{i}). Then, with probability at least1−δ1-\delta, for allf∈ℱf\in\mathcal{F}simultaneously,

R​(f)≤R^S​(f)+2​Radn⁡(ℓ∘ℱ)+3​log⁡(2/δ)2​n,R(f)\leq\widehat{R}_{S}(f)+2\operatorname{Rad}_{n}(\ell\circ\mathcal{F})+3\sqrt{\frac{\log(2/\delta)}{2n}},(77)whereRadn⁡(ℓ∘ℱ)\operatorname{Rad}_{n}(\ell\circ\mathcal{F})is an upper bound on the empirical Rademacher complexity of the loss-composed class.

Proof.

This is a direct result of the standard symmetrization and McDiarmid/Hoeffding concentration inequalities for bounded loss function classes. Specifically, let

Z​(S)=supf∈ℱ|R​(f)−R^S​(f)|.Z(S)=\sup_{f\in\mathcal{F}}\left\lvert R(f)-\widehat{R}_{S}(f)\right\rvert.(78)By symmetrization,𝔼​Z​(S)≤2​Radn⁡(ℓ∘ℱ)\mathbb{E}Z(S)\leq 2\operatorname{Rad}_{n}(\ell\circ\mathcal{F}). Moreover, since changing one sample can changeZ​(S)Z(S)by at most1/n1/n, McDiarmid’s inequality gives

Z​(S)≤𝔼​Z​(S)+3​log⁡(2/δ)2​nZ(S)\leq\mathbb{E}Z(S)+3\sqrt{\frac{\log(2/\delta)}{2n}}(79)with probability at least1−δ1-\delta. Combining the two inequalities gives the conclusion. ∎

Theorem C.7(Schema prior reduces the sample complexity of target-graph adaptation).

Fix a target graphGG, and suppose the number of supervised samples available for reader calibration on the target graph isnGn_{G}. Letℋfull\mathcal{H}_{\mathrm{full}}be the full reader class that needs to be learned on the target graph when no schema prior is used; letℋres\mathcal{H}_{\mathrm{res}}be the residual class that only needs to be learned given a schema priorfsf_{s}, with the combined modelf=fs+frf=f_{s}+f_{r},fr∈ℋresf_{r}\in\mathcal{H}_{\mathrm{res}}. Letf^full\hat{f}_{\mathrm{full}}andf^res\hat{f}_{\mathrm{res}}be the empirical risk minimizers over the two classes, respectively. Then, with probability at least1−δ1-\delta,

ℛG​(f^full)\displaystyle\mathcal{R}_{G}(\hat{f}_{\mathrm{full}})≤inff∈ℋfullℛG​(f)+4​RadnG⁡(ℓ∘ℋfull)+6​log⁡(4/δ)2​nG,\displaystyle\leq\inf_{f\in\mathcal{H}_{\mathrm{full}}}\mathcal{R}_{G}(f)+4\operatorname{Rad}_{n_{G}}(\ell\circ\mathcal{H}_{\mathrm{full}})+6\sqrt{\frac{\log(4/\delta)}{2n_{G}}},(80)ℛG​(fs+f^res)\displaystyle\mathcal{R}_{G}(f_{s}+\hat{f}_{\mathrm{res}})≤inffr∈ℋresℛG​(fs+fr)+4​RadnG⁡(ℓ∘(fs+ℋres))+6​log⁡(4/δ)2​nG.\displaystyle\leq\inf_{f_{r}\in\mathcal{H}_{\mathrm{res}}}\mathcal{R}_{G}(f_{s}+f_{r})+4\operatorname{Rad}_{n_{G}}(\ell\circ(f_{s}+\mathcal{H}_{\mathrm{res}}))+6\sqrt{\frac{\log(4/\delta)}{2n_{G}}}.(81)If the complexities satisfy

RadnG⁡(ℓ∘ℋfull)=O~​(dfullnG),RadnG⁡(ℓ∘(fs+ℋres))=O~​(dresnG),\operatorname{Rad}_{n_{G}}(\ell\circ\mathcal{H}_{\mathrm{full}})=\widetilde{O}\left(\sqrt{\frac{d_{\mathrm{full}}}{n_{G}}}\right),\qquad\operatorname{Rad}_{n_{G}}(\ell\circ(f_{s}+\mathcal{H}_{\mathrm{res}}))=\widetilde{O}\left(\sqrt{\frac{d_{\mathrm{res}}}{n_{G}}}\right),(82)anddres≪dfulld_{\mathrm{res}}\ll d_{\mathrm{full}}, then the schema prior reduces target-graph learning from estimating the full reading function to estimating the residual, and lowers the estimation error term for target-graph adaptation.

Proof.

For any function classℱ\mathcal{F}, letf^\hat{f}be the empirical risk minimizer and letf∘∈arg​inff∈ℱR​(f)f^{\circ}\in\arg\inf_{f\in\mathcal{F}}R(f)be the true risk minimizer. By empirical optimality,R^​(f^)≤R^​(f∘)\widehat{R}(\hat{f})\leq\widehat{R}(f^{\circ}), so

R​(f^)−R​(f∘)\displaystyle R(\hat{f})-R(f^{\circ})=R​(f^)−R^​(f^)+R^​(f^)−R^​(f∘)+R^​(f∘)−R​(f∘)\displaystyle=R(\hat{f})-\widehat{R}(\hat{f})+\widehat{R}(\hat{f})-\widehat{R}(f^{\circ})+\widehat{R}(f^{\circ})-R(f^{\circ})≤2​supf∈ℱ|R​(f)−R^​(f)|.\displaystyle\leq 2\sup_{f\in\mathcal{F}}\left\lvert R(f)-\widehat{R}(f)\right\rvert.(83)Applying Lemma 2.7 toℱ=ℋfull\mathcal{F}=\mathcal{H}_{\mathrm{full}}andℱ=fs+ℋres\mathcal{F}=f_{s}+\mathcal{H}_{\mathrm{res}}, respectively, and combining the two probability events by a union bound, gives Eq. (80) and Eq. (81). The complexity-order conclusion follows by substituting the corresponding Rademacher upper bounds. ∎

Appendix DWriter-induced Graph Distribution Shift and Target Graph Calibration

D.1Writer-induced Dynamic Graph Distribution

The writer parameterθ\thetainduces a graph distributionPθ​(G∣q,D)P_{\theta}(G\mid q,D)on the sample(q,D)(q,D). For notational simplicity, denote the joint distribution of(q,D,D+,y,G)(q,D,D^{+},y,G)byΠθ\Pi_{\theta}. Given the reader parameterϕ\phi, define the reader risk as

ℒR​(ϕ;θ)=𝔼(q,D,D+,y,G)∼Πθ​[ℓR​(Rϕ​(q,G,D),D+,y)].\mathcal{L}_{R}(\phi;\theta)=\mathbb{E}_{(q,D,D^{+},y,G)\sim\Pi_{\theta}}\left[\ell_{R}(R_{\phi}(q,G,D),D^{+},y)\right].(84)Here,ℓR\ell_{R}can be supporting-entity BCE, multi-positive ranking loss, document recall loss, or a combination thereof.

Proposition 7(Writer updates cause reader distribution shift).

Assume0≤ℓR≤10\leq\ell_{R}\leq 1. For any fixedϕ\phiand any writer parametersθ,θ′\theta,\theta^{\prime}, we have

|ℒR​(ϕ;θ′)−ℒR​(ϕ;θ)|≤TV⁡(Πθ′,Πθ),\left\lvert\mathcal{L}_{R}(\phi;\theta^{\prime})-\mathcal{L}_{R}(\phi;\theta)\right\rvert\leq\operatorname{TV}(\Pi_{\theta^{\prime}},\Pi_{\theta}),(85)whereTV\operatorname{TV}is the total variation distance. If, further,ℓR​(Rϕ​(q,G,D),D+,y)\ell_{R}(R_{\phi}(q,G,D),D^{+},y)isLℓL_{\ell}-Lipschitz with respect to the graph variable under some graph metricd𝒢d_{\mathcal{G}}, then

|ℒR​(ϕ;θ′)−ℒR​(ϕ;θ)|≤Lℓ​W1​(Πθ′,Πθ),\left\lvert\mathcal{L}_{R}(\phi;\theta^{\prime})-\mathcal{L}_{R}(\phi;\theta)\right\rvert\leq L_{\ell}W_{1}(\Pi_{\theta^{\prime}},\Pi_{\theta}),(86)whereW1W_{1}is the first-order Wasserstein distance induced byd𝒢d_{\mathcal{G}}.

Proof.

For the bounded loss case, lethϕ​(q,D,D+,y,G)=ℓR​(Rϕ​(q,G,D),D+,y)∈[0,1]h_{\phi}(q,D,D^{+},y,G)=\ell_{R}(R_{\phi}(q,G,D),D^{+},y)\in[0,1]. Then

|ℒR​(ϕ;θ′)−ℒR​(ϕ;θ)|\displaystyle\left\lvert\mathcal{L}_{R}(\phi;\theta^{\prime})-\mathcal{L}_{R}(\phi;\theta)\right\rvert=|∫hϕ​𝑑Πθ′−∫hϕ​𝑑Πθ|≤TV⁡(Πθ′,Πθ),\displaystyle=\left\lvert\int h_{\phi}\,d\Pi_{\theta^{\prime}}-\int h_{\phi}\,d\Pi_{\theta}\right\rvert\leq\operatorname{TV}(\Pi_{\theta^{\prime}},\Pi_{\theta}),(87)where the last step follows from the dual definition of total variation. Ifhϕh_{\phi}isLℓL_{\ell}-Lipschitz, the Wasserstein upper bound follows from the Kantorovich–Rubinstein duality. ∎

Corollary D.1(Necessity of target graph calibration).

If updating the writer fromθ\thetatoθ′\theta^{\prime}causesTV⁡(Πθ′,Πθ)\operatorname{TV}(\Pi_{\theta^{\prime}},\Pi_{\theta})to be non-negligible, then the risk of a fixed readerϕ\phion the new graph distribution may increase. Therefore, target graph calibration of the reader, namelyϕ↦ϕ′\phi\mapsto\phi^{\prime}to reduceℒR​(ϕ′;θ′)\mathcal{L}_{R}(\phi^{\prime};\theta^{\prime}), is a necessary mechanism for handling writer-induced graph distribution shift.

Proof.

By Proposition 3.1, writer distribution shift can directly change the new-distribution risk of the fixed reader. If the reader is not updated, there is no optimization mechanism to offset this drift term. Target graph calibration is precisely re-optimization ofℒR​(⋅;θ′)\mathcal{L}_{R}(\cdot;\theta^{\prime}), and is therefore a natural step for reducing the risk on the new graph. ∎

Appendix EReader Stability under Dynamic Graph Evolution

E.1Realistic Graph Evolution Distance

Real graphs often contain hubs, node additions and deletions, alias merges, anchor rewrites, and structural statistics that are not globally Lipschitz. Therefore, we use the augmented graph drift actually perceived by the reader to measure graph evolution.

Definition E.1(Padding alignment and presence bit).

Given two consecutive-round graphsGGandG′G^{\prime}, align them through persistent memory ids to a common node universeV¯=V​(G)∪V​(G′)\bar{V}=V(G)\cup V(G^{\prime}). If a node exists only in one graph, it is treated as an isolated padding node in the other graph, and a presence bit is added to its features. The aligned node feature matrices are still denoted byX,X′X,X^{\prime}.

Definition E.2(Augmented graph drift).

LetA,A′A,A^{\prime}be self-looped row-normalized adjacency matrices, letSq,Sq′S_{q},S_{q}^{\prime}be the entry score vectors before soft addressing, and letB,B′B,B^{\prime}be row-normalized entity-to-document anchoring matrices. Define

ΔX=‖X−X′‖2,∞,ΔA=‖A−A′‖∞,Δseed=‖Sq−Sq′‖∞,ΔB=‖B−B′‖∞,\Delta_{X}=\left\lVert X-X^{\prime}\right\rVert_{2,\infty},\qquad\Delta_{A}=\left\lVert A-A^{\prime}\right\rVert_{\infty},\qquad\Delta_{\mathrm{seed}}=\left\lVert S_{q}-S_{q}^{\prime}\right\rVert_{\infty},\qquad\Delta_{B}=\left\lVert B-B^{\prime}\right\rVert_{\infty},(88)where

‖H‖2,∞=maxv∈V¯⁡‖hv‖2,‖A‖∞=maxv​∑u|Av​u|.\left\lVert H\right\rVert_{2,\infty}=\max_{v\in\bar{V}}\left\lVert h_{v}\right\rVert_{2},\qquad\left\lVert A\right\rVert_{\infty}=\max_{v}\sum_{u}\left\lvert A_{vu}\right\rvert.(89)For the gate inputzu​v(l)z_{uv}^{(l)}at layerll, define the weighted structural drift as

ΔZ(l)=maxv​∑uAv​u′​‖zu​v(l)−zu​v′⁣(l)‖2,ΔZ=max1≤l≤L⁡ΔZ(l).\Delta_{Z}^{(l)}=\max_{v}\sum_{u}A^{\prime}_{vu}\left\lVert z_{uv}^{(l)}-z_{uv}^{\prime(l)}\right\rVert_{2},\qquad\Delta_{Z}=\max_{1\leq l\leq L}\Delta_{Z}^{(l)}.(90)The total augmented graph drift is defined as

Δaug​(G,G′;q)=ΔX+ΔA+Δseed+ΔZ+ΔB.\Delta_{\mathrm{aug}}(G,G^{\prime};q)=\Delta_{X}+\Delta_{A}+\Delta_{\mathrm{seed}}+\Delta_{Z}+\Delta_{B}.(91)

E.2Stability Assumptions

Assumption E.3(Normalized adjacency).

For all considered graphs,Av​u≥0A_{vu}\geq 0and

∑uAv​u=1,∀v∈V¯.\sum_{u}A_{vu}=1,\qquad\forall v\in\bar{V}.(92)Therefore,‖A‖∞=1\left\lVert A\right\rVert_{\infty}=1. This assumption allows high-degree hubs to exist, but prevents single-layer propagation from being unboundedly amplified by node degree.

Assumption E.4(Trajectory-local boundedness).

For graph pairsG,G′G,G^{\prime}on the training and inference trajectories, there exist constantsBlB_{l}such that

‖H(l)​(q,G)‖2,∞≤Bl,‖H(l)​(q,G′)‖2,∞≤Bl,l=0,…,L.\left\lVert H^{(l)}(q,G)\right\rVert_{2,\infty}\leq B_{l},\qquad\left\lVert H^{(l)}(q,G^{\prime})\right\rVert_{2,\infty}\leq B_{l},\qquad l=0,\dots,L.(93)

Assumption E.5(Locally Lipschitz modules).

In a neighborhood of the training trajectory, thell-th layer satisfies

‖Wm(l)‖2≤Ml,\left\lVert W_{m}^{(l)}\right\rVert_{2}\leq M_{l},(94)‖MLPg(l)⁡(z)−MLPg(l)⁡(z′)‖∞≤Lg,l​‖z−z′‖2.\left\lVert\operatorname{MLP}_{g}^{(l)}(z)-\operatorname{MLP}_{g}^{(l)}(z^{\prime})\right\rVert_{\infty}\leq L_{g,l}\left\lVert z-z^{\prime}\right\rVert_{2}.(95)The Lipschitz constant of PReLU isLσL_{\sigma}, and the Lipschitz constant of LayerNorm with a numerical stabilizer in this trajectory neighborhood isLLN,lL_{\operatorname{LN},l}.

Assumption E.6(Local Lipschitzness of score head and projection).

The entity score head satisfies

‖sE​(q,G)−sE​(q,G′)‖∞≤LE​‖H​(q,G)−H​(q,G′)‖2,∞.\left\lVert s_{E}(q,G)-s_{E}(q,G^{\prime})\right\rVert_{\infty}\leq L_{E}\left\lVert H(q,G)-H(q,G^{\prime})\right\rVert_{2,\infty}.(96)Meanwhile,‖sE​(q,G)‖∞≤SE\left\lVert s_{E}(q,G)\right\rVert_{\infty}\leq S_{E}, andBBis row-normalized, so‖B‖∞≤1\left\lVert B\right\rVert_{\infty}\leq 1.

E.3Stability of Soft Addressing and Initial Representation

Lemma E.7(Softmax and pre-activation stability).

Let

p=softmax⁡(S/T0),p′=softmax⁡(S′/T0).p=\operatorname{softmax}(S/T_{0}),\qquad p^{\prime}=\operatorname{softmax}(S^{\prime}/T_{0}).(97)Then

‖p−p′‖∞≤1T0​‖S−S′‖∞.\left\lVert p-p^{\prime}\right\rVert_{\infty}\leq\frac{1}{T_{0}}\left\lVert S-S^{\prime}\right\rVert_{\infty}.(98)Furthermore, letav=(pv+ϵp)ηa_{v}=(p_{v}+\epsilon_{p})^{\eta}andav′=(pv′+ϵp)ηa_{v}^{\prime}=(p_{v}^{\prime}+\epsilon_{p})^{\eta}, where0<η≤10<\eta\leq 1. Then

‖a−a′‖∞≤η​ϵpη−1T0​‖S−S′‖∞.\left\lVert a-a^{\prime}\right\rVert_{\infty}\leq\frac{\eta\epsilon_{p}^{\eta-1}}{T_{0}}\left\lVert S-S^{\prime}\right\rVert_{\infty}.(99)

Proof.

The Jacobian of softmax isJ​(z)=diag⁡(p)−p​p⊤J(z)=\operatorname{diag}(p)-pp^{\top}. For any rowii,

∑j|Ji​j​(z)|=2​pi​(1−pi)≤1.\sum_{j}\left\lvert J_{ij}(z)\right\rvert=2p_{i}(1-p_{i})\leq 1.(100)Thus,‖J​(z)‖∞→∞≤1\left\lVert J(z)\right\rVert_{\infty\to\infty}\leq 1. By the mean value theorem and settingz=S/T0z=S/T_{0}, we obtain

‖p−p′‖∞≤1T0​‖S−S′‖∞.\left\lVert p-p^{\prime}\right\rVert_{\infty}\leq\frac{1}{T_{0}}\left\lVert S-S^{\prime}\right\rVert_{\infty}.(101)The functiont↦(t+ϵp)ηt\mapsto(t+\epsilon_{p})^{\eta}is Lipschitz on[0,1][0,1], with constant at mostη​ϵpη−1\eta\epsilon_{p}^{\eta-1}. Composing the two inequalities gives the conclusion. ∎

Lemma E.8(Initial node representation stability).

Letuq=Wq​Emb⁡(q)u_{q}=W_{q}\operatorname{Emb}(q), and define

hv(0)=av​(q)​uq+Wx​xv.h_{v}^{(0)}=a_{v}(q)u_{q}+W_{x}x_{v}.(102)Then

‖H(0)​(q,G)−H(0)​(q,G′)‖2,∞≤Cinit​(Δseed+ΔX),\left\lVert H^{(0)}(q,G)-H^{(0)}(q,G^{\prime})\right\rVert_{2,\infty}\leq C_{\mathrm{init}}(\Delta_{\mathrm{seed}}+\Delta_{X}),(103)where

Cinit=η​ϵpη−1T0​‖uq‖2+‖Wx‖2.C_{\mathrm{init}}=\frac{\eta\epsilon_{p}^{\eta-1}}{T_{0}}\left\lVert u_{q}\right\rVert_{2}+\left\lVert W_{x}\right\rVert_{2}.(104)

Proof.

For any nodevv,

hv(0)−hv′⁣(0)=(av−av′)​uq+Wx​(xv−xv′).h_{v}^{(0)}-h_{v}^{\prime(0)}=(a_{v}-a_{v}^{\prime})u_{q}+W_{x}(x_{v}-x_{v}^{\prime}).(105)Taking the norm and applying Lemma 4.8 gives

‖hv(0)−hv′⁣(0)‖2≤η​ϵpη−1T0​‖uq‖2​Δseed+‖Wx‖2​ΔX.\left\lVert h_{v}^{(0)}-h_{v}^{\prime(0)}\right\rVert_{2}\leq\frac{\eta\epsilon_{p}^{\eta-1}}{T_{0}}\left\lVert u_{q}\right\rVert_{2}\Delta_{\mathrm{seed}}+\left\lVert W_{x}\right\rVert_{2}\Delta_{X}.(106)Taking the maximum overvvgives the conclusion. ∎

E.4Single-layer Stability of Structurally Gated Propagation

Lemma E.9(Boundedness and stability of structural gate).

The structural gate of layerll,

gu​v(l)=1+δ​tanh⁡(MLPg(l)⁡(zu​v(l))),g_{uv}^{(l)}=1+\delta\tanh(\operatorname{MLP}_{g}^{(l)}(z_{uv}^{(l)})),(107)satisfies

‖gu​v(l)‖∞≤1+δ,\left\lVert g_{uv}^{(l)}\right\rVert_{\infty}\leq 1+\delta,(108)and

‖gu​v(l)−gu​v′⁣(l)‖∞≤δ​Lg,l​‖zu​v(l)−zu​v′⁣(l)‖2.\left\lVert g_{uv}^{(l)}-g_{uv}^{\prime(l)}\right\rVert_{\infty}\leq\delta L_{g,l}\left\lVert z_{uv}^{(l)}-z_{uv}^{\prime(l)}\right\rVert_{2}.(109)

Proof.

Since the range oftanh\tanhis contained in[−1,1][-1,1], the first statement follows immediately. Moreover, becausetanh\tanhis 1-Lipschitz andMLPg(l)\operatorname{MLP}_{g}^{(l)}isLg,lL_{g,l}-Lipschitz in the trajectory neighborhood,

‖gu​v(l)−gu​v′⁣(l)‖∞\displaystyle\left\lVert g_{uv}^{(l)}-g_{uv}^{\prime(l)}\right\rVert_{\infty}≤δ​‖MLPg(l)⁡(zu​v(l))−MLPg(l)⁡(zu​v′⁣(l))‖∞\displaystyle\leq\delta\left\lVert\operatorname{MLP}_{g}^{(l)}(z_{uv}^{(l)})-\operatorname{MLP}_{g}^{(l)}(z_{uv}^{\prime(l)})\right\rVert_{\infty}≤δ​Lg,l​‖zu​v(l)−zu​v′⁣(l)‖2.\displaystyle\leq\delta L_{g,l}\left\lVert z_{uv}^{(l)}-z_{uv}^{\prime(l)}\right\rVert_{2}.(110)∎

Lemma E.10(Single-layer stability of structurally gated propagation).

Define

Dl=‖H(l)​(q,G)−H(l)​(q,G′)‖2,∞.D_{l}=\left\lVert H^{(l)}(q,G)-H^{(l)}(q,G^{\prime})\right\rVert_{2,\infty}.(111)Under Assumptions 4.5–4.7, thell-th propagation layer satisfies

Dl≤αl​Dl−1+βlA​ΔA+βlZ​ΔZ(l),D_{l}\leq\alpha_{l}D_{l-1}+\beta_{l}^{A}\Delta_{A}+\beta_{l}^{Z}\Delta_{Z}^{(l)},(112)where one can take

αl=LLN,l​(1+Lσ​(1+δ)​Ml),\alpha_{l}=L_{\operatorname{LN},l}\bigl(1+L_{\sigma}(1+\delta)M_{l}\bigr),(113)βlA=LLN,l​Lσ​(1+δ)​Ml​Bl−1,βlZ=LLN,l​Lσ​δ​Lg,l​Ml​Bl−1.\beta_{l}^{A}=L_{\operatorname{LN},l}L_{\sigma}(1+\delta)M_{l}B_{l-1},\qquad\beta_{l}^{Z}=L_{\operatorname{LN},l}L_{\sigma}\delta L_{g,l}M_{l}B_{l-1}.(114)

Proof.

Write

Mv=∑uAv​u​gu​v⊙W​hu,Mv′=∑uAv​u′​gu​v′⊙W​hu′,M_{v}=\sum_{u}A_{vu}g_{uv}\odot Wh_{u},\qquad M_{v}^{\prime}=\sum_{u}A^{\prime}_{vu}g^{\prime}_{uv}\odot Wh^{\prime}_{u},(115)where the layer indexllis omitted. Adding and subtracting intermediate terms gives

Mv−Mv′=\displaystyle M_{v}-M_{v}^{\prime}={}∑uAv​u​gu​v⊙W​(hu−hu′)\displaystyle\sum_{u}A_{vu}g_{uv}\odot W(h_{u}-h^{\prime}_{u})+∑u(Av​u−Av​u′)​gu​v⊙W​hu′\displaystyle+\sum_{u}(A_{vu}-A^{\prime}_{vu})g_{uv}\odot Wh^{\prime}_{u}+∑uAv​u′​(gu​v−gu​v′)⊙W​hu′.\displaystyle+\sum_{u}A^{\prime}_{vu}(g_{uv}-g^{\prime}_{uv})\odot Wh^{\prime}_{u}.(116)The first term is controlled by row-normalization,‖gu​v‖∞≤1+δ\left\lVert g_{uv}\right\rVert_{\infty}\leq 1+\delta, and‖W‖2≤Ml\left\lVert W\right\rVert_{2}\leq M_{l}:

‖∑uAv​u​gu​v⊙W​(hu−hu′)‖2≤(1+δ)​Ml​Dl−1.\left\lVert\sum_{u}A_{vu}g_{uv}\odot W(h_{u}-h^{\prime}_{u})\right\rVert_{2}\leq(1+\delta)M_{l}D_{l-1}.(117)The second term satisfies

‖∑u(Av​u−Av​u′)​gu​v⊙W​hu′‖2≤(1+δ)​Ml​Bl−1​∑u|Av​u−Av​u′|≤(1+δ)​Ml​Bl−1​ΔA.\left\lVert\sum_{u}(A_{vu}-A^{\prime}_{vu})g_{uv}\odot Wh^{\prime}_{u}\right\rVert_{2}\leq(1+\delta)M_{l}B_{l-1}\sum_{u}\left\lvert A_{vu}-A^{\prime}_{vu}\right\rvert\leq(1+\delta)M_{l}B_{l-1}\Delta_{A}.(118)For the third term, by Lemma 4.10,

‖∑uAv​u′​(gu​v−gu​v′)⊙W​hu′‖2\displaystyle\left\lVert\sum_{u}A^{\prime}_{vu}(g_{uv}-g^{\prime}_{uv})\odot Wh^{\prime}_{u}\right\rVert_{2}≤δ​Lg,l​Ml​Bl−1​∑uAv​u′​‖zu​v(l)−zu​v′⁣(l)‖2\displaystyle\leq\delta L_{g,l}M_{l}B_{l-1}\sum_{u}A^{\prime}_{vu}\left\lVert z_{uv}^{(l)}-z_{uv}^{\prime(l)}\right\rVert_{2}≤δ​Lg,l​Ml​Bl−1​ΔZ(l).\displaystyle\leq\delta L_{g,l}M_{l}B_{l-1}\Delta_{Z}^{(l)}.(119)Therefore,

‖Mv−Mv′‖2≤(1+δ)​Ml​Dl−1+(1+δ)​Ml​Bl−1​ΔA+δ​Lg,l​Ml​Bl−1​ΔZ(l).\left\lVert M_{v}-M_{v}^{\prime}\right\rVert_{2}\leq(1+\delta)M_{l}D_{l-1}+(1+\delta)M_{l}B_{l-1}\Delta_{A}+\delta L_{g,l}M_{l}B_{l-1}\Delta_{Z}^{(l)}.(120)By theLσL_{\sigma}-Lipschitz property of PReLU, the residual structure, and theLLN,lL_{\operatorname{LN},l}-Lipschitz property of LayerNorm,

‖hv(l)−hv′⁣(l)‖2≤LLN,l​(‖hv(l−1)−hv′⁣(l−1)‖2+Lσ​‖Mv−Mv′‖2).\left\lVert h_{v}^{(l)}-h_{v}^{\prime(l)}\right\rVert_{2}\leq L_{\operatorname{LN},l}\left(\left\lVert h_{v}^{(l-1)}-h_{v}^{\prime(l-1)}\right\rVert_{2}+L_{\sigma}\left\lVert M_{v}-M_{v}^{\prime}\right\rVert_{2}\right).(121)Taking the maximum overvvgives Eq. (112). ∎

E.5Stability of Representations, Scores, and Retrieval Sets

Theorem E.11(Local stability of structurally gated representations to augmented graph drift).

Under Assumptions 4.5–4.7,LL-layer structurally gated propagation satisfies

DL≤\displaystyle D_{L}\leq{}(∏l=1Lαl)​D0+∑t=1L(∏l=t+1Lαl)​(βtA​ΔA+βtZ​ΔZ(t)).\displaystyle\left(\prod_{l=1}^{L}\alpha_{l}\right)D_{0}+\sum_{t=1}^{L}\left(\prod_{l=t+1}^{L}\alpha_{l}\right)\left(\beta_{t}^{A}\Delta_{A}+\beta_{t}^{Z}\Delta_{Z}^{(t)}\right).(122)Therefore, there exists a constantCH>0C_{H}>0such that

‖H(L)​(q,G)−H(L)​(q,G′)‖2,∞≤CH​(ΔX+Δseed+ΔA+ΔZ).\left\lVert H^{(L)}(q,G)-H^{(L)}(q,G^{\prime})\right\rVert_{2,\infty}\leq C_{H}(\Delta_{X}+\Delta_{\mathrm{seed}}+\Delta_{A}+\Delta_{Z}).(123)

Proof.

By Lemma 4.11, the recursion in Eq. (112) holds. Unrolling the recursion layer by layer gives Eq. (122). By Lemma 4.9,

D0≤Cinit​(ΔX+Δseed).D_{0}\leq C_{\mathrm{init}}(\Delta_{X}+\Delta_{\mathrm{seed}}).(124)SubstitutingΔZ(t)≤ΔZ\Delta_{Z}^{(t)}\leq\Delta_{Z}and merging all layer-related constants intoCHC_{H}gives the conclusion. ∎

Theorem E.12(Stability of context/schema dual channels).

If the two channels respectively satisfy

‖Hctx​(q,G)−Hctx​(q,G′)‖2,∞≤Cctx​Δaug,\left\lVert H_{\mathrm{ctx}}(q,G)-H_{\mathrm{ctx}}(q,G^{\prime})\right\rVert_{2,\infty}\leq C_{\mathrm{ctx}}\Delta_{\mathrm{aug}},(125)‖Hsch​(q,G)−Hsch​(q,G′)‖2,∞≤Csch​Δaug,\left\lVert H_{\mathrm{sch}}(q,G)-H_{\mathrm{sch}}(q,G^{\prime})\right\rVert_{2,\infty}\leq C_{\mathrm{sch}}\Delta_{\mathrm{aug}},(126)then the additive fusion in Eq.(LABEL:eq:additive_fusion)satisfies

‖H​(q,G)−H​(q,G′)‖2,∞≤(Cctx+|βsch|​Csch)​Δaug.\left\lVert H(q,G)-H(q,G^{\prime})\right\rVert_{2,\infty}\leq(C_{\mathrm{ctx}}+\left\lvert\beta_{\mathrm{sch}}\right\rvert C_{\mathrm{sch}})\Delta_{\mathrm{aug}}.(127)If a normalized or gated convex fusion theoretical form is adopted,

Hλ​(q,G)=(1−λ)​Hctx​(q,G)+λ​Hsch​(q,G),0≤λ≤1,H_{\lambda}(q,G)=(1-\lambda)H_{\mathrm{ctx}}(q,G)+\lambda H_{\mathrm{sch}}(q,G),\qquad 0\leq\lambda\leq 1,(128)then

‖Hλ​(q,G)−Hλ​(q,G′)‖2,∞≤((1−λ)​Cctx+λ​Csch)​Δaug.\left\lVert H_{\lambda}(q,G)-H_{\lambda}(q,G^{\prime})\right\rVert_{2,\infty}\leq\bigl((1-\lambda)C_{\mathrm{ctx}}+\lambda C_{\mathrm{sch}}\bigr)\Delta_{\mathrm{aug}}.(129)In particular, ifCsch<CctxC_{\mathrm{sch}}<C_{\mathrm{ctx}}, increasingλ\lambdadecreases this worst-case stability upper bound.

Proof.

The additive fusion case follows directly from the triangle inequality:

‖H​(G)−H​(G′)‖2,∞\displaystyle\left\lVert H(G)-H(G^{\prime})\right\rVert_{2,\infty}≤‖Hctx​(G)−Hctx​(G′)‖2,∞+|βsch|​‖Hsch​(G)−Hsch​(G′)‖2,∞\displaystyle\leq\left\lVert H_{\mathrm{ctx}}(G)-H_{\mathrm{ctx}}(G^{\prime})\right\rVert_{2,\infty}+\left\lvert\beta_{\mathrm{sch}}\right\rvert\left\lVert H_{\mathrm{sch}}(G)-H_{\mathrm{sch}}(G^{\prime})\right\rVert_{2,\infty}≤(Cctx+|βsch|​Csch)​Δaug.\displaystyle\leq(C_{\mathrm{ctx}}+\left\lvert\beta_{\mathrm{sch}}\right\rvert C_{\mathrm{sch}})\Delta_{\mathrm{aug}}.(130)The convex fusion case is analogous:

‖Hλ​(G)−Hλ​(G′)‖2,∞\displaystyle\left\lVert H_{\lambda}(G)-H_{\lambda}(G^{\prime})\right\rVert_{2,\infty}≤(1−λ)​Cctx​Δaug+λ​Csch​Δaug.\displaystyle\leq(1-\lambda)C_{\mathrm{ctx}}\Delta_{\mathrm{aug}}+\lambda C_{\mathrm{sch}}\Delta_{\mathrm{aug}}.(131)IfCsch<CctxC_{\mathrm{sch}}<C_{\mathrm{ctx}}, the right-hand side is monotonically decreasing inλ\lambda. ∎

Theorem E.14(Stability of entity scores and document scores).

Under Assumption 4.7, there exist constantsCE,CD>0C_{E},C_{D}>0such that

‖sE​(q,G)−sE​(q,G′)‖∞≤CE​Δaug,\left\lVert s_{E}(q,G)-s_{E}(q,G^{\prime})\right\rVert_{\infty}\leq C_{E}\Delta_{\mathrm{aug}},(132)‖sD​(q,G)−sD​(q,G′)‖∞≤CD​Δaug.\left\lVert s_{D}(q,G)-s_{D}(q,G^{\prime})\right\rVert_{\infty}\leq C_{D}\Delta_{\mathrm{aug}}.(133)For additive fusion, one can take

CE=LE​(Cctx+|βsch|​Csch),CD=CE+SE.C_{E}=L_{E}(C_{\mathrm{ctx}}+\left\lvert\beta_{\mathrm{sch}}\right\rvert C_{\mathrm{sch}}),\qquad C_{D}=C_{E}+S_{E}.(134)

Proof.

The entity score upper bound follows from the Lipschitz property of the score head and Theorem 4.13. For document projection,sD=B​sEs_{D}=Bs_{E}, and therefore

sD​(G)−sD​(G′)=BG​(sE​(G)−sE​(G′))+(BG−BG′)​sE​(G′).s_{D}(G)-s_{D}(G^{\prime})=B_{G}(s_{E}(G)-s_{E}(G^{\prime}))+(B_{G}-B_{G^{\prime}})s_{E}(G^{\prime}).(135)Taking theℓ∞\ell_{\infty}norm and using‖BG‖∞≤1\left\lVert B_{G}\right\rVert_{\infty}\leq 1gives

‖sD​(G)−sD​(G′)‖∞≤‖sE​(G)−sE​(G′)‖∞+‖BG−BG′‖∞​‖sE​(G′)‖∞.\left\lVert s_{D}(G)-s_{D}(G^{\prime})\right\rVert_{\infty}\leq\left\lVert s_{E}(G)-s_{E}(G^{\prime})\right\rVert_{\infty}+\left\lVert B_{G}-B_{G^{\prime}}\right\rVert_{\infty}\left\lVert s_{E}(G^{\prime})\right\rVert_{\infty}.(136)Using‖sE​(G′)‖∞≤SE\left\lVert s_{E}(G^{\prime})\right\rVert_{\infty}\leq S_{E}andΔB≤Δaug\Delta_{B}\leq\Delta_{\mathrm{aug}}gives the document score stability. ∎

Theorem E.15(Boundary stability of hard top-kk).

Lets=sD​(q,G)s=s_{D}(q,G)ands′=sD​(q,G′)s^{\prime}=s_{D}(q,G^{\prime}), and suppose

‖s−s′‖∞≤ϵs.\left\lVert s-s^{\prime}\right\rVert_{\infty}\leq\epsilon_{s}.(137)Lettk=s(k)t_{k}=s_{(k)}be thekk-th largest score inss, and define the boundary set

ℬk,2​ϵs​(s)={d:|sd−tk|≤2​ϵs}.\mathcal{B}_{k,2\epsilon_{s}}(s)=\{d:\left\lvert s_{d}-t_{k}\right\rvert\leq 2\epsilon_{s}\}.(138)Then

Top​-​k​(s)​△​Top​-​k​(s′)⊆ℬk,2​ϵs​(s),\mathrm{Top}\text{-}k(s)\triangle\mathrm{Top}\text{-}k(s^{\prime})\subseteq\mathcal{B}_{k,2\epsilon_{s}}(s),(139)and hence

|Top​-​k​(s)​△​Top​-​k​(s′)|≤|ℬk,2​ϵs​(s)|.\left\lvert\mathrm{Top}\text{-}k(s)\triangle\mathrm{Top}\text{-}k(s^{\prime})\right\rvert\leq\left\lvert\mathcal{B}_{k,2\epsilon_{s}}(s)\right\rvert.(140)In particular, ifs(k)−s(k+1)>2​ϵss_{(k)}-s_{(k+1)}>2\epsilon_{s}, thenTop​-​k​(s)=Top​-​k​(s′)\mathrm{Top}\text{-}k(s)=\mathrm{Top}\text{-}k(s^{\prime}).

Proof.

Take anyi∈Top​-​k​(s)∖Top​-​k​(s′)i\in\mathrm{Top}\text{-}k(s)\setminus\mathrm{Top}\text{-}k(s^{\prime}). Sinceiidrops out of the top-kk, there existsj∉Top​-​k​(s)j\notin\mathrm{Top}\text{-}k(s)such thatj∈Top​-​k​(s′)j\in\mathrm{Top}\text{-}k(s^{\prime})andsj′≥si′s_{j}^{\prime}\geq s_{i}^{\prime}. By the perturbation bound,

sj+ϵs≥sj′≥si′≥si−ϵs,s_{j}+\epsilon_{s}\geq s_{j}^{\prime}\geq s_{i}^{\prime}\geq s_{i}-\epsilon_{s},(141)sosi≤sj+2​ϵs≤tk+2​ϵss_{i}\leq s_{j}+2\epsilon_{s}\leq t_{k}+2\epsilon_{s}. Sincei∈Top​-​k​(s)i\in\mathrm{Top}\text{-}k(s), we havesi≥tks_{i}\geq t_{k}, and hencei∈ℬk,2​ϵs​(s)i\in\mathcal{B}_{k,2\epsilon_{s}}(s). A symmetric argument forj∈Top​-​k​(s′)∖Top​-​k​(s)j\in\mathrm{Top}\text{-}k(s^{\prime})\setminus\mathrm{Top}\text{-}k(s)givesj∈ℬk,2​ϵs​(s)j\in\mathcal{B}_{k,2\epsilon_{s}}(s). Therefore, the symmetric difference is contained in the boundary set. Ifs(k)−s(k+1)>2​ϵss_{(k)}-s_{(k+1)}>2\epsilon_{s}, boundary exchange cannot occur, and the top-kkset remains unchanged. ∎

Corollary E.16(Top-kkboundary stability under graph evolution).

By Theorem 4.15, takingϵs=CD​Δaug​(G,G′;q)\epsilon_{s}=C_{D}\Delta_{\mathrm{aug}}(G,G^{\prime};q)yields

Pk​(q,G)​△​Pk​(q,G′)⊆ℬk,2​CD​Δaug​(sD​(q,G)).P_{k}(q,G)\triangle P_{k}(q,G^{\prime})\subseteq\mathcal{B}_{k,2C_{D}\Delta_{\mathrm{aug}}}(s_{D}(q,G)).(142)Therefore, the instability of hard top-kkis restricted to candidates near the original score boundary.

Theorem E.17(Stability of soft retrieval distribution).

Let

πD​(q,G)=softmax⁡(sD​(q,G)/τ).\pi_{D}(q,G)=\operatorname{softmax}(s_{D}(q,G)/\tau).(143)If‖sD​(q,G)−sD​(q,G′)‖∞≤ϵs\left\lVert s_{D}(q,G)-s_{D}(q,G^{\prime})\right\rVert_{\infty}\leq\epsilon_{s}, then

‖πD​(q,G)−πD​(q,G′)‖1≤2τ​ϵs.\left\lVert\pi_{D}(q,G)-\pi_{D}(q,G^{\prime})\right\rVert_{1}\leq\frac{2}{\tau}\epsilon_{s}.(144)Therefore,

‖πD​(q,G)−πD​(q,G′)‖1≤2​CDτ​Δaug​(G,G′;q).\left\lVert\pi_{D}(q,G)-\pi_{D}(q,G^{\prime})\right\rVert_{1}\leq\frac{2C_{D}}{\tau}\Delta_{\mathrm{aug}}(G,G^{\prime};q).(145)

Proof.

The Jacobian of softmax isJ​(z)=diag⁡(π)−π​π⊤J(z)=\operatorname{diag}(\pi)-\pi\pi^{\top}. For any perturbationrr,

J​(z)​r=π⊙(r−𝔼π​r).J(z)r=\pi\odot(r-\mathbb{E}_{\pi}r).(146)If‖r‖∞≤1\left\lVert r\right\rVert_{\infty}\leq 1, then|ri−𝔼π​r|≤2\left\lvert r_{i}-\mathbb{E}_{\pi}r\right\rvert\leq 2, so

‖J​(z)​r‖1≤∑iπi​|ri−𝔼π​r|≤2.\left\lVert J(z)r\right\rVert_{1}\leq\sum_{i}\pi_{i}\left\lvert r_{i}-\mathbb{E}_{\pi}r\right\rvert\leq 2.(147)Thus, theℓ∞→ℓ1\ell_{\infty}\to\ell_{1}Lipschitz constant of softmax is at most22. Since the input issD/τs_{D}/\tau, we obtain

‖πD​(s)−πD​(s′)‖1≤2τ​‖s−s′‖∞.\left\lVert\pi_{D}(s)-\pi_{D}(s^{\prime})\right\rVert_{1}\leq\frac{2}{\tau}\left\lVert s-s^{\prime}\right\rVert_{\infty}.(148)Substituting Theorem 4.15 gives the conclusion. ∎

Theorem E.18(High-probability graph evolution stability).

If the writer’s single-round graph update satisfies

ℙ​[Δaug​(G,G′;q)>ϵ]≤δ,\mathbb{P}[\Delta_{\mathrm{aug}}(G,G^{\prime};q)>\epsilon]\leq\delta,(149)then

ℙ​[‖sD​(q,G)−sD​(q,G′)‖∞>CD​ϵ]≤δ.\mathbb{P}\left[\left\lVert s_{D}(q,G)-s_{D}(q,G^{\prime})\right\rVert_{\infty}>C_{D}\epsilon\right]\leq\delta.(150)If𝔼​[Δaug​(G,G′;q)]≤ϵ¯\mathbb{E}[\Delta_{\mathrm{aug}}(G,G^{\prime};q)]\leq\bar{\epsilon}, then

𝔼​[‖sD​(q,G)−sD​(q,G′)‖∞]≤CD​ϵ¯.\mathbb{E}\left[\left\lVert s_{D}(q,G)-s_{D}(q,G^{\prime})\right\rVert_{\infty}\right]\leq C_{D}\bar{\epsilon}.(151)

Proof.

By Theorem 4.15, for any graph pair, we have

‖sD​(q,G)−sD​(q,G′)‖∞≤CD​Δaug​(G,G′;q).\left\lVert s_{D}(q,G)-s_{D}(q,G^{\prime})\right\rVert_{\infty}\leq C_{D}\Delta_{\mathrm{aug}}(G,G^{\prime};q).(152)Therefore, the event{‖sD​(q,G)−sD​(q,G′)‖∞>CD​ϵ}\{\left\lVert s_{D}(q,G)-s_{D}(q,G^{\prime})\right\rVert_{\infty}>C_{D}\epsilon\}implies the event{Δaug​(G,G′;q)>ϵ}\{\Delta_{\mathrm{aug}}(G,G^{\prime};q)>\epsilon\}, so the probability upper bound follows immediately. The expectation conclusion follows by taking expectations on both sides of the deterministic inequality. ∎

E.6Local Influence Cone

Proposition 8(Influence cone of local graph updates).

Suppose the writer only changes nodes, edges, anchors, or attributes on a primitive set𝒰\mathcal{U}. Suppose that the structural gate inputzu​v(l)z_{uv}^{(l)}, except for the graph-level summary, only depends on a local neighborhood of radiusrzr_{z}, and thatGGandG′G^{\prime}are exactly the same outside𝒩L+rz​(𝒰)\mathcal{N}_{L+r_{z}}(\mathcal{U}). If graph-level summary drift is ignored, then for any

v∉𝒩L+rz​(𝒰),v\notin\mathcal{N}_{L+r_{z}}(\mathcal{U}),(153)we have

hv(L)​(q,G)=hv(L)​(q,G′).h_{v}^{(L)}(q,G)=h_{v}^{(L)}(q,G^{\prime}).(154)If the graph-level summary drift isρg=‖rG−rG′‖2\rho_{g}=\left\lVert r_{G}-r_{G^{\prime}}\right\rVert_{2}, then there exists a constantCgC_{g}such that

‖hv(L)​(q,G)−hv(L)​(q,G′)‖2≤Cg​ρg.\left\lVert h_{v}^{(L)}(q,G)-h_{v}^{(L)}(q,G^{\prime})\right\rVert_{2}\leq C_{g}\rho_{g}.(155)

Proof.

First consider the case without graph-level summary drift. We induct on the layer indexll. Forl=0l=0, ifv∉𝒩L+rz​(𝒰)v\notin\mathcal{N}_{L+r_{z}}(\mathcal{U}), then its node features, presence bit, and seed score are all identical, sohv(0)​(G)=hv(0)​(G′)h_{v}^{(0)}(G)=h_{v}^{(0)}(G^{\prime}). Suppose that at layerl−1l-1, all nodes whose distance from𝒰\mathcal{U}exceedsL+rz−(l−1)L+r_{z}-(l-1)have identical representations. Ifv∉𝒩L+rz−l​(𝒰)v\notin\mathcal{N}_{L+r_{z}-l}(\mathcal{U}), then all its one-hop neighborsuudo not belong to𝒩L+rz−(l−1)​(𝒰)\mathcal{N}_{L+r_{z}-(l-1)}(\mathcal{U}); by the induction hypothesis,hu(l−1)​(G)=hu(l−1)​(G′)h_{u}^{(l-1)}(G)=h_{u}^{(l-1)}(G^{\prime}). Meanwhile, the radius-rzr_{z}local structural contexts of all relevant edges are also identical, so the gate, message multiset, and aggregation result are identical, and hencehv(l)​(G)=hv(l)​(G′)h_{v}^{(l)}(G)=h_{v}^{(l)}(G^{\prime}). Takingl=Ll=Lgives the first conclusion. If graph-level summary drift exists, then the gate input has an additional uniform perturbation termρg\rho_{g}, and aCg​ρgC_{g}\rho_{g}-type upper bound follows from Lemma 4.10 and the recursion in Theorem 4.12. ∎

Appendix FTheoretical Motivation of the Self-evolving Writer–Reader Loop

F.1Joint Memory Utility

The reader-aware writer reward can consist of evidence coverage, precision, deducibility, and answer utility. Abstractly, define the joint memory utility as

𝒥​(θ,ϕ)=𝔼(q,D,D+,y)​[U​(Rϕ​(q,Wθ​(q,D),D),D+,y)],\mathcal{J}(\theta,\phi)=\mathbb{E}_{(q,D,D^{+},y)}\left[U\left(R_{\phi}(q,W_{\theta}(q,D),D),D^{+},y\right)\right],(156)whereUUcan be taken as

U=α​rrec+β​rpre+γ​rdedα+β+γ−λrep​ρrep+λfmt​rfmt,U=\frac{\alpha r_{\mathrm{rec}}+\beta r_{\mathrm{pre}}+\gamma r_{\mathrm{ded}}}{\alpha+\beta+\gamma}-\lambda_{\mathrm{rep}}\rho_{\mathrm{rep}}+\lambda_{\mathrm{fmt}}r_{\mathrm{fmt}},(157)or an extended form including answer-level reward. This definition places the writer’s graph construction quality and the reader’s graph reading ability under the same objective.

F.2Approximate Coordinate Improvement

Theorem F.1(The self-evolution process is approximate coordinate improvement on joint utility).

Suppose that the writer update at roundrrsatisfies

𝒥​(θ(r+1),ϕ(r))≥𝒥​(θ(r),ϕ(r))+ΔW(r)−ϵW(r),\mathcal{J}(\theta^{(r+1)},\phi^{(r)})\geq\mathcal{J}(\theta^{(r)},\phi^{(r)})+\Delta_{W}^{(r)}-\epsilon_{W}^{(r)},(158)and the reader update satisfies

𝒥​(θ(r+1),ϕ(r+1))≥𝒥​(θ(r+1),ϕ(r))+ΔR(r)−ϵR(r).\mathcal{J}(\theta^{(r+1)},\phi^{(r+1)})\geq\mathcal{J}(\theta^{(r+1)},\phi^{(r)})+\Delta_{R}^{(r)}-\epsilon_{R}^{(r)}.(159)Then one full round of writer–reader self-evolution satisfies

𝒥​(θ(r+1),ϕ(r+1))−𝒥​(θ(r),ϕ(r))≥ΔW(r)+ΔR(r)−ϵW(r)−ϵR(r).\mathcal{J}(\theta^{(r+1)},\phi^{(r+1)})-\mathcal{J}(\theta^{(r)},\phi^{(r)})\geq\Delta_{W}^{(r)}+\Delta_{R}^{(r)}-\epsilon_{W}^{(r)}-\epsilon_{R}^{(r)}.(160)Therefore, as long asΔW(r)+ΔR(r)>ϵW(r)+ϵR(r)\Delta_{W}^{(r)}+\Delta_{R}^{(r)}>\epsilon_{W}^{(r)}+\epsilon_{R}^{(r)}, the joint memory utility improves in that round.

Proof.

By telescoping decomposition,

𝒥​(θ(r+1),ϕ(r+1))−𝒥​(θ(r),ϕ(r))\displaystyle\mathcal{J}(\theta^{(r+1)},\phi^{(r+1)})-\mathcal{J}(\theta^{(r)},\phi^{(r)})=\displaystyle={}[𝒥​(θ(r+1),ϕ(r+1))−𝒥​(θ(r+1),ϕ(r))]+[𝒥​(θ(r+1),ϕ(r))−𝒥​(θ(r),ϕ(r))].\displaystyle\left[\mathcal{J}(\theta^{(r+1)},\phi^{(r+1)})-\mathcal{J}(\theta^{(r+1)},\phi^{(r)})\right]+\left[\mathcal{J}(\theta^{(r+1)},\phi^{(r)})-\mathcal{J}(\theta^{(r)},\phi^{(r)})\right].(161)Substituting Eq. (158) and Eq. (159), respectively, gives the conclusion. ∎

F.3Reader Reward Bias and Calibration Benefit

Definition F.2(True utility and reader surrogate reward).

LetU⋆​(G)U^{\star}(G)denote the true utility of graph memoryGGwith respect to the downstream task, and letU^ϕ​(G)\widehat{U}_{\phi}(G)denote the surrogate reward constructed from the readout result of readerRϕR_{\phi}. We say that the reader reward bias is at mostϵϕ\epsilon_{\phi}if, for all considered graphsGG,

|U^ϕ​(G)−U⋆​(G)|≤ϵϕ.\left\lvert\widehat{U}_{\phi}(G)-U^{\star}(G)\right\rvert\leq\epsilon_{\phi}.(162)

Theorem F.3(Surrogate reward improvement to true utility improvement).

If the reader reward bias is at mostϵϕ\epsilon_{\phi}, and the writer update improves the surrogate reward by

U^ϕ​(Gθ′)−U^ϕ​(Gθ)≥Δ,\widehat{U}_{\phi}(G_{\theta^{\prime}})-\widehat{U}_{\phi}(G_{\theta})\geq\Delta,(163)then the true utility satisfies

U⋆​(Gθ′)−U⋆​(Gθ)≥Δ−2​ϵϕ.U^{\star}(G_{\theta^{\prime}})-U^{\star}(G_{\theta})\geq\Delta-2\epsilon_{\phi}.(164)

Proof.

By the bias assumption,

U⋆​(Gθ′)≥U^ϕ​(Gθ′)−ϵϕ,U⋆​(Gθ)≤U^ϕ​(Gθ)+ϵϕ.U^{\star}(G_{\theta^{\prime}})\geq\widehat{U}_{\phi}(G_{\theta^{\prime}})-\epsilon_{\phi},\qquad U^{\star}(G_{\theta})\leq\widehat{U}_{\phi}(G_{\theta})+\epsilon_{\phi}.(165)Subtracting the two inequalities gives

U⋆​(Gθ′)−U⋆​(Gθ)≥U^ϕ​(Gθ′)−U^ϕ​(Gθ)−2​ϵϕ≥Δ−2​ϵϕ.U^{\star}(G_{\theta^{\prime}})-U^{\star}(G_{\theta})\geq\widehat{U}_{\phi}(G_{\theta^{\prime}})-\widehat{U}_{\phi}(G_{\theta})-2\epsilon_{\phi}\geq\Delta-2\epsilon_{\phi}.(166)∎

Corollary F.4(Reader calibration reduces writer optimization bias).

If the reader is calibrated fromϕ\phitoϕ′\phi^{\prime}and reduces the reward bias fromϵϕ\epsilon_{\phi}toϵϕ′\epsilon_{\phi^{\prime}}, whereϵϕ′<ϵϕ\epsilon_{\phi^{\prime}}<\epsilon_{\phi}, then for the same surrogate reward improvementΔ\Delta, the lower bound on true utility improvement increases by

2​(ϵϕ−ϵϕ′).2(\epsilon_{\phi}-\epsilon_{\phi^{\prime}}).(167)

Proof.

By Theorem 5.3, the true utility improvement lower bound before calibration isΔ−2​ϵϕ\Delta-2\epsilon_{\phi}, and after calibration it isΔ−2​ϵϕ′\Delta-2\epsilon_{\phi^{\prime}}. Subtracting the two gives the result. ∎

F.4Irreducible Bottlenecks of Single-sided Updates

Proposition 9(Lower-bound bottlenecks of single-sided updates).

Assume that the overall error can be decomposed as

ℰ​(θ,ϕ)=ℰwrite​(θ)+ℰread​(ϕ;θ)+ϵint​(θ,ϕ),\mathcal{E}(\theta,\phi)=\mathcal{E}_{\mathrm{write}}(\theta)+\mathcal{E}_{\mathrm{read}}(\phi;\theta)+\epsilon_{\mathrm{int}}(\theta,\phi),(168)where all terms are nonnegative. If only the reader is updated, i.e.,ϕ↦ϕ′\phi\mapsto\phi^{\prime}whileθ\thetais fixed, then

ℰ​(θ,ϕ′)≥ℰwrite​(θ).\mathcal{E}(\theta,\phi^{\prime})\geq\mathcal{E}_{\mathrm{write}}(\theta).(169)If only the writer is updated, i.e.,θ↦θ′\theta\mapsto\theta^{\prime}whileϕ\phiis fixed, then

ℰ​(θ′,ϕ)≥ℰread​(ϕ;θ′).\mathcal{E}(\theta^{\prime},\phi)\geq\mathcal{E}_{\mathrm{read}}(\phi;\theta^{\prime}).(170)Therefore, reader-only updates cannot compensate for evidence chains that the writer has not written; writer-only updates cannot guarantee that a fixed reader can read out the evidence structures in the new graph distribution.

Proof.

By the decomposition in Eq. (168) and the nonnegativity of all terms,

ℰ​(θ,ϕ′)=ℰwrite​(θ)+ℰread​(ϕ′;θ)+ϵint​(θ,ϕ′)≥ℰwrite​(θ).\mathcal{E}(\theta,\phi^{\prime})=\mathcal{E}_{\mathrm{write}}(\theta)+\mathcal{E}_{\mathrm{read}}(\phi^{\prime};\theta)+\epsilon_{\mathrm{int}}(\theta,\phi^{\prime})\geq\mathcal{E}_{\mathrm{write}}(\theta).(171)The second inequality is analogous. ∎

F.5Stability of Closed-loop Graph Evolution and Parameter Updates

Theorem F.5(Score drift control under multi-round self-evolution).

Let the graph at roundrrbeG(r)G^{(r)}, and the reader parameter beϕ(r)\phi^{(r)}. If single-step graph stability satisfies

‖sD​(q,G(r+1);ϕ(r))−sD​(q,G(r);ϕ(r))‖∞≤CD​Δr,\left\lVert s_{D}(q,G^{(r+1)};\phi^{(r)})-s_{D}(q,G^{(r)};\phi^{(r)})\right\rVert_{\infty}\leq C_{D}\Delta_{r},(172)whereΔr=Δaug​(G(r),G(r+1);q)\Delta_{r}=\Delta_{\mathrm{aug}}(G^{(r)},G^{(r+1)};q); and if the score is locally Lipschitz with respect to the parameter:

‖sD​(q,G;ϕ)−sD​(q,G;ϕ′)‖∞≤Cϕ​‖ϕ−ϕ′‖2,\left\lVert s_{D}(q,G;\phi)-s_{D}(q,G;\phi^{\prime})\right\rVert_{\infty}\leq C_{\phi}\left\lVert\phi-\phi^{\prime}\right\rVert_{2},(173)then

‖sD​(q,G(T);ϕ(T))−sD​(q,G(0);ϕ(0))‖∞\displaystyle\left\lVert s_{D}(q,G^{(T)};\phi^{(T)})-s_{D}(q,G^{(0)};\phi^{(0)})\right\rVert_{\infty}(174)≤∑r=0T−1(CD​Δr+Cϕ​‖ϕ(r+1)−ϕ(r)‖2).\displaystyle\qquad\leq\sum_{r=0}^{T-1}\left(C_{D}\Delta_{r}+C_{\phi}\left\lVert\phi^{(r+1)}-\phi^{(r)}\right\rVert_{2}\right).

Proof.

For eachrr, adding and subtracting the intermediate termsD​(q,G(r+1);ϕ(r))s_{D}(q,G^{(r+1)};\phi^{(r)})gives

‖sD​(q,G(r+1);ϕ(r+1))−sD​(q,G(r);ϕ(r))‖∞\displaystyle\left\lVert s_{D}(q,G^{(r+1)};\phi^{(r+1)})-s_{D}(q,G^{(r)};\phi^{(r)})\right\rVert_{\infty}≤‖sD​(q,G(r+1);ϕ(r+1))−sD​(q,G(r+1);ϕ(r))‖∞+‖sD​(q,G(r+1);ϕ(r))−sD​(q,G(r);ϕ(r))‖∞\displaystyle\quad\leq\left\lVert s_{D}(q,G^{(r+1)};\phi^{(r+1)})-s_{D}(q,G^{(r+1)};\phi^{(r)})\right\rVert_{\infty}+\left\lVert s_{D}(q,G^{(r+1)};\phi^{(r)})-s_{D}(q,G^{(r)};\phi^{(r)})\right\rVert_{\infty}≤Cϕ​‖ϕ(r+1)−ϕ(r)‖2+CD​Δr.\displaystyle\quad\leq C_{\phi}\left\lVert\phi^{(r+1)}-\phi^{(r)}\right\rVert_{2}+C_{D}\Delta_{r}.(175)Summing overr=0,…,T−1r=0,\dots,T-1and using the triangle inequality gives the conclusion. ∎

Corollary F.6(High-probability multi-round stability).

Ifℙ​[Δr>ϵr]≤δr\mathbb{P}[\Delta_{r}>\epsilon_{r}]\leq\delta_{r}, then with probability at least1−∑r=0T−1δr1-\sum_{r=0}^{T-1}\delta_{r},

‖sD​(q,G(T);ϕ(T))−sD​(q,G(0);ϕ(0))‖∞\displaystyle\left\lVert s_{D}(q,G^{(T)};\phi^{(T)})-s_{D}(q,G^{(0)};\phi^{(0)})\right\rVert_{\infty}(176)≤∑r=0T−1(CD​ϵr+Cϕ​‖ϕ(r+1)−ϕ(r)‖2).\displaystyle\qquad\leq\sum_{r=0}^{T-1}\left(C_{D}\epsilon_{r}+C_{\phi}\left\lVert\phi^{(r+1)}-\phi^{(r)}\right\rVert_{2}\right).

Proof.

By a union bound, the event{∀r,Δr≤ϵr}\{\forall r,\Delta_{r}\leq\epsilon_{r}\}holds with probability at least1−∑rδr1-\sum_{r}\delta_{r}. Applying Theorem 5.7 on this event gives the conclusion. ∎

Appendix GAnalysis of the Memory Writer

This section further analyzes the memory writer while keeping the memory reader fixed. The experiments mainly use HotpotQA and MuSiQue. To further examine domain transfer capability, we also evaluate the trained writing policy on GRBench-Amazon, HaluMem-Medium, and LongMemEval-Oracle[longmemeval2024,halumem2025]. We primarily report Precision, Recall, and Deducible: the first two measure whether the text retrieved by the reader covers the gold supporting contexts, while Deducible is determined by a judge as to whether the standard answer can be inferred from the retrieved context, thus more directly reflecting whether the graph memory is usable for reasoning.

Table 5:Training results for the memory writer.GFM-pretrained-onlyrefers to using rewards fed back only by the pretrained memory reader, whileGFM-finetunedfurther refers to using the fine-tuned memory reader.MethodsPrec.↑\uparrowRecall↑\uparrowDeducible↑\uparrowGFM-pretrained-only0.8380.8180.510GFM-finetuned0.8240.8130.512RL-Recall0.8890.8350.502RL-F10.8390.8810.497RL-Deduce0.8610.8920.517RL-Hybrid0.9020.9170.522Hybrid + frozen answer API0.8320.8740.526Table5shows that different rewards have different preferences. Overall, RL-Hybrid achieves the best overall results, indicating that hybrid rewards can simultaneously constrain the selectivity and coverage of graph writing.Hybrid + frozen answer APIachieves the highest Deducible but slightly lower retrieval Precision/Recall, suggesting that answer-side feedback helps improve reasoning usability, but may also make the writer more conservatively inclined to write evidence that directly supports the answer. Table6shows that a writer learned on HotpotQA/MuSiQue can transfer to GRBench, HaluMem, and LongMemEval, but continued training on the target domain still brings significant improvements. This indicates that the memory structure in agent memory scenarios is not entirely similar to that in traditional multi-hop QA, so target-domain feedback remains crucial. Table7further shows that the writing protocol and interaction budget affect the trade-off between coverage and noise. Relaxing the tight prompt can improve Recall, but reduces Deducible; increasing iterative turns helps complete cross-document bridging paths, but when the budget is too large or the protocol is too loose. More detailed reader-side sensitivity, training stability, and regularization analyses are provided in AppendixA.

Table 6:Cross-dataset memory writing results. “Base→\rightarrowTarget” indicates direct evaluation on the target domain after training on the HotpotQA/MuSiQue base; “Target train→\rightarrowval” indicates training and validation on the target domain.SettingsPrec.↑\uparrowRecall↑\uparrowDeducible↑\uparrowBase→\rightarrowGRBench0.5750.6090.411GRBench train→\rightarrowval0.7940.8330.596Base→\rightarrowHaluMem0.2300.4480.299HaluMem train→\rightarrowval0.3120.7080.438Base→\rightarrowLongMemEval0.2320.3760.475LongMem train→\rightarrowLongMemEval0.3770.4390.531Table 7:Ablation of writing protocols and interaction budgets. Tight=True indicates that the writer performs a single-round graph write under a stricter evidence budget, meaning the reader exposes only fewer, higher-confidence candidate pieces of evidence to the writer; Tight=False indicates that this evidence budget is relaxed, allowing the writer to access a broader candidate context. Iterative indicates that multi-round interactive writer–reader writing is enabled: the writer first writes the initial graph memory, the reader then returns retrieval feedback based on the current graph, and the writer continues to supplement or revise the graph structure. Here, 12/20/24 turns indicates the maximum number of interaction rounds allowed, and tight/loose indicates that strict or relaxed evidence budget constraints are still used during this multi-round interaction process.SettingsPrec.↑\uparrowRecall↑\uparrowDeducible↑\uparrowTight=True0.8360.8060.515Tight=False0.8450.8510.506Iterative, 12 turns, tight0.8520.8290.516Iterative, 20 turns, tight0.8350.8810.522Iterative, 24 turns, loose0.8630.8260.531

Appendix HAblation Study of the Memory Reader

We conduct ablation studies to isolate the contribution of each major component in the memory reader. All variants use the same writer-produced graph memory and the same retrieval budget unless the ablated component directly changes the retrieval mechanism. The ablations are organized around four questions: (1) whether structured query planning and global soft addressing are necessary for recovering evidence from fragmented cues; (2) whether structurally gated propagation improves over uniform graph propagation; (3) whether cross-graph structural priors and target-graph calibration are both needed for evolving graph memory; and (4) whether reader training and entity-to-document projection are important for converting entity-level activation into document-level retrieval.

Table8summarizes the results. The first group evaluates how the reader handles fragmented cues. Removing structured query planning, alias/constraint cues, or global soft addressing forces the reader to rely more heavily on surface-level query matches or a small number of anchor entities, directly testing whether the complete evidence chain can still be recovered when distant bridge nodes are not initially activated. The second group studies whether the reader uses graph structure in a learned and selective way. Removing structural gates or replacing them with uniform message passing tests whether treating hub edges, bridge edges, redundant edges, and noisy shortcuts similarly harms retrieval. The third group examines the context–schema decomposition: the schema channel captures transferable structural reading patterns, while the context channel adapts the reader to the current writer-produced graph. The last group evaluates the selector, entity-to-document projection, GFM pre-training, and supervised retrieval fine-tuning.

Table 8:Ablation study of the memory reader on multi-hop QA retrieval. We report document-level Recall (%) at top-2 and top-5. All variants use the same writer-produced graph memory unless otherwise specified.Reader VariantHotpotQAMuSiQue2WikiMultiHopQAR@2R@5R@2R@5R@[email protected] planning and global addressingSAGEw/o Structured Query Planning62.775.140.450.180.786.6SAGEw/o Global Soft Addressing59.372.537.647.475.983.1SAGEw/o Alias and Constraint Cues63.075.841.050.880.886.9SAGEw/ Anchor-only Initialization58.671.436.846.574.282.4Structurally conditioned propagationSAGEw/o Structural Gate60.473.239.248.778.184.9SAGEw/o Node Structural Features62.174.840.550.080.086.0SAGEw/o Edge-pair Structural Features61.574.040.149.479.185.6SAGEw/o Graph-level Summary63.275.941.651.081.387.0SAGEw/ Uniform Message Passing58.971.837.546.975.382.9Cross-graph priors and target-graph calibrationSAGEw/o Schema Prior Channel62.475.040.950.680.486.4SAGEw/o Context Calibration Channel61.874.340.049.879.585.9SAGEw/o Context–Schema Fusion60.773.539.148.677.884.7Selector, projection, and reader trainingSAGEw/o Controlled Entity-to-Document Projection60.973.938.748.277.284.4SAGEw/o Query-conditioned Selector63.976.541.952.082.287.6SAGEw/ Vanilla GNN Reader57.270.636.345.272.880.7

The ablation results show the relative contribution of each memory-reader component. First, structured query planning and global soft addressing are important for fragmented-cue retrieval: removing them noticeably weakens performance, especially on MuSiQue and 2WikiMultiHopQA, where evidence chains are more likely to depend on implicit bridge entities. Second, structurally gated propagation consistently improves over uniform message passing, indicating that graph structure should be used selectively rather than as a fixed expansion rule. Among the structural inputs, graph-level summaries have a relatively smaller effect, while node-level and edge-pair features are more important for recognizing hubs, bridges, and cross-community evidence paths. Third, both the schema prior and context calibration channels contribute to performance, suggesting that the reader benefits from preserving transferable structural priors while adapting to the current writer-produced graph. Finally, supervised retrieval fine-tuning is essential for aligning the GFM reader with document-level evidence retrieval, whereas GFM pre-training provides transferable structural initialization that improves stability across datasets.

Appendix IImplementation Details of the Query-conditioned Subgraph Selection Regularizer

We provide the implementation details of the query-conditioned subgraph selector. In addition to the base entity scoring, it further learns a soft gating probabilityπe​(q)\pi_{e}(q), which characterizes whether entityeeshould enter the reading subgraph of the current queryqq. This module performs lightweight reweighting of the final entity score, and constrains the reading subgraph through several structural regularizers during training.

Query-conditioned Selection Probability.

Given a queryqqand a graph𝒢=(𝒱,ℰ)\mathcal{G}=(\mathcal{V},\mathcal{E}), let𝐡e∈ℝd\mathbf{h}_{e}\in\mathbb{R}^{d}denote the representation of entity nodee∈𝒱e\in\mathcal{V}after propagation by the GFM backbone, and let𝐳q∈ℝd\mathbf{z}_{q}\in\mathbb{R}^{d}denote the query representation output by the query encoder and then linearly projected. Here,ddis the hidden dimension. The selector first projects the node representation and the query representation into the same selector space:

𝐮e=Wn​𝐡e,𝐯q=Ws​𝐳q,\mathbf{u}_{e}=W_{n}\mathbf{h}_{e},\qquad\mathbf{v}_{q}=W_{s}\mathbf{z}_{q},(177)whereWn∈ℝds×dW_{n}\in\mathbb{R}^{d_{s}\times d}is the node-side projection matrix,Ws∈ℝds×dW_{s}\in\mathbb{R}^{d_{s}\times d}is the query-side projection matrix, anddsd_{s}is the hidden dimension of the selector space. In our implementation, we setds=dd_{s}=d, but the two do not have to be equal. Then, the selector obtains the selection logit of entityeewith respect to queryqqthrough a scaled inner product:

ζe​(q)=𝐮e⊤​𝐯qTs,\zeta_{e}(q)=\frac{\mathbf{u}_{e}^{\top}\mathbf{v}_{q}}{T_{s}},(178)whereTs>0T_{s}>0is the selector temperature coefficient. The final soft selection probability is defined as

πe​(q)=sigmoid⁡(ζe​(q))=11+exp⁡(−ζe​(q)).\pi_{e}(q)=\operatorname{sigmoid}(\zeta_{e}(q))=\frac{1}{1+\exp(-\zeta_{e}(q))}.(179)Here,πe​(q)∈(0,1)\pi_{e}(q)\in(0,1)can be understood as the soft probability that entityeeis included in the reading subgraph of the current query. During training, we directly useπe​(q)\pi_{e}(q)for differentiable optimization; during inference, we can either continue to use the soft probability for reweighting, or obtain a discrete subgraph according to a thresholdτπ\tau_{\pi}:

𝒱q={e∈𝒱∣πe​(q)>τπ},ℰq={(u,v)∈ℰ∣u,v∈𝒱q}.\mathcal{V}_{q}=\{e\in\mathcal{V}\mid\pi_{e}(q)>\tau_{\pi}\},\qquad\mathcal{E}_{q}=\{(u,v)\in\mathcal{E}\mid u,v\in\mathcal{V}_{q}\}.(180) Letae​(q)a_{e}(q)denote the base entity score given by the GFM backbone reader. This score is usually obtained from the similarity between the node representation and the query representation, for example

ae​(q)=𝐡e⊤​𝐳q.a_{e}(q)=\mathbf{h}_{e}^{\top}\mathbf{z}_{q}.(181)Finally, we have:

aefinal​(q)=ae​(q)+λs​ζe​(q),a_{e}^{\mathrm{final}}(q)=a_{e}(q)+\lambda_{s}\zeta_{e}(q),(182)whereλs≥0\lambda_{s}\geq 0controls the influence of the selector logit on the final entity ranking.

Query–Subgraph Contrastive Regularizer.

Using only Eq. (182) to fuse the selector score can easily lead to two types of degeneration: first, the selector may assign high probabilities to most nodes, thereby degenerating into full-graph activation; second, the selector may only learn local high-frequency entities, without forming a subgraph representation that is consistent with the overall semantics of the query. To this end, we first construct a query-conditioned subgraph representation weighted by the selection probabilities:

𝐡¯π​(q)=∑e∈𝒱πe​(q)​𝐡e∑e∈𝒱πe​(q)+ϵ,\bar{\mathbf{h}}_{\pi}(q)=\frac{\sum_{e\in\mathcal{V}}\pi_{e}(q)\mathbf{h}_{e}}{\sum_{e\in\mathcal{V}}\pi_{e}(q)+\epsilon},(183)whereϵ>0\epsilon>0is a numerical stability term, which avoids an excessively small denominator when allπe​(q)\pi_{e}(q)are close to0.𝐡¯π​(q)\bar{\mathbf{h}}_{\pi}(q)can be understood as the semantic center of the soft subgraph activated by the current selector.

For a mini-batchℬ={qi}i=1B\mathcal{B}=\{q_{i}\}_{i=1}^{B}, we treat(𝐡¯π​(qi),𝐳qi)(\bar{\mathbf{h}}_{\pi}(q_{i}),\mathbf{z}_{q_{i}})within the same sample as a positive pair, and treat(𝐡¯π​(qi),𝐳qj)(\bar{\mathbf{h}}_{\pi}(q_{i}),\mathbf{z}_{q_{j}}),j≠ij\neq i, as in-batch negative pairs. The query–subgraph contrastive loss is defined as

Ωnce=−1B​∑i=1Blog⁡exp⁡(sim​(𝐡¯π​(qi),𝐳qi)/Tn)∑j=1Bexp⁡(sim​(𝐡¯π​(qi),𝐳qj)/Tn),\Omega_{\mathrm{nce}}=-\frac{1}{B}\sum_{i=1}^{B}\log\frac{\exp\left(\mathrm{sim}(\bar{\mathbf{h}}_{\pi}(q_{i}),\mathbf{z}_{q_{i}})/T_{n}\right)}{\sum_{j=1}^{B}\exp\left(\mathrm{sim}(\bar{\mathbf{h}}_{\pi}(q_{i}),\mathbf{z}_{q_{j}})/T_{n}\right)},(184)whereTn>0T_{n}>0is the contrastive learning temperature coefficient, andsim​(⋅,⋅)\mathrm{sim}(\cdot,\cdot)is the similarity function. In implementation, we usually applyℓ2\ell_{2}normalization to𝐡¯π​(q)\bar{\mathbf{h}}_{\pi}(q)and𝐳q\mathbf{z}_{q}, and use inner-product similarity, so thatsim​(𝐡¯π​(q),𝐳q)=𝐡¯π​(q)⊤​𝐳q‖𝐡¯π​(q)‖2​‖𝐳q‖2\mathrm{sim}(\bar{\mathbf{h}}_{\pi}(q),\mathbf{z}_{q})=\frac{\bar{\mathbf{h}}_{\pi}(q)^{\top}\mathbf{z}_{q}}{\|\bar{\mathbf{h}}_{\pi}(q)\|_{2}\|\mathbf{z}_{q}\|_{2}}. This term encourages the soft subgraph activated by the selector to semantically represent the current query, rather than only selecting nodes with high frequency or high centrality in the graph.

Size Regularizer.

To prevent the selector from improving recall by activating a large number of nodes, we use the average selection probability as a size penalty:

Ωsize=1|𝒱|​∑e∈𝒱πe​(q).\Omega_{\mathrm{size}}=\frac{1}{|\mathcal{V}|}\sum_{e\in\mathcal{V}}\pi_{e}(q).(185)This term approximately represents the expected proportion of activated nodes. MinimizingΩsize\Omega_{\mathrm{size}}pushes the model to select a smaller reading subgraph. However, this term cannot be used alone; otherwise, the selector may degenerate into selecting too few nodes or even no nodes. Therefore, it needs to be jointly optimized withΩnce\Omega_{\mathrm{nce}}and the main retrieval loss: the former ensures query relevance, while the latter ensures that the selected structure can still support correct entity and document recall.

Connectivity Smoothing Regularizer.

In addition to controllable size, an effective reading subgraph should also have local structural coherence. If the selection probabilities of adjacent nodes differ too much, the model may form several isolated activated points, making multi-hop paths difficult to explicitly utilize. To this end, we use a smoothing penalty on edges:

Ωcon=1|ℰ|​∑(u,v)∈ℰ(πu​(q)−πv​(q))2.\Omega_{\mathrm{con}}=\frac{1}{|\mathcal{E}|}\sum_{(u,v)\in\mathcal{E}}\left(\pi_{u}(q)-\pi_{v}(q)\right)^{2}.(186)Here,(u,v)(u,v)is a directed or undirected edge in the graph, depending on whether edge directions are preserved during graph construction. If an undirected graph is used,ℰ\mathcal{E}can be viewed as the symmetrized edge set. This term does not force all selected nodes to be strictly connected, but encourages adjacent nodes to have similar selection probabilities. In matrix form, if𝝅​(q)∈ℝ|𝒱|\bm{\pi}(q)\in\mathbb{R}^{|\mathcal{V}|}is the selection probability vector composed ofπe​(q)\pi_{e}(q), and𝐋\mathbf{L}is the graph Laplacian matrix, then this term is equivalent to Laplacian smoothing:

Ωcon∝𝝅​(q)⊤​𝐋​𝝅​(q).\Omega_{\mathrm{con}}\propto\bm{\pi}(q)^{\top}\mathbf{L}\bm{\pi}(q).(187)Therefore, it is consistent with the classical assumption of graph signal smoothing: query relevance, as a soft signal on the graph, should maintain a certain degree of continuity within local neighborhoods.

Computational Complexity of the Selector Itself.

Let the batch size beBB, the number of nodes ben=|𝒱|n=|\mathcal{V}|, the number of edges bem=|ℰ|m=|\mathcal{E}|, and the hidden dimension bedd. The cost of computingWn​𝐡eW_{n}\mathbf{h}_{e}isO​(B​n​d2)O(Bnd^{2}), the cost of computingWs​𝐳qW_{s}\mathbf{z}_{q}isO​(B​d2)O(Bd^{2}), and the cost of the inner-product logits isO​(B​n​d)O(Bnd). If we cacheWn​𝐡eW_{n}\mathbf{h}_{e}in advance, this term can be reduced toO​(B​n​d)O(Bnd). For the contrastive term, the cost of the subgraph pooling in Eq. (183) isO​(B​n​d)O(Bnd), and the cost of the in-batch NCE similarity matrix isO​(B2​d)O(B^{2}d); the size term has costO​(B​n)O(Bn); the connectivity term needs to traverse edges and has costO​(B​m)O(Bm). Therefore, the additional complexity of the selector during training is

O​(B​n​d2+B​n​d+B2​d+B​m),O\left(Bnd^{2}+Bnd+B^{2}d+Bm\right),(188)and if the quadratic term of the linear projection is ignored or cached, it can be approximated as

O​(B​n​d+B2​d+B​m).O\left(Bnd+B^{2}d+Bm\right).(189)During inference, if only the selector logit is used to fuse entity scores, without computing NCE, size, and connectivity regularizers, then the additional cost is mainlyO​(B​n​d2+B​n​d)O(Bnd^{2}+Bnd), orO​(B​n​d)O(Bnd)under caching/lightweight projection.

Appendix JTraining and Inference Complexity

For ease of exposition, suppose that the graph𝒢=(𝒱,ℰ)\mathcal{G}=(\mathcal{V},\mathcal{E})hasn=|𝒱|n=|\mathcal{V}|entity nodes andm=|ℰ|m=|\mathcal{E}|entity-relation edges. If self-loops are added in GCN propagation, we denotem~=m+n\tilde{m}=m+n. Let the hidden dimension bedd, the number of propagation layers beLL, the batch size beBB, and the number of pseudo-queries beMM. Therefore, one real query together withMMpseudo-queries requiresM+1M+1graph reads. Let𝐌∈{0,1}n×ND\mathbf{M}\in\{0,1\}^{n\times N_{D}}denote the sparse entity–document association matrix, whereNDN_{D}is the number of documents, andnnz⁡(𝐌)\operatorname{nnz}(\mathbf{M})is the number of nonzero entity–document links. LetKeK_{e}denote the number of top entities used for document projection, and letf¯\bar{f}denote the average number of documents linked to the top entities.

J.1Offline Structural Feature and Indexing Cost

Let the dimension of node structural features bepnp_{n}, the dimension of edge structural features bepep_{e}, and the dimension of graph-level summaries bepgp_{g}. In the current implementation,pnp_{n},pep_{e}, andpgp_{g}are all small constants.

Given that the adjacency list has been constructed, degrees and average neighbor degrees can be computed inO​(n+m)O(n+m)time. Clustering coefficients and the number of common neighbors require computing intersections of neighbor sets, whose complexity can be written as

O​(n+m+∑(u,v)∈ℰmin⁡{deg⁡(u),deg⁡(v)}).O\left(n+m+\sum_{(u,v)\in\mathcal{E}}\min\{\deg(u),\deg(v)\}\right).(190)In sparse graphs or graphs with bounded average degree, Eq. (190) is approximatelyO​(n+m)O(n+m); in extremely dense graphs, the worst case can reachO​(n3)O(n^{3}). These structural features and the entity–document matrix can both be precomputed and cached offline, with space cost

O​(n​pn+m​pe+pg+nnz⁡(𝐌)).O\left(np_{n}+mp_{e}+p_{g}+\operatorname{nnz}(\mathbf{M})\right).(191)Since this part does not depend on a specific query, when evaluating multiple queries on the same candidate graph in the self-evolving memory loop, the same set of structural features and entity–document indices can be reused.

J.2Forward Propagation Complexity of the Structurally Gated GFM

We first consider a single forward propagation for one query on one graph. A standard GCN layer contains two parts: node linear transformation and sparse adjacency aggregation. The cost of node linear transformation isO​(n​d2)O(nd^{2}), and the cost of edge-level message aggregation isO​(m~​d)O(\tilde{m}d). Therefore, the complexity of a standard GCN layer is

Cplain=O​(n​d2+m~​d).C_{\mathrm{plain}}=O\left(nd^{2}+\tilde{m}d\right).(192) Beyond ordinary message propagation, a structurally gated layer generates a vector gate for each edge. Its message form is

𝐦u→v=≫u​v⊙W𝐡u,\mathbf{m}_{u\rightarrow v}=\gg_{uv}\odot W\mathbf{h}_{u},(193)where𝐡u\mathbf{h}_{u}is the source node representation,W∈ℝd×dW\in\mathbb{R}^{d\times d}is the node linear transformation matrix,≫u​v∈ℝd\gg_{uv}\in\mathbb{R}^{d}is the structural gate vector of edge(u,v)(u,v), and⊙\odotdenotes element-wise multiplication. Letdgd_{g}denote the encoding dimension of structural features, and lethgh_{g}denote the hidden dimension of the gating MLP. If the gate uses four types of inputs, namely source-node structure, target-node structure, edge-pair structure, and graph-level summary, then the gate generation cost can be written as

Cgate=O(\displaystyle C_{\mathrm{gate}}=O\Big(n​pn​dg+m​pe​dg+pg​dg\displaystyle np_{n}d_{g}+mp_{e}d_{g}+p_{g}d_{g}(194)+m(4dghg+hgd)).\displaystyle+m(4d_{g}h_{g}+h_{g}d)\Big).Here,n​pn​dgnp_{n}d_{g}comes from node structural feature encoding,m​pe​dgmp_{e}d_{g}comes from edge structural feature encoding,pg​dgp_{g}d_{g}comes from graph-level summary encoding, andm​(4​dg​hg+hg​d)m(4d_{g}h_{g}+h_{g}d)comes from the per-edge gating MLP. If edge-pair features or graph-level summaries are disabled, the corresponding terms in Eq. (194) can be removed. Ifdgd_{g}andhgh_{g}are regarded as being of the same order asdd, then gate generation isO​(m​d2)O(md^{2})in the worst case; if the gating MLP is regarded as a small constant-width module, or if low-rank/dimension-wise gating is adopted, it can be approximated asO​(m​d)O(md). Therefore, the complexity of a structurally gated layer is

Cgated=O​(n​d2+m~​d+Cgate).C_{\mathrm{gated}}=O\left(nd^{2}+\tilde{m}d+C_{\mathrm{gate}}\right).(195) The current implementation supports dual structural prompts: one is a holistic gated branch, and the other is a specific prompt branch. If only a standard GCN is used, the per-layer cost isCplainC_{\mathrm{plain}}; if only a structurally gated GCN is used, the per-layer cost isCgatedC_{\mathrm{gated}}; if one gated branch and one standard branch are used simultaneously, the per-layer cost is approximatelyCgated+CplainC_{\mathrm{gated}}+C_{\mathrm{plain}}. Letρplain∈{0,1}\rho_{\mathrm{plain}}\in\{0,1\}denote whether the standard prompt branch is enabled, and letρgated∈{0,1}\rho_{\mathrm{gated}}\in\{0,1\}denote whether the structurally gated branch is enabled. Then the GFM encoding cost for one batch can be uniformly written as

Cenc​(B)=O​(B​L​(ρplain​Cplain+ρgated​Cgated)).C_{\mathrm{enc}}(B)=O\left(BL\left(\rho_{\mathrm{plain}}C_{\mathrm{plain}}+\rho_{\mathrm{gated}}C_{\mathrm{gated}}\right)\right).(196)The factorBBappears because, in the current implementation, each query in a batch separately constructs query-conditioned node inputs and performs graph encoding. If query-independent structural gates for a fixed graph are cached during inference, part of the gate cost can be reduced; however, based on the current code implementation, Eq. (196) is a more conservative upper bound.

In the most common simplified analysis, we setB=1B=1,M=0M=0, disable dual branches, and regard the gating MLP as a lightweight constant-width module. Then Eq. (196) degenerates to

O​(L​(m​d+d2​n)),O\left(L(md+d^{2}n)\right),(197)which is exactly the core propagation complexity given in the main text. Here,m​dmdcorresponds to edge-level messages, structural gating, and sparse aggregation, whiled2​nd^{2}ncorresponds to node linear projection.

J.3Entity Scoring, Selector Regularization, and Document Projection Complexity

After GFM encoding obtains node representations, entity scoring is usually obtained by

ae​(q)=𝐡e⊤​𝐳qa_{e}(q)=\mathbf{h}_{e}^{\top}\mathbf{z}_{q}(198)For one batch, the complexity of this step is

Cscore​(B)=O​(B​n​d).C_{\mathrm{score}}(B)=O(Bnd).(199)If the query-conditioned subgraph selector in AppendixIis enabled, then the additional inference-stage cost is

Csel,infer​(B)=O​(B​n​d2+B​n​d),C_{\mathrm{sel,infer}}(B)=O(Bnd^{2}+Bnd),(200)which can be approximated asO​(B​n​d)O(Bnd)if the node-side projection is cached or a lightweight projection is used. During training, NCE, the size term, and the connectivity term also need to be computed, with additional complexity

Csel,train​(B)=O​(B​n​d2+B​n​d+B2​d+B​m).C_{\mathrm{sel,train}}(B)=O(Bnd^{2}+Bnd+B^{2}d+Bm).(201)Here,B2​dB^{2}dcomes from the in-batch query–subgraph contrastive matrix, andB​mBmcomes from the edge-level connectivity smoothing term.

Entity-to-document projection is performed by the entity–document matrix𝐌\mathbf{M}. If full sparse matrix multiplication is used, the complexity is

Cdoc,full​(B)=O​(B​nnz⁡(𝐌)).C_{\mathrm{doc,full}}(B)=O\left(B\operatorname{nnz}(\mathbf{M})\right).(202)TheIDFWeightedRankerin the current code belongs to this type: it first constructs IDF weights according to entity occurrence frequency, and then performs sparse matrix multiplication. If top-KeK_{e}entity projection is used, conceptually only the inverted lists corresponding to these entities need to be accessed, so the complexity can be written as

Cdoc,top​K​(B)=O​(B​n​log⁡Ke+B​Ke​f¯),C_{\mathrm{doc,top}K}(B)=O\left(Bn\log K_{e}+BK_{e}\bar{f}\right),(203)whereB​n​log⁡KeBn\log K_{e}comes from top-KeK_{e}entity selection, andB​Ke​f¯BK_{e}\bar{f}comes from accessing the documents linked on average by the top entities. If the final document top-KKranking is performed over allNDN_{D}documents, the complexity isO​(B​ND​log⁡K)O(BN_{D}\log K); if it is performed only over the candidate document pool, it isO​(B​Ncand​log⁡K)O(BN_{\mathrm{cand}}\log K), whereNcand≪NDN_{\mathrm{cand}}\ll N_{D}.

J.4Training Complexity

The main costs in the training stage come from GFM forward propagation, the entity-level retrieval loss, the optional selector regularizer, and backpropagation. Letκbw\kappa_{\mathrm{bw}}denote the constant-factor cost of backpropagation relative to forward propagation, which can usually be regarded as a constant between22and33. If the entity-level training loss is BCE, ranking loss, or ListCE, then because the predicted scores ofnnentities need to be supervised or ranked, the loss computation complexity is

Closs​(B)=O​(B​n).C_{\mathrm{loss}}(B)=O(Bn).(204)Therefore, when the selector is not enabled, the complexity of a single training batch is

Ctrain=O​(κbw​[Cenc​(B)+Cscore​(B)+Closs​(B)]).C_{\mathrm{train}}=O\left(\kappa_{\mathrm{bw}}\left[C_{\mathrm{enc}}(B)+C_{\mathrm{score}}(B)+C_{\mathrm{loss}}(B)\right]\right).(205)After enabling the query-conditioned subgraph selector, the training complexity becomes

Ctrainsel=O​(κbw​[Cenc​(B)+Cscore​(B)+Closs​(B)+Csel,train​(B)]).C_{\mathrm{train}}^{\mathrm{sel}}=O\left(\kappa_{\mathrm{bw}}\left[C_{\mathrm{enc}}(B)+C_{\mathrm{score}}(B)+C_{\mathrm{loss}}(B)+C_{\mathrm{sel,train}}(B)\right]\right).(206)If a document-level loss is also explicitly added during training, then entity-to-document projection needs to be additionally performed, with costCdoc,full​(B)C_{\mathrm{doc,full}}(B)orCdoc,top​K​(B)C_{\mathrm{doc,top}K}(B).

J.5Inference Complexity

The inference stage first performs query encoding, named entity recognition, and entity linking, whose total cost is denoted asCprep​(q)C_{\mathrm{prep}}(q). This part depends on the adopted text encoder, NER model, and entity linking model, and does not belong to the graph propagation backbone. Given the prepared query embedding and query entity mask, the core reading complexity of a single query is

Cinfer​(q)=\displaystyle C_{\mathrm{infer}}(q)=(M+1)​[Cenc​(1)+Cscore​(1)+Csel,infer​(1)+Cdoc​(1)]\displaystyle(M+1)\Big[C_{\mathrm{enc}}(1)+C_{\mathrm{score}}(1)+C_{\mathrm{sel,infer}}(1)+C_{\mathrm{doc}}(1)\Big](207)+Cfuse​(M,K),\displaystyle+C_{\mathrm{fuse}}(M,K),whereMMis the number of pseudo-queries,Cdoc​(1)C_{\mathrm{doc}}(1)can take the complexity of full sparse projection or top-KeK_{e}inverted projection, andCfuse​(M,K)C_{\mathrm{fuse}}(M,K)is the cost of fusing the results from the main query and pseudo-queries. If each query keepsKKcandidate documents, then the cost of simple weighted merging is

Cfuse​(M,K)=O​((M+1)​K​log⁡((M+1)​K)),C_{\mathrm{fuse}}(M,K)=O((M+1)K\log((M+1)K)),(208)

J.6Space Complexity

The model parameter space mainly comes from the GFM backbone, structural prompts, the structurally gated MLP, selector projections, and text projection layers. If we only discuss graph- related runtime space, offline graph storage requires

O​(n+m+nnz⁡(𝐌)+n​pn+m​pe+pg).O\left(n+m+\operatorname{nnz}(\mathbf{M})+np_{n}+mp_{e}+p_{g}\right).(209)During training, node activations of each layer need to be saved, and the space complexity is on the order of

If structurally gated edge messages are fully materialized, they requireO​(m​d)O(md)GPU memory; the current implementation adopts edge chunk streaming. Let the chunk size becc, then the peak memory of gated messages can be reduced to

wherec≪mc\ll m. This is also one of the key engineering designs that makes the current implementation suitable for large-graph reading. Ifreturn_gateis enabled and the gate vectors of all edges are saved for visualization or interpretation, then the space will rise again toO​(m​d)O(md). If document scoring materializes allNDN_{D}document scores, it requiresO​(B​ND)O(BN_{D})space; if only a candidate document heap is maintained, it can be reduced toO​(B​K)O(BK)orO​(B​Ke​f¯)O(BK_{e}\bar{f}).

J.7Complexity Comparison with Related Work

Overall Comparison.

From the perspective of complexity, standard dense RAG has the lightest online retrieval cost, but it is difficult to explicitly model cross-document relations; multi-step RAG improves complex reasoning ability through multiple rounds of retrieval, but its cost grows linearly with the number of LLM calls; GraphRAG-style methods shift a large amount of cost to offline graph construction and summary generation; SubgraphRAG reduces online cost through lightweight triple scoring, but its effectiveness depends on the candidate triple set and structural distance features; GFM-RAG and our reader concentrate the main computation on one or a small number of query-conditioned graph propagations. Therefore, when the self-evolving memory loop needs to repeatedly evaluate the retrievability of different written graphs, the advantage of our design lies in the following: each evaluation does not need to start multi-round LLM agentic search, but instead quickly obtains differentiable or scoreable retrieval feedback through a fixed GFM reader, structurally gated propagation, and sparse document projection. This allows the graph writing strategy to perform high-frequency comparison and optimization over a large number of candidate memory graphs.

Appendix KImplementation Details of Structured Query Planning

K.1Detailed definitions of the notation and additional information

In𝒫ω​(q)=(ℰexp,𝒜,𝒞rel,𝒞hard,τ,{(q~m,αm,tm)}m=1M)\mathcal{P}_{\omega}(q)=\Big(\mathcal{E}_{\mathrm{exp}},\mathcal{A},\mathcal{C}_{\mathrm{rel}},\mathcal{C}_{\mathrm{hard}},\tau,\{(\tilde{q}_{m},\alpha_{m},t_{m})\}_{m=1}^{M}\Big),ℰexp\mathcal{E}_{\mathrm{exp}}acts as a direct anchor for memory (explicit entities);𝒜\mathcal{A}maps the brain’s multiple representational habits for the same concept (aliases);𝒞rel\mathcal{C}_{\mathrm{rel}}simulates the relational network in semantic memory;𝒞hard\mathcal{C}_{\mathrm{hard}}serves as the spatiotemporal and logical boundaries of episodic memory (such as hard constraints like time and location);τ\taupresets the cognitive template of the target memory (answer type); and the pseudo-queriesq~m\tilde{q}_{m}with confidenceαm\alpha_{m}and intenttmt_{m}are analogous to the multiple exploratory recalls conducted in the human mind (Simulated Recall).

K.2Two-stage Planning: Extraction and Inference

Natural-language questions often compress key retrieval cues into implicit relations, such as “the birthplace of the author”, “the publication year of the only mystery novel of a certain work”, or “the death date of the father”. If the original question is sent as a whole to the entity linker, the system can easily hit only surface entities while missing bridge entities or answer-type constraints. Therefore, the query planner is defined as a structured function

𝒫​(q)=(ℰexp,𝒜,𝒞rel,𝒞hard,τ,{(q~m,αm)}m=1M).\mathcal{P}(q)=\left(\mathcal{E}_{\mathrm{exp}},\mathcal{A},\mathcal{C}_{\mathrm{rel}},\mathcal{C}_{\mathrm{hard}},\tau,\{(\tilde{q}_{m},\alpha_{m})\}_{m=1}^{M}\right).(212)It consists of two stages:ExtractorK.2extracts explicit entities, aliases, relation clues, hard constraints, and the answer type;InfererK.2generates at mostMMretrieval intents based on the extraction results.

Extractor prompt template.You are a retrieval planner for graph-based multi-hop QA.

Question:

{QUESTION}

Extract structured retrieval signals.

Return JSON only with keys:

{

“explicit_entities”: [string],

“candidate_aliases”: {“entity”: [alias]},

“relation_clues”: [string],

“constraints”: {},

“answer_type”: “string”

}

Rules: keep entries short, avoid explanations, keep empty fields as [] or {}.

Figure 6:Metadata of the case study fromHotpotQA. Inferer prompt template.You are a retrieval planner for graph-based multi-hop QA.

Question:

{QUESTION}

Structured extraction:

{EXTRACTOR_JSON}

Generate at most M retrieval intents that help locate:

- evidence directly supporting the target relation;

- bridge entities required for multi-hop reasoning;

- documents likely to contain the target attribute;

- evidence satisfying temporal, spatial, type, comparison or negation constraints;

- evidence using aliases or alternative mentions.

Return JSON only with keys:

{

“pseudo_queries”: [string],

“rewriter_confidence”: [number]

}

Figure 7:Metadata of the case study fromHotpotQA.

Appendix LComputation Details of Topological Structural Features

L.1Normalized Structural Graph

Structural features are computed on an undirected, self-loop-free, binarized adjacency matrix𝒜s\mathcal{A}_{s}:

𝒜s=𝕀​[(𝒜+𝒜⊤)>0],diag⁡(𝒜s)=0.\mathcal{A}_{s}=\mathbb{I}[(\mathcal{A}+\mathcal{A}^{\top})>0],\qquad\operatorname{diag}(\mathcal{A}_{s})=0.(213)This avoids drastic fluctuations in topological statistics caused by unstable relation-extraction directions. Message propagation can still use the original bidirectional edges or relation-aware graph; structural statistics are only used as gating conditions.

L.2Node-level Structural Features

For nodevv, let𝒩​(v)={u:𝒜s,u​v=1}\mathcal{N}(v)=\{u:\mathcal{A}_{s,uv}=1\}anddv=|𝒩​(v)|d_{v}=|\mathcal{N}(v)|. The node-level features are

ϕ​(v)=[log⁡(1+dv),cv,κv,d¯𝒩​(v)].\phi(v)=\big[\log(1+d_{v}),c_{v},\kappa_{v},\bar{d}_{\mathcal{N}(v)}\big].(214)The local clustering coefficient is

cv={2​Tvdv​(dv−1),dv≥2,0,dv<2,c_{v}=\begin{cases}\dfrac{2T_{v}}{d_{v}(d_{v}-1)},&d_{v}\geq 2,\\[3.99994pt] 0,&d_{v}<2,\end{cases}(215)whereTvT_{v}is the number of undirected edges inside the neighborhood ofvv;κv\kappa_{v}is the core number; and the average neighbor degree is

d¯𝒩​(v)={1dv​∑u∈𝒩​(v)du,dv>0,0,dv=0.\bar{d}_{\mathcal{N}(v)}=\begin{cases}\dfrac{1}{d_{v}}\sum_{u\in\mathcal{N}(v)}d_{u},&d_{v}>0,\\[3.99994pt] 0,&d_{v}=0.\end{cases}(216)These quantities respectively characterize node frequency, local clustering, core/peripheral position, and neighborhood density. For RAG memory, they correspond to four common structural risks: over-propagation by high-frequency hubs, redundant diffusion inside clustered regions, ignored peripheral bridge entities, and scale mismatch between sparse and dense regions.

L.3Edge-pair Structural Features

For an undirected structural edge(u,v)(u,v), the pairwise features are

ψ​(u,v)=[|du−dv|,CN⁡(u,v),Jac⁡(u,v)],\psi(u,v)=\big[|d_{u}-d_{v}|,\operatorname{CN}(u,v),\operatorname{Jac}(u,v)\big],(217)where

CN⁡(u,v)=|𝒩​(u)∩𝒩​(v)|,Jac⁡(u,v)=|𝒩​(u)∩𝒩​(v)||𝒩​(u)∪𝒩​(v)|+ε.\operatorname{CN}(u,v)=|\mathcal{N}(u)\cap\mathcal{N}(v)|,\qquad\operatorname{Jac}(u,v)=\frac{|\mathcal{N}(u)\cap\mathcal{N}(v)|}{|\mathcal{N}(u)\cup\mathcal{N}(v)|+\varepsilon}.(218)Degree difference reflects cross-level connections, while common neighbors and Jaccard reflect local community overlap. Based on these features, the gate can distinguish intra-community evidence aggregation edges from cross-community bridge edges.

L.4Graph-level Summary and Normalization

The graph-level summary concatenates the mean, standard deviation, and density of node features:

𝐫𝒢=[meanv∈𝒱⁡ϕ​(v);stdv∈𝒱⁡ϕ​(v);dens⁡(𝒢)],\mathbf{r}_{\mathcal{G}}=\big[\operatorname{mean}_{v\in\mathcal{V}}\phi(v);\operatorname{std}_{v\in\mathcal{V}}\phi(v);\operatorname{dens}(\mathcal{G})\big],(219)where

dens⁡(𝒢)={2​msn​(n−1),n≥2,0,n<2,\operatorname{dens}(\mathcal{G})=\begin{cases}\dfrac{2m_{s}}{n(n-1)},&n\geq 2,\\[3.99994pt] 0,&n<2,\end{cases}(220)n=|𝒱|n=|\mathcal{V}|, andmsm_{s}is the number of undirected structural edges. To remove graph-size differences, node and edge features are z-scored within each graph, and the graph-level summary computes global mean and standard deviation over the set of training graphs:

𝐫¯𝒢=𝐫𝒢−μrσr+ε.\bar{\mathbf{r}}_{\mathcal{G}}=\frac{\mathbf{r}_{\mathcal{G}}-\mu_{r}}{\sigma_{r}+\varepsilon}.(221)If the standard deviation of a certain dimension is close to zero, we only perform centering to avoid division by an unstable small value.

L.5Gating Input Encoding

For each message edgeu→vu\to v, the structural gate reads the source node, target node, pairwise features, and graph-level summary:

ϕ¯​(u)\displaystyle\bar{\phi}(u)=NormNode⁡(ϕ​(u)),\displaystyle=\operatorname{NormNode}(\phi(u)),ψ¯​(u,v)\displaystyle\bar{\psi}(u,v)=NormPair⁡(ψ​(u,v)),\displaystyle=\operatorname{NormPair}(\psi(u,v)),(222)uu(l)\displaystyle u_{u}^{(l)}=En(l)​(ϕ¯​(u)),\displaystyle=E_{n}^{(l)}(\bar{\phi}(u)),uv(l)\displaystyle u_{v}^{(l)}=En(l)​(ϕ¯​(v)),\displaystyle=E_{n}^{(l)}(\bar{\phi}(v)),(223)vu​v(l)\displaystyle v_{uv}^{(l)}=Ep(l)​(ψ¯​(u,v)),\displaystyle=E_{p}^{(l)}(\bar{\psi}(u,v)),r𝒢(l)\displaystyle r_{\mathcal{G}}^{(l)}=Eg(l)​(𝐫¯𝒢).\displaystyle=E_{g}^{(l)}(\bar{\mathbf{r}}_{\mathcal{G}}).(224)The encodersEn,Ep,EgE_{n},E_{p},E_{g}are all two-layer MLPs. The concatenated gating input is

𝐳u​v(l)=[𝐮u(l);𝐮v(l);𝐯u​v(l);𝐫𝒢(l)].\mathbf{z}_{uv}^{(l)}=[\mathbf{u}_{u}^{(l)};\mathbf{u}_{v}^{(l)};\mathbf{v}_{uv}^{(l)};\mathbf{r}_{\mathcal{G}}^{(l)}].(225)The gate itself is a vector rather than a scalar:

𝐠u​v(l)=𝟏+δ​tanh⁡(MLPg(l)⁡(𝐳u​v(l))),δ=0.1.\mathbf{g}_{uv}^{(l)}=\mathbf{1}+\delta\tanh\big(\operatorname{MLP}_{g}^{(l)}(\mathbf{z}_{uv}^{(l)})\big),\qquad\delta=0.1.(226)The last layer of the gating MLP is initialized to zero, so initially𝐠u​v(l)=𝟏\mathbf{g}_{uv}^{(l)}=\mathbf{1}. At the beginning of training, the model does not destroy the original propagation scale; the learned structural bias gradually emerges in a residual manner.

L.6Message Propagation with Normalized Weights

Let𝔼~\tilde{\mathbb{E}}be the edge set after adding self-loops. Structural gates are used for non-self-loop edges, and unit gates are used for self-loops. The GCN normalization coefficient is

ηu​v=wu​vd~u​d~v,d~v=∑u:(u,v)∈𝔼~wu​v,\eta_{uv}=\frac{w_{uv}}{\sqrt{\tilde{d}_{u}\tilde{d}_{v}}},\qquad\tilde{d}_{v}=\sum_{u:(u,v)\in\tilde{\mathbb{E}}}w_{uv},(227)wherewu​vw_{uv}defaults to11, but can also come from edge weights. The propagation at layerllis

𝐦u→v(l)\displaystyle\mathbf{m}_{u\to v}^{(l)}=ηu​v​𝐠u​v(l)⊙W(l)​𝐡u(l−1),\displaystyle=\eta_{uv}\,\mathbf{g}_{uv}^{(l)}\odot W^{(l)}\mathbf{h}_{u}^{(l-1)},(228)𝐡v(l)\displaystyle\mathbf{h}_{v}^{(l)}=σ​(𝐛(l)+∑u:(u,v)∈𝔼~𝐦u→v(l)).\displaystyle=\sigma\left(\mathbf{b}^{(l)}+\sum_{u:(u,v)\in\tilde{\mathbb{E}}}\mathbf{m}_{u\to v}^{(l)}\right).(229)The multi-layer wrapper also contains inter-layer residuals: whenl>1l>1and the dimensions are consistent,

𝐇(l)←𝐇(l)+𝐇(l−1).\mathbf{H}^{(l)}\leftarrow\mathbf{H}^{(l)}+\mathbf{H}^{(l-1)}.(230)This residual and Eq. (226) form a dual stability mechanism: the former stabilizes deep propagation, while the latter stabilizes structural modulation.

L.7Chunked Gating and GPU Memory Complexity

Explicitly storing all gates requiresO​(|𝔼|​d)O(|\mathbb{E}|d)GPU memory. For large graphs, gates are computed by edge chunks:

𝔼=⋃b=1Be𝔼b,|𝔼b|≤Ce.\mathbb{E}=\bigcup_{b=1}^{B_{e}}\mathbb{E}_{b},\qquad|\mathbb{E}_{b}|\leq C_{e}.(231)Each edge chunk sequentially executes

𝐠b→𝐦b→scatter​_​add⁡(𝐦b),\mathbf{g}_{b}\rightarrow\mathbf{m}_{b}\rightarrow\operatorname{scatter\_add}(\mathbf{m}_{b}),(232)and immediately releases the intermediate gate tensor. Online GPU memory is reduced fromO​(|𝔼|​d)O(|\mathbb{E}|d)toO​(Ce​d)O(C_{e}d), while the time complexity remains linear,O​(|𝔼|​d)O(|\mathbb{E}|d). This is especially important for self-evolving memory, because the same reader needs to repeatedly evaluate candidate graphs produced by different writers.

Appendix MPretraining Objective and Augmented Views

M.1GraphCL View Construction

The goal of the pretraining stage is to learn cross-graph transferable structural–semantic propagation, rather than fitting specific question-answering labels. Given the original graph view(𝒢0,X0)(\mathcal{G}_{0},X_{0}), we construct two augmented views(𝒢1,X1)(\mathcal{G}_{1},X_{1}),(𝒢2,X2)(\mathcal{G}_{2},X_{2})and one negative feature view(𝒢0,X−)(\mathcal{G}_{0},X^{-}). The augmentation types include edge perturbation, feature masking, node perturbation, and subgraph sampling; let the augmentation operators be𝒜1,𝒜2\mathcal{A}_{1},\mathcal{A}_{2}, then

(𝒢j,Xj)=𝒜j​(𝒢0,X0),j∈{1,2}.(\mathcal{G}_{j},X_{j})=\mathcal{A}_{j}(\mathcal{G}_{0},X_{0}),\qquad j\in\{1,2\}.(233)If structural gating is enabled, each view precomputes its own node structural features, edge-pair features, and graph-level summary; the negative feature view shares the base graph structure, but its node features are shuffled or replaced.

M.2Graph-level Contrastive Objective

The encoder outputs four sets of node representations:

H0=fθ​(X0,𝒢0),H1=fθ​(X1,𝒢1),H2=fθ​(X2,𝒢2),H−=fθ​(X−,𝒢0).H_{0}=f_{\theta}(X_{0},\mathcal{G}_{0}),\quad H_{1}=f_{\theta}(X_{1},\mathcal{G}_{1}),\quad H_{2}=f_{\theta}(X_{2},\mathcal{G}_{2}),\quad H^{-}=f_{\theta}(X^{-},\mathcal{G}_{0}).(234)The graph readout of each augmented view is

cj=sigmoid⁡(1|𝒱j|​∑v∈𝒱jHj,v),j∈{1,2}.c_{j}=\operatorname{sigmoid}\left(\frac{1}{|\mathcal{V}_{j}|}\sum_{v\in\mathcal{V}_{j}}H_{j,v}\right),\qquad j\in\{1,2\}.(235)The bilinear discriminator

D​(c,h)=h⊤​WD​cD(c,h)=h^{\top}W_{D}c(236)determines whether the node representation comes from the same graph semantics. The pretraining loss is

ℒGCL=12​∑j=12[BCE⁡(D​(cj,H0),𝟏)+BCE⁡(D​(cj,H−),𝟎)].\mathcal{L}_{\mathrm{GCL}}=\frac{1}{2}\sum_{j=1}^{2}\left[\operatorname{BCE}\big(D(c_{j},H_{0}),\mathbf{1}\big)+\operatorname{BCE}\big(D(c_{j},H^{-}),\mathbf{0}\big)\right].(237)When edge-level gating is enabled, traditional static structural prompts are neutralized into identity mappings to avoid scale confusion caused by two sets of structural modulations acting simultaneously; the structural bias is mainly carried by the target edge’s𝐠u​v(l)\mathbf{g}_{uv}^{(l)}.

M.3Feature Alignment Layer

When the input dimensions produced by different graphs or different text encoders are consistent but their distributions have large shifts, the feature alignment layer can be enabled:

Align⁡(x)=Dropout⁡(LayerNorm⁡(PReLU⁡(Wa​x+ba))).\operatorname{Align}(x)=\operatorname{Dropout}\left(\operatorname{LayerNorm}\left(\operatorname{PReLU}(W_{a}x+b_{a})\right)\right).(238)WaW_{a}is initialized as the identity matrix, andbab_{a}is initialized as zero. Therefore, this layer is initially an approximately identity transformation; after training, it absorbs inter-graph feature-scale differences without changing the core structure of the graph propagator.

Appendix NSupervised Fine-tuning Objective

N.1Entity-level Supervision

For each questionqbq_{b}, the data provide a supporting-entity maskyb,e∈{0,1}y_{b,e}\in\{0,1\}. The model outputs entity logitsab,ea_{b,e}. The weighted BCE is defined as

ℒbce=1B​∑b=1B∑ewb,e​BCEWithLogits⁡(ab,e,yb,e)∑ewb,e+ε.\mathcal{L}_{\mathrm{bce}}=\frac{1}{B}\sum_{b=1}^{B}\frac{\sum_{e}w_{b,e}\,\operatorname{BCEWithLogits}(a_{b,e},y_{b,e})}{\sum_{e}w_{b,e}+\varepsilon}.(239)Positive weights are uniformly normalized within the positive set; if the adversarial temperatureTaT_{a}is enabled for negative weights, they are computed by applying softmax to the current model scores:

wb,e−=exp⁡(ab,e/Ta)∑v:yb,v=0exp⁡(ab,v/Ta),yb,e=0.w_{b,e}^{-}=\frac{\exp(a_{b,e}/T_{a})}{\sum_{v:y_{b,v}=0}\exp(a_{b,v}/T_{a})},\qquad y_{b,e}=0.(240)IfTa=0T_{a}=0, the negative weights degenerate into a uniform distribution. This design makes training focus more on high-scoring hard negatives, rather than being dominated by a large number of obviously irrelevant entities.

N.2Multi-positive List Cross-Entropy

Using only BCE treats each entity as an independent binary classification problem, lacking the constraint that “supporting entities should collectively rank near the top of the same candidate list”. To this end, we introduce a multi-positive list loss. Let

pb,e=sigmoid⁡(ab,e)∑vsigmoid⁡(ab,v)+ε.p_{b,e}=\frac{\operatorname{sigmoid}(a_{b,e})}{\sum_{v}\operatorname{sigmoid}(a_{b,v})+\varepsilon}.(241)If samplebbhas at least one supporting entity, the list loss is

ℒlist=−1|ℬ+|​∑b∈ℬ+1|YE​(qb)|​∑e∈YE​(qb)log⁡(pb,e+ε).\mathcal{L}_{\mathrm{list}}=-\frac{1}{|\mathcal{B}_{+}|}\sum_{b\in\mathcal{B}_{+}}\frac{1}{|Y_{E}(q_{b})|}\sum_{e\in Y_{E}(q_{b})}\log(p_{b,e}+\varepsilon).(242)Samples with empty supporting-entity sets are skipped. The final entity fine-tuning objective is

ℒent=λbce​ℒbce+λlist​ℒlist,(λbce,λlist)=(0.3,0.7).\mathcal{L}_{\mathrm{ent}}=\lambda_{\mathrm{bce}}\mathcal{L}_{\mathrm{bce}}+\lambda_{\mathrm{list}}\mathcal{L}_{\mathrm{list}},\qquad(\lambda_{\mathrm{bce}},\lambda_{\mathrm{list}})=(0.3,0.7).(243)

N.3Optional Document-level Supervision

If the training configuration provides a document-level loss, entity logits are first projected into document logits:

S~b=ab⊤​𝐌,\tilde{S}_{b}=a_{b}^{\top}\mathbf{M},(244)and then the same type of BCE or list loss is computed with the supporting-document maskzb,iz_{b,i}. This term is suitable for tasks where entity annotations are noisy but the document support set is reliable; if it is not enabled, training is entirely driven by the entity-level support set, and document ranking is obtained through projection only during inference or validation.

Appendix OMemory Writer Implementation Details

O.1The Markov Decision Process for Multi-turn Graph Construction

Specifically, the training of our graph constructor is implemented through VeRL’s multi-turn GRPO loop. The state machine of the interactor can be abstracted as a finite-horizon MDP:

ℳ=(𝒮,𝒜,P,ℛ,ρ0,H).\mathcal{M}=(\mathcal{S},\mathcal{A},P,\mathcal{R},\rho_{0},H).(245)Given a samplexx, at roundtt, the state can be written as

st=(q,𝒢t,𝒟tproc,𝒟trem,ζt),s_{t}=(q,\mathcal{G}_{t},\mathcal{D}^{\mathrm{proc}}_{t},\mathcal{D}^{\mathrm{rem}}_{t},\zeta_{t}),(246)where𝒢t\mathcal{G}_{t}is the current partially written graph,𝒟tproc\mathcal{D}^{\mathrm{proc}}_{t}and𝒟trem\mathcal{D}^{\mathrm{rem}}_{t}denote the processed and remaining documents, respectively, andζt\zeta_{t}is an interaction control flag, such as whether the process is still in the graph-construction stage or has already switched to the RAG stage. The action is generated by the language model in JSON format:

at∼πθ(⋅∣st),a_{t}\sim\pi_{\theta}(\cdot\mid s_{t}),(247)and is restricted to two types of legal actions:

1. 1.Triple action: output a JSON array, where each element is of the form{subject,relation,object}\{\texttt{subject},\texttt{relation},\texttt{object}\}, representing the set of facts𝒯t\mathcal{T}_{t}written in the current round;
2. 2.Termination action: after graph construction is completed, output a JSON object carrying the terminal fields required by the reader side, such asanswer,recall,precision,deducible, and so on.

In implementation, the environment first checks whether the action can be parsed byjson_repair, and strictly cleans the triples: items with missing keys, empty strings, or non-dictionary entries are all removed. If illegal JSON is output during the graph-construction stage, the interaction terminates immediately and returns zero reward; if legal triples are output, the environment proceeds to the next round and returns a round-level format reward. The corresponding environment transition can be written as

st+1=P​(st,at)={(q,𝒢t⊕𝒯t,𝒟tproc∪{dt},𝒟trem∖{dt},ζt+1),at​is legal,(q,𝒢t,𝒟tproc,𝒟trem,STOP),at​is illegal,(q,𝒢t,𝒟tproc,𝒟trem,RAG),at​triggers the reading stage.s_{t+1}=P(s_{t},a_{t})=\begin{cases}(q,\mathcal{G}_{t}\oplus\mathcal{T}_{t},\mathcal{D}^{\mathrm{proc}}_{t}\cup\{d_{t}\},\mathcal{D}^{\mathrm{rem}}_{t}\setminus\{d_{t}\},\zeta_{t+1}),&a_{t}\text{ is legal },\\ (q,\mathcal{G}_{t},\mathcal{D}^{\mathrm{proc}}_{t},\mathcal{D}^{\mathrm{rem}}_{t},\mathrm{STOP}),&a_{t}\text{ is illegal},\\ (q,\mathcal{G}_{t},\mathcal{D}^{\mathrm{proc}}_{t},\mathcal{D}^{\mathrm{rem}}_{t},\mathrm{RAG}),&a_{t}\text{ triggers the reading stage}.\end{cases}(248)

Iterative and non-iterative writing.

Two strategies are supported. In the non-iterative mode, the model reads the entire context𝒟\mathcal{D}at once and outputs all triples. In the iterative mode, the environment reads the documents segment by segment in document order, and in each round the model is only allowed to write triples for the current document. After all documents have been processed, the environment then switches to the RAG stage. If𝒯i\mathcal{T}_{i}denotes the set of triples output for documentdid_{i}, then the final graph constructed in the iterative mode is𝒢=⨁i=1m𝒯i\mathcal{G}=\bigoplus_{i=1}^{m}\mathcal{T}_{i}, where⊕\oplusdenotes edge-set union and node deduplication. We adopt the iterative strategy by default, because it decomposes the long-context problem into a sequence of local writing decisions, significantly reducing the difficulty of performing global planning in advance. At the same time, it also allows the source document of each triple to be precisely recorded, providing explicit source edges for subsequent text-graph retrieval.

Constructing text-graph memory from output triples.

To enable the frozen retriever to operate under thegraph-guided text retrievalsetting, the environment does not directly pass the raw triple strings to the retriever. Instead, it first constructs a text graph with document nodes:

𝒢=(𝒱e∪𝒱d,ℰe​e∪ℰe​d),\mathcal{G}=(\mathcal{V}_{e}\cup\mathcal{V}_{d},\mathcal{E}_{ee}\cup\mathcal{E}_{ed}),(249)where the entity node set𝒱e\mathcal{V}_{e}comes from the subjects and objects in the triples, and the document node set𝒱d={d1,…,dm}\mathcal{V}_{d}=\{d_{1},\dots,d_{m}\}corresponds to the original documents in the context. The entity-entity edges are defined as

ℰe​e={(u,r,v)∣(u,r,v)∈𝒯},\mathcal{E}_{ee}=\{(u,r,v)\mid(u,r,v)\in\mathcal{T}\},(250)and the entity-document source edges are defined as

ℰe​d={(u,source,di),(v,source,di)∣(u,r,v)∈𝒯i}.\mathcal{E}_{ed}=\{(u,\texttt{source},d_{i}),(v,\texttt{source},d_{i})\mid(u,r,v)\in\mathcal{T}_{i}\}.(251)In the iterative mode, the source edges are explicit, because the environment already knows that the triples in each round come from the current document. In the non-iterative mode, we use a heuristic alignment method based on tokenizer token overlap to map each triple to the most similar document. The significance of this design is that, after separating writing from reading, the graph constructor is only responsible for deciding “what to write into memory”; as for how the reader aggregates entities on the graph and retrieves documents, this is entirely determined by the frozenfϕf_{\phi}.

Frozen GFM retrieval environment.

When training the graph constructor, the readerfϕf_{\phi}is fixed as the already trained GFM retriever. Let the entity set be𝒱e={e1,…,en}\mathcal{V}_{e}=\{e_{1},\dots,e_{n}\}and the document set be𝒱d={d1,…,dM}\mathcal{V}_{d}=\{d_{1},\dots,d_{M}\}. We then construct:

1. 1.the relation-edge index𝐄\mathbf{E}with both forward and reverse directions, together with the relation types𝐫\mathbf{r};
2. 2.the sparse entity-document matrix𝐌∈{0,1}n×M\mathbf{M}\in\{0,1\}^{n\times M}, whereMi​j=1M_{ij}=1if entityeie_{i}appears in documentdjd_{j};
3. 3.the question-related entity mask𝐦q∈{0,1}n\mathbf{m}_{q}\in\{0,1\}^{n}, which is obtained preferentially through lexical matching with the question; if lexical matching fails, it falls back to a heuristic seed set ranked by entity degree.

After encoding the question as a vector𝐪\mathbf{q}and the relation names as a matrix𝐑\mathbf{R}, the frozen GFM forward pass computes the entity relevance scores:

𝐬e=fϕ​(𝒢,𝐪,𝐦q;ϕ)∈ℝn.\mathbf{s}_{e}=f_{\phi}(\mathcal{G},\mathbf{q},\mathbf{m}_{q};\phi)\in\mathbb{R}^{n}.(252)The entity scores are then projected into document scores. Let𝐌Top​-​k​(𝐬e)\mathbf{M}_{\mathrm{Top}\text{-}k}(\mathbf{s}_{e})denote the masking operation that retains only the top-KKentity scores, and let𝐰idf\mathbf{w}_{\mathrm{idf}}denote the inverse-frequency weights defined according to the document frequency of each entity. The four document-scoring modes can be written uniformly as

𝐬~e={𝐬e,raw,𝐌Top​-​k​(𝐬e),topk,𝐰idf⊙𝐬e,idf,𝐰idf⊙𝐌Top​-​k​(𝐬e),idf_topk,\tilde{\mathbf{s}}_{e}=\begin{cases}\mathbf{s}_{e},&\texttt{raw},\\ \mathbf{M}_{\mathrm{Top}\text{-}k}(\mathbf{s}_{e}),&\texttt{topk},\\ \mathbf{w}_{\mathrm{idf}}\odot\mathbf{s}_{e},&\texttt{idf},\\ \mathbf{w}_{\mathrm{idf}}\odot\mathbf{M}_{\mathrm{Top}\text{-}k}(\mathbf{s}_{e}),&\texttt{idf\_topk},\end{cases}(253)and the document scores are obtained by

𝐬d=𝐌⊤​𝐬~e.\mathbf{s}_{d}=\mathbf{M}^{\top}\tilde{\mathbf{s}}_{e}.(254)We then takeTop​-​k​(𝐬d)\mathrm{Top}\text{-}k(\mathbf{s}_{d})as the retrieval result. In actual use, we also enableinit_entities_weight, that is, during the GFM forward pass, a1/f​(e)1/f(e)weight is applied to high-frequency entities to suppress the dominance of entities connected to too many documents in the retrieval results.

Appendix PAdditional Detailed Experimental Results

The results of retrieval performance on multi-hop QA benchmarks are in Table9.

Table 9:Results of retrieval performance on multi-hop QA benchmarks. We report document-level Recall (%) at top-2 and top-5. Best results are inboldand runner-ups areunderlined. The darker the cell, the better.DatasetHotpotQAMuSiQue2WikiMultiHopQAAvg. RankMethodR@2R@5R@2R@5R@2R@5BM25(⊳\rhdSIGIR’94) 55.472.232.341.251.861.918.2Contriever(⊳\rhdTMLR’22) 57.275.534.846.646.657.517.2GTR(⊳\rhdEMNLP’22) 59.473.337.449.160.267.913.8ColBERTv2(⊳\rhdNAACL’22) 64.779.337.949.259.268.211.5RAPTOR(⊳\rhdICLR’24) 58.171.235.745.346.353.817.7Proposition(⊳\rhdEMNLP’24) 58.771.137.649.356.463.114.8GraphRAG(⊳\rhdarXiv’24) 58.376.635.449.361.677.312.2G-Retriever(⊳\rhdNeurIPS’24) 53.365.538.845.160.867.815.7LightRAG(⊳\rhdarXiv’24) 38.854.724.834.745.159.120.5HippoRAG(⊳\rhdNeurIPS’24) 60.178.541.253.268.487.08.7HippoRAG2(⊳\rhdICML’25) 80.588.147.056.788.990.12.4SubgraphRAG(⊳\rhdICLR’25) 61.573.042.149.370.785.59.7PropRAG(⊳\rhdEMNLP’25) 81.988.047.757.987.990.11.9GFM-RAG(⊳\rhdNeurIPS’25) 75.689.643.557.679.192.42.9FLARE(⊳\rhdEMNLP’23) 73.181.344.355.167.173.16.5Adaptive-RAG(⊳\rhdNAACL’24) 61.076.435.144.744.761.416.6BM25+IRCoT(⊳\rhdACL’23) 65.679.034.244.761.275.612.4Contriever+IRCoT(⊳\rhdACL’23) 65.981.639.152.251.663.810.7ColBERTv2+IRCoT(⊳\rhdACL’23) 67.982.041.753.764.174.47.2HippoRAG+IRCoT(⊳\rhdACL’23) 67.083.045.357.675.893.93.6SAGE(ours)65.177.643.253.183.688.67.0

The results on AmazonQA are in Table10.

Table 10:Performance of representative baselines on the originalAmazonQAfull-test protocol. BLEU-1/2/3/4 are denoted as B-1/2/3/4, and R denotes ROUGE. Best results are inboldand runner-ups areunderlined.Only rows marked with0-shotare our zero-shot transfer results; baseline rows and trained variants are not marked as zero-shot.Zero-shot setting applies only toOursrows marked with0-shotonAmazonQA.MethodB-1B-2B-3B-4RHeuristic baselines from the originalAmazonQAprotocolRandom Sentence(⊳\rhdIJCAI’19) 78.5663.9544.3729.8749.12Top-1 using IR(⊳\rhdIJCAI’19) 89.4974.8056.7643.5261.48Top-1 Using BLEU(⊳\rhdIJCAI’19) 92.7478.4360.9148.0862.68Top-1 Helpfulness(⊳\rhdIJCAI’19) 20.6619.7816.3912.5437.01Top-1 Wilson Score(⊳\rhdIJCAI’19) 20.7419.8416.4412.5837.26Neural baseline from the originalAmazonQAprotocolR-Net(⊳\rhdIJCAI’19) 47.0440.3231.4823.9240.22Human answers under the originalAmazonQAprotocolAmazon User Community(⊳\rhdIJCAI’19) 80.8868.8654.3642.0162.18Expert (Spans)(⊳\rhdIJCAI’19) 68.3357.7944.6134.4351.09Expert (Descriptive)(⊳\rhdIJCAI’19) 53.6746.5637.8130.7653.31Our methodOurs(0-shot)0-shot61.840-shot49.360-shot37.820-shot28.410-shot46.730-shotOurs(trained)74.9261.5847.6335.8654.92Ours+1 round82.7668.9152.7439.6858.83

The HaluMem results are shown in Table11.

Table 11:Results onHaluMem-Medium. We report memory extraction metrics, memory updating metrics, and memory question-answering metrics. R denotes Recall, W-R denotes Weighted Recall, T-P denotes Target Memory Precision, Acc. denotes Memory Accuracy, FMR denotes False Memory Resistance, F1 denotes Memory Extraction F1-score, C denotes Correct Rate, H denotes Hallucination Rate, and O denotes Omission Rate. For R, W-R, T-P, Acc., FMR, F1, and C, higher is better; for H and O, lower is better. Best results are inboldand runner-ups areunderlined. The darker the cell, the better. For systems whose public reports only provide a subset of metrics, missing entries are denoted by “–”.Only rows marked with0-shotare our zero-shot results; baseline rows and trainedSAGErows are not marked as zero-shot.Zero-shot setting applies only toSAGEvariants marked with0-shotonHaluMem-Medium.DatasetMemory ExtractionMemory UpdatingMemory QAMethodR↑\uparrowW-R↑\uparrowT-P↑\uparrowAcc.↑\uparrowFMR↑\uparrowF1↑\uparrowC↑\uparrowH↓\downarrowO↓\downarrowC↑\uparrowH↓\downarrowO↓\downarrowMemory-system baselines from the originalHaluMembenchmarkMemobase(https://github.com/memodb-io/memobase) 14.5525.8892.2432.2980.7825.135.200.5594.2535.3329.9734.71Supermemory(https://github.com/supermemoryai/supermemory) 41.5364.7690.3260.8351.7756.9016.371.1582.4754.0722.2423.69Mem0(⊳\rhdarXiv’25) 42.9165.0386.2660.8656.8057.3125.500.4574.0253.0219.1727.81Zep(⊳\rhdarXiv’25) ––––––47.280.4252.3155.4721.9222.62MemOS(⊳\rhdarXiv’25) 74.0784.8186.2559.5544.9479.7062.110.4237.4867.2315.1717.59SAGE(ours, 0-shot)0-shot13.1220.9131.3622.8032.5919.3821.671.6184.5330.1433.6836.17SAGE(ours, trained)16.4229.3671.8835.4144.9628.527.340.6891.9838.2628.7333.01SAGE+1 round20.1835.7471.0240.6340.2833.4710.860.7688.3842.9126.6430.45

P.1Path Interpretations

We provide path interpretations ofSAGEfor multi-hop reasoning inTable˜12. The importance of each path to the final prediction can be measured by the partial derivative of the prediction score with respect to the triples at each reasoning layer. The top-kkpath interpretations are then obtained by selecting the top-kklongest paths with beam search.

As shown inTable˜12,SAGEsuccessfully identifies the answer by connecting two key constraints in the question: the person who presented the Australia 2022 FIFA World Cup bid and the person born on October 22, 1930. Specifically, the first path starts from the entity “the bid for the 2022 FIFA World Cup” and follows the inverse relation of “was one of the representatives of” to reach “Frank Lowy”. Then, through an entity-equivalence relation, it links “Frank Lowy” to “Sir Frank P. Lowy”, whose birth date is “22 October 1930”. The second path verifies the reasoning in the reverse direction by starting from the birth date and tracing back to the representative of the World Cup bid. These paths demonstrate thatSAGEcan effectively align different surface forms of the same entity and integrate multiple question constraints within a single-step retrieval process, showing its ability to perform interpretable multi-hop reasoning.

Table 12:Path interpretations of SAMGPT for multi-hop reasoning, wherer−1r^{-1}denotes the inverse of original relation.QuestionWhich man who presented theAustralia 2022 FIFA World Cup bidwas born onOctober 22, 1930?AnswerFrank LowySup. Doc.[ “Frank Lowy”, “Australia 2022 FIFA World Cup bid”]Paths1: (the bid for the 2022 fifa world cup, was one of the representatives of-1, frank lowy)→\to(frank lowy, equivalent, sir frank p lowy)→\to(sir frank p lowy, was born on, 22 october 1930)2: (22 october 1930, was born on-1, sir frank p lowy)→\to(sir frank p lowy, equivalent, frank lowy)→\to(frank lowy, was one of the representatives of, the bid for the 2022 fifa world cup)

Appendix QDataset Details

Table13summarizes the details of each dataset.

Table 13:Dataset statistics and evaluation scenarios. We evaluateSAGEon three complementary categories: general and multi-hop QA, practical e-commerce review QA, and long-term agent memory. “Train/Dev/Test” denotes the standard split when available. For benchmark-only datasets without a conventional supervised training split, we report the total number of evaluation instances or benchmark scale.CategoryDatasetScale / SplitEvidence SourceTask TypeKey CapabilitiesMain MetricsGeneral /Multi-hop QANQ-Open(⊳\rhd;⊳\rhd) 79,168 / 8,757 / 3,610English WikipediaOpen-domain short-answer QAFactual retrieval; entity-level knowledge access; open-domain answer generationEM / F1 / Acc.; Recall@kPopQA(⊳\rhd) 14,267 QA pairsWikidata triples + Wikipedia page-view popularityEntity-centric open-domain QALong-tail factual recall; parametric vs. non-parametric memory; retrieval under entity popularity shiftAcc. / EM; long-tail breakdownHotpotQA(⊳\rhd) 90,447 / 7,405 / 7,405Wikipedia paragraphs; 10-paragraph distractor settingExplainable 2-hop QABridge-entity recovery; comparison reasoning; sentence-level supporting factsAnswer EM/F1; Support EM/F1; Joint EM/F12WikiMultiHopQA(⊳\rhd) 167,454 / 12,576 / 12,576Wikipedia + Wikidata; 10 passages per instance2–4 hop multi-hop QAReasoning-path recovery; comparison, bridge, and bridge-comparison reasoningAnswer EM/F1; Evidence / path recallMuSiQue(⊳\rhd) 19,938 / 2,417 / 2,459 (24,814 total)Composed single-hop QA over textual passages2–4 hop connected multi-hop QAConnected reasoning; shortcut-resistant evidence aggregation; multi-hop compositionalityAnswer EM/F1; Support / evidence recallE-commerceReview QAAmazonQA(⊳\rhd) 923K questions; 3.6M answers; 14M reviews; 156K productsAmazon product reviews, questions, answers, and product metadataReview-based QA with answerability annotationNoisy review retrieval; answerable / unanswerable detection; evidence synthesis from user-generated reviewsBLEU / ROUGE; answerability Acc./F1; groundednessLong-termAgent MemoryLongMemEval(⊳\rhd) 500 eval. instances per file;S:∼\sim115K tokens / 30–40 sessions;M:∼\sim1.5M tokens /∼\sim500 sessions;Oracle: evidence sessions onlyLong multi-session human–AI chat historiesLong-term interactive memory QAInformation extraction; multi-session reasoning; temporal reasoning; knowledge update; abstentionOverall Acc.; category-wise Acc.; context tokens; latencyHaluMem(⊳\rhd) Medium: 20 users, 30,073 dialogue rounds,∼\sim160K tokens/user, 14,948 memory points, 3,467 QA pairs;Long: 53,516 rounds,∼\sim1M tokens/userSynthetic long-term human–AI interaction histories with memory points and multi-type questionsOperation-level memory hallucination benchmarkMemory extraction; memory updating; memory QA; hallucination, omission, and conflict propagation across memory operationsExtraction R/P/F1; Updating C/H/O; QA C/H/O

General and Multi-hop QA.

We first evaluateSAGEon a set of general open-domain and multi-hop QA benchmarks that stress different aspects of retrieval-augmented reasoning.NQ-Openis derived from Natural Questions and is widely used as a standard open-domain short-answer QA benchmark; it evaluates whether a system can retrieve and ground factual answers from a large Wikipedia-scale corpus.PopQAcomplements NQ by focusing on entity-centric factual questions whose subjects span different popularity levels, making it particularly useful for testing whether a retrieval or memory system can recover long-tail factual knowledge rather than relying only on parametric memorization.HotpotQAcontains Wikipedia-based multi-hop questions with sentence-level supporting facts, allowing us to evaluate not only answer correctness but also whether the system can recover bridge evidence and produce interpretable reasoning chains.2WikiMultiHopQAfurther stresses structured multi-hop reasoning by combining Wikipedia text with Wikidata-derived relations and providing evidence paths for 2–4 hop questions. Finally,MuSiQueis designed to reduce shortcut reasoning by composing connected single-hop questions into 2–4 hop questions, making it a strong testbed for evaluating whetherSAGEcan retrieve and integrate multiple pieces of evidence in a genuinely compositional manner.

E-commerce Review-based QA.

We useAmazonQAto evaluateSAGEin a practical, noisy, user-generated e-commerce setting. Unlike Wikipedia-style QA benchmarks,AmazonQAconsists of real product questions, community answers, product reviews, and product metadata, and includes answerability annotations indicating whether a question can be answered from available reviews. This makes it a suitable benchmark for testing whether a memory system can identify useful evidence from noisy review collections, distinguish answerable from unanswerable questions, and synthesize grounded answers from multiple user-generated snippets. From the perspective of self-evolving memory, AmazonQA is especially valuable because the system must learn which review facts, product attributes, and user opinions are worth indexing for future retrieval, rather than simply matching a question to a clean encyclopedic passage.

Long-term Agent Memory.

To move beyond conventional RAG evaluation, we further evaluateSAGEon long-term agent memory benchmarks.LongMemEvalis designed to assess the long-term memory abilities of chat assistants over extended multi-session interaction histories. It covers five core memory abilities: information extraction, multi-session reasoning, temporal reasoning, knowledge updates, and abstention. This benchmark directly tests whetherSAGEcan retrieve sparse but relevant memory traces from long histories, combine evidence across sessions, respect temporal order, and update previously stored information when new interactions supersede old memories. We useHaluMemas a complementary benchmark for evaluating hallucination in memory systems. Rather than only measuring end-to-end QA accuracy, HaluMem decomposes memory evaluation into memory extraction, memory updating, and memory question answering, thereby revealing at which operational stage hallucinations, omissions, or conflicts arise. This is particularly important for our setting because errors introduced during graph construction or memory updating may propagate to graph-guided retrieval and final answer generation.

Evaluation Rationale.

Together, these datasets form a progressively broader evaluation suite. NQ and PopQA test factual open-domain retrieval; HotpotQA, 2WikiMultiHopQA, and MuSiQue test multi-hop evidence composition; AmazonQA evaluates noisy real-world review memory in an e-commerce domain; LongMemEval tests long-horizon interactive memory; and HaluMem diagnoses operation-level hallucinations in memory systems. This combination allows us to evaluateSAGEnot merely as a retrieval-augmented QA pipeline, but as a self-evolving memory system that must decide what to store, how to organize stored information, how to retrieve it under different query conditions, and how to update or suppress unreliable memories over time.

Appendix RBaselines and Metrics

Baselines.

We evaluateSAGEagainst state-of-the-art baselines, including their combined variants, which are grouped intofourcategories:

• •Base LLM:GPT-4o-miniHurstet al.[2024].
• •Single-step RAGs:includingBM25Robertson and Walker [1994],ContrieverIzacardet al.[2021],GTRNiet al.[2022],ColBERTv2Santhanamet al.[2022],RAPTORSarthiet al.[2024], andPropositionChenet al.[2024].
• •Graph-enhanced RAGs:includingGraphRAGEdgeet al.[2024],G-RetrieverHeet al.[2024],LightRAGGuoet al.[2024],HippoRAGGutiérrezet al.[2024],HippoRAG2Gutiérrezet al.[2024],SubgraphRAGLiet al.[2024],PropRAGWang and Han [2025], and the closely relatedGFM-RAGLuoet al.[2025].
• •Multi-step RAGs:IRCoTTrivediet al.[2023],FLAREJianget al.[2023], andAdaptive-RAGJeonget al.[2024].

In particular,IRCoTTrivediet al.[2023]is a general multi-step reasoning framework that can be integrated with non-iterative retrievers, allowing both single-step RAG and graph-based methods to conduct multi-hop reasoning through interleaved retrieval and generation. TableLABEL:tab:comparepresents a comprehensive comparison between all baselines andSAGE.

Metrics.

To evaluate retrieval quality, we report Recall@2 and Recall@5 for both retrieved entities and documents, denoted as R@2/5E{}_{\textsf{E}}and R@2/5D{}_{\textsf{D}}, respectively.. For end-to-end QA evaluation, we use standard metrics, including Exact Match (EM), F1 score, Precision (P), and Recall (R), in the main experiments to comprehensively measure answer correctness and coverage.

Limitations

SAGE treats graph memory as a dynamic substrate for writing, reading, and self-evolution, but its effectiveness still depends on the quality of entity extraction, relation writing, source anchoring, and reader feedback. Errors introduced during graph construction may propagate to retrieval and final answer generation, especially in long-term memory settings involving temporal updates, conflicting user preferences, or sparse evidence. Our experiments show promising results across multi-hop QA, open-domain retrieval, review-based QA, and long-term agent-memory benchmarks, but the current system still leaves room for improvement on memory updating, high-coverage extraction, and hallucination control in more realistic deployments. The theoretical analysis also relies on assumptions such as bounded graph drift, aggregate signal propagation, and local Lipschitz stability, which provide useful intuition but may not capture all failure modes of large-scale, noisy, continuously evolving memory graphs.

Broader Impact

This work may have positive societal impact by improving the reliability and grounding of long-horizon language agents. A structure-aware and self-evolving memory system can help agents recover evidence chains from fragmented cues, maintain more consistent long-term interactions, and reduce unsupported answers in applications such as knowledge assistance, research support, customer support, and review-based question answering. At the same time, long-term agent memory raises important risks. If deployed on personal or sensitive interaction histories, such systems may store private information, infer user preferences, preserve outdated or incorrect memories, or enable profiling and surveillance. Incorrect graph writes or retrieval failures may also lead to confidently grounded but wrong answers. Practical deployments should therefore use consent-based data collection, data minimization, access control, deletion and forgetting mechanisms, provenance tracking, auditing, and human oversight for high-stakes use cases.

Compute Resources

All experiments were run on a server equipped with 8 NVIDIA A100 GPUs. The main computational cost of SAGE comes from graph-memory construction, GFM-based graph propagation, selector regularization, and entity-to-document projection. Appendix J analyzes the training and inference complexity in terms of the number of graph nodesnn, edgesmm, hidden dimensiondd, propagation layersLL, batch sizeBB, pseudo-queriesMM, and entity-document links. In our implementation, structural features and entity-document indices can be precomputed and cached, while edge-level gates are computed in chunks to reduce peak GPU memory fromO​(|E|​d)O(|E|d)toO​(Ce​d)O(C_{e}d)for chunk sizeCeC_{e}. The dominant inference cost is one or a small number of query-conditioned graph propagations followed by sparse document projection, making the reader suitable for repeated evaluation inside the self-evolving writer–reader loop.

Licenses and Existing Assets

This paper uses existing public benchmarks and baselines, including NQ-Open, PopQA, HotpotQA, 2WikiMultiHopQA, MuSiQue, AmazonQA, LongMemEval, HaluMem, BM25, Contriever, GTR, ColBERTv2, RAPTOR, GraphRAG, G-Retriever, LightRAG, HippoRAG, HippoRAG 2, SubgraphRAG, PropRAG, GFM-RAG, IRCoT, FLARE, and Adaptive-RAG. We cite the original papers or repositories for these assets and use them only for research evaluation under their stated licenses and terms of use. We do not redistribute modified versions of the datasets beyond the preprocessing scripts and instructions needed for reproducibility. The released code is intended for research use and includes documentation for environment setup, data preparation, training, and evaluation.

NeurIPS Paper Checklist

The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist:The papers not including the checklist will be desk rejected.The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.

Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:

• •You should answer[Yes],[No], or[N/A].
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The checklist answers are an integral part of your paper submission.They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will also be asked to include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.

The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While[Yes]is generally preferable to[No], it is perfectly acceptable to answer[No]provided a proper justification is given (e.g., error bars are not reported because it would be too computationally expensive” or “we were unable to find the license for the dataset we used”). In general, answering[No]or[N/A]is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer[Yes]to a question, in the justification please point to the section(s) where related material for the question can be found.

IMPORTANT, please:

• •Delete this instruction block, but keep the section heading “NeurIPS Paper Checklist“,
• •Keep the checklist subsection headings, questions/answers and guidelines below.
• •Do not modify the questions and only use the provided macros for your answers.

1. 1.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:[Yes]
4. Justification: The abstract and introduction state the scope of SAGE as a self-evolving agentic graph-memory engine for long-term memory, structure-aware retrieval, and evidence-chain recovery. The theoretical results and experiments across multi-hop QA, open-domain retrieval, review-based QA, and long-term agent-memory benchmarks support the stated contributions.
5. Guidelines: - •The answer[N/A]means that the abstract and introduction do not include the claims made in the paper. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A[No]or[N/A]answer to this question will not be perceived well by the reviewers. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
6. 2.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:[Yes]
9. Justification: The paper includes a limitations discussion covering the dependence on graph-writing quality, memory updating, domain adaptation, hallucination control, and assumptions used in the theoretical analysis.
10. Guidelines: - •The answer[N/A]means that the paper has no limitation while the answer[No]means that the paper has limitations, but those are not discussed in the paper. - •The authors are encouraged to create a separate “Limitations” section in their paper. - •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. - •The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. - •The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. - •The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. - •If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. - •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
11. 3.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
13. Answer:[Yes]
14. Justification: The paper states theoretical results in the main text and provides complete assumptions, theorem statements, lemmas, and proofs in the appendix, including analyses of signal-to-noise ratio, retrieval budget, target-graph calibration, stability, and self-evolution.
15. Guidelines: - •The answer[N/A]means that the paper does not include theoretical results. - •All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. - •All assumptions should be clearly stated or referenced in the statement of any theorems. - •The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. - •Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. - •Theorems and Lemmas that the proof relies upon should be properly referenced.
16. 4.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
18. Answer:[Yes]
19. Justification: The paper describes the architecture, training procedure, datasets, baselines, evaluation metrics, ablations, and implementation details needed to reproduce the main experimental claims. Additional details are provided in the appendix and the released code.
20. Guidelines: - •The answer[N/A]means that the paper does not include experiments. - •If the paper includes experiments, a[No]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. - •Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example 1. (a)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. 2. (b)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. 3. (c)If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). 4. (d)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
21. 5.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:[Yes]
24. Justification: The paper provides open access to the code and data instructions, together with scripts and documentation for reproducing the main experimental results.
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26. 6.Experimental setting/details
27. Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer) necessary to understand the results?
28. Answer:[Yes]
29. Justification: The paper specifies the datasets, evaluation scenarios, baselines, metrics, training procedures, and implementation details in the experimental section and appendices. Dataset statistics, reader training, writer implementation, and ablation settings are also reported.
30. Guidelines: - •The answer[N/A]means that the paper does not include experiments. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. - •The full details can be provided either with the code, in appendix, or as supplemental material.
31. 7.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:[Yes]
34. Justification: The paper reports error bars or statistical significance information for the experiments supporting the main empirical claims, and states how the variability is computed.
35. Guidelines: - •The answer[N/A]means that the paper does not include experiments. - •The authors should answer[Yes]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. - •The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). - •The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) - •The assumptions made should be given (e.g., Normally distributed errors). - •It should be clear whether the error bar is the standard deviation or the standard error of the mean. - •It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g., negative error rates). - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
36. 8.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
38. Answer:[Yes]
39. Justification: The paper reports computational complexity and compute-resource information in the Compute Resources section and Appendix J. The experiments were run on a server equipped with 8 NVIDIA A100 GPUs.
40. Guidelines: - •The answer[N/A]means that the paper does not include experiments. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn’t make it into the paper).
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45. 10.Broader impacts
46. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
47. Answer:[Yes]
48. Justification: The paper discusses positive impacts such as improving grounded long-term memory for language agents, as well as negative impacts such as privacy, profiling, surveillance, outdated memory retention, and hallucination risks.
49. Guidelines: - •The answer[N/A]means that there is no societal impact of the work performed. - •If the authors answer[N/A]or[No], they should explain why their work has no societal impact or why the paper does not address societal impact. - •Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
50. 11.Safeguards
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57. Answer:[Yes]
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77. Answer:[Yes]
78. Justification: The paper describes the use of LLMs in the system, including the memory writer, structured query planning prompts, and answer generation. This usage is part of the proposed method rather than only writing, editing, or formatting assistance.
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