AgentGFM: A Graph Foundation Model with Node-Agent Information-Flow Control

# AgentGFM: A Graph Foundation Model with Node-Agent Information-Flow Control
Source: [https://arxiv.org/html/2607.26533](https://arxiv.org/html/2607.26533)
Jingbo Cui,Jitao ZhaoSchool of Computer Science and Technology, Tianjin UniversityTianjinChina[zjtao@tju\.edu\.cn](https://arxiv.org/html/2607.26533v1/mailto:[email protected]),Di JinSchool of Computer Science and Technology, Tianjin UniversityTianjinChina[jindi@tju\.edu\.cn](https://arxiv.org/html/2607.26533v1/mailto:[email protected])andDongxiao HeSchool of Computer Science and Technology, Tianjin UniversityTianjinChina[hedongxiao@tju\.edu\.cn](https://arxiv.org/html/2607.26533v1/mailto:[email protected])

###### Abstract\.

Graph Foundation Models \(GFMs\) aim to learn transferable knowledge from multi\-domain graphs and adapt to unseen scenarios\. As a fundamental source of relational semantics in graphs, the transferability of topological patterns has long been central to GFM research\. However, local structural patterns may vary across graphs and even among nodes within the same graph\. Despite such structural variation, most existing GFMs rely on manually designed propagation schemes and apply them to new graphs largely unchanged\. Such fixed schemes may not suit the diverse structural patterns of different nodes\. This raises a key question: can each node autonomously determine how information should be propagated through the graph? We refer to this capability as information\-flow control\. Inspired by recent advances in agent technology, we formulate this problem as agent\-based decision making and treat each node as an agent\. Accordingly, we propose AgentGFM, in which all node agents follow a shared end\-to\-end trainable policy rather than using independent models\. For adaptive information\-flow control, each node interacts with the graph through a predict–act–observe–correct process\. During the act stage, the node makes three decisions: source reception, signal\-channel selection and gain\-aware node\-wise halting\. The resulting observation is compared with the prediction and their discrepancy is used to correct the node state and guide subsequent interactions\. Extensive experiments across node\-level, graph\-level and large\-scale transfer scenarios demonstrate the effectiveness of AgentGFM across diverse graph topologies\.

Graph Representation Learning, Graph Neural Networks, Graph Foundation Models

††copyright:none††conference:; ;††ccs:Computing methodologies Neural networks††ccs:Information systems Data mining††ccs:Computing methodologies Transfer learning## 1\.Introduction

Graph\-structured data are ubiquitous in real\-world applications, including social network analysis\(Wu et al\.,[2019](https://arxiv.org/html/2607.26533#bib.bib26)\), recommendation systems\(Palomares et al\.,[2021](https://arxiv.org/html/2607.26533#bib.bib16)\), protein interaction modeling\(Vlaic et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib24)\)and knowledge graph reasoning\(Schlichtkrull et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib18)\)\. However, current graph learning models often remain task\-specific, which limits their generality across domains\. Inspired by the success of foundation models in natural language processing and computer vision, recent studies have begun to explore Graph Foundation Models \(GFMs\)\. These models aim to learn generalizable graph knowledge from multi\-domain graphs, enabling rapid adaptation to new scenarios\(Liu et al\.,[2023a](https://arxiv.org/html/2607.26533#bib.bib13)\)\.

As a fundamental source of relational semantics in graphs, the transferability of topological patterns has long been central to GFM research\. However, this transferability is challenged by the fact that local structural patterns may vary across graphs and even among nodes within the same graph\. Existing studies have explored topology\-aware designs to improve generalization under such structural diversity\. Prompt\-based methods use topology\-aware prompts or structural encodings to adapt pretrained models to graph\-specific contexts\(Fang et al\.,[2023](https://arxiv.org/html/2607.26533#bib.bib7); Liu et al\.,[2023b](https://arxiv.org/html/2607.26533#bib.bib15); Yang et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib32); Yu et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib34)\)\. Other methods improve transferability through structural routing, topology alignment or adaptive aggregation\(Zhao et al\.,[2025b](https://arxiv.org/html/2607.26533#bib.bib39); Wang et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib25)\)\. Recent studies further use Riemannian representations or mixture\-of\-experts architectures to model heterogeneous topologies\(Sun et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib21); Liu et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib14),[2026](https://arxiv.org/html/2607.26533#bib.bib12)\)\. Transformer\-based GFMs also encode graph structures as tokens or graph sequences for general\-purpose modeling\(Xia et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib27); Zhao et al\.,[2025a](https://arxiv.org/html/2607.26533#bib.bib40)\)\. These efforts have advanced GFMs toward topology\-aware generalization\.

Despite this progress, most existing GFMs rely on manually designed propagation schemes and apply them to new graphs largely unchanged\. Such fixed schemes may not adapt to the diverse local structural patterns of different nodes\. As illustrated in Figure[1](https://arxiv.org/html/2607.26533#S1.F1)\(a\), a fixed scheme may stop before useful information is reached or continue to aggregate irrelevant neighboring signals\. This mismatch between uniform propagation and node\-specific structural patterns limits adaptation under topology shift\. This raises a key question:can each node autonomously determine how information should be propagated through the graph?We refer to this capability as*information\-flow control*\.

However, realizing information\-flow control is nontrivial\. Propagation decisions must adapt to node\-specific structures while remaining transferable across unseen graphs\. In addition, expanding the propagation range may capture useful structural dependencies but can also introduce irrelevant signals, making it difficult to determine an appropriate propagation process for each node\.

![Refer to caption](https://arxiv.org/html/2607.26533v1/x1.png)The figure contains two side\-by\-side diagrams\. The left diagram illustrates preset graph propagation, where a target node may fail to reach useful distant information or may aggregate misleading neighboring signals\. The right diagram illustrates AgentGFM, where a node performs a predict–act–observe–correct interaction and selectively controls information sources, signal channels and halting decisions before updating its state through feedback\.

Figure 1\.Motivation of AgentGFM\. Preset propagation may miss useful evidence or absorb noisy signals under topology shifts\. AgentGFM instead enables node\-level information\-flow control through source reception, signal\-channel selection, gain\-aware node\-wise halting and prediction–observation feedback\.Prior adaptive propagation studies have shown that graph propagation can be conditioned on graph signals rather than following a uniform routine\(Xue et al\.,[2021](https://arxiv.org/html/2607.26533#bib.bib29); Lai et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib11)\)\. However, these methods are typically optimized for a specific graph distribution or supervision objective\. They are not designed to learn transferable decision mechanisms across graphs\. Existing GFMs adapt to target graphs mainly through topology\-aware prompts or expert routing, while their propagation schemes remain largely fixed\. Thus, how to enable node\-specific propagation decisions that transfer across unseen graphs remains underexplored\.

Inspired by recent advances in agent technology, we connect this problem with agent\-based decision making, where agents adapt their actions according to internal states and environmental feedback\. This paradigm naturally aligns with graph information\-flow control, where each node should determine how information is received and propagated\. Motivated by this perspective, we treat each node as an agent that makes propagation decisions according to its current state and refines them through feedback from the graph\. Importantly, node agents do not use independent models\. Instead, all nodes follow a shared end\-to\-end trainable policy, enabling node\-specific decisions while preserving cross\-graph transferability\.

Accordingly, we propose AgentGFM, a Graph Foundation Model with node\-agent information\-flow control\. As illustrated in Figure[1](https://arxiv.org/html/2607.26533#S1.F1)\(b\), each node interacts with the graph through a predict–act–observe–correct process\. It first predicts the contextual information expected from its current state and then makes three decisions during the act stage: source reception, signal\-channel selection and gain\-aware node\-wise halting\. The graph returns an observation induced by these decisions and the discrepancy between the observation and prediction is used to correct the node state and guide subsequent interactions\. Through this process, different nodes follow distinct information\-flow trajectories under the same transferable policy\.

Our contributions are summarized as follows:

- •We identify the limitation of transferring manually designed propagation schemes to new graphs largely unchanged and formulate topology generalization as node\-agent information\-flow control\.
- •We propose AgentGFM, where each node acts as an agent under a shared trainable policy and controls information flow through a predict–act–observe–correct process with source reception, signal\-channel selection and gain\-aware halting\.
- •Extensive experiments across node\-level, graph\-level and large\-scale transfer scenarios demonstrate the effectiveness of AgentGFM across diverse graph topologies\.

## 2\.Related Work

### 2\.1\.Graph Foundation Models

Graph Foundation Models \(GFMs\) aim to learn transferable graph knowledge that generalizes across graphs\. Existing GFMs mainly address cross\-graph heterogeneity from two perspectives: feature alignment and structural adaptation\.

For feature heterogeneity, GCOPE\(Zhao et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib37)\)introduces coordinator modules for cross\-graph interaction and latent\-space alignment, MDGPT\(Yu et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib35)\)uses domain\-specific tokens to incorporate domain information, TIG\(Zhao et al\.,[2026](https://arxiv.org/html/2607.26533#bib.bib38)\)learns transfer\-invariant node features and BRIDGE\(Yuan et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib36)\)combines feature unification with semantic alignment\. These methods mainly improve transferability by learning compatible feature spaces\.

For structural heterogeneity, GraphControl\(Zhu et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib41)\)constructs structure\-aware positional encodings from spectral information, GFT\(Wang et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib25)\)learns transferable propagation patterns, R\-GFM\(Liu et al\.,[2026](https://arxiv.org/html/2607.26533#bib.bib12)\)captures geometry\-adaptive representations and GraphAny\(Zhao et al\.,[2025b](https://arxiv.org/html/2607.26533#bib.bib39)\)adopts mixture\-of\-experts to adapt encoders to different graph structures\.

Although these methods improve GFMs through feature alignment, structural encoding, or component adaptation, they mainly adapt model representations or components to target graphs\. How individual nodes should adjust their information propagation process under diverse structural conditions remains largely unexplored\. AgentGFM addresses this limitation by enabling node\-agent information\-flow control through a shared transferable policy\.

### 2\.2\.Adaptive Graph Propagation and Decision\-based Reasoning

Adaptive graph propagation and decision\-based reasoning methods learn how information should be acquired over graph structures instead of relying on fixed message\-passing rules\. Existing studies mainly adapt information sources or propagation depths\. DeepPath\(Xiong et al\.,[2017](https://arxiv.org/html/2607.26533#bib.bib28)\)and MINERVA\(Das et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib5)\)formulate knowledge graph reasoning as path navigation, GDPNet\(Xue et al\.,[2021](https://arxiv.org/html/2607.26533#bib.bib29)\)learns adaptive neighborhood selection and Policy\-GNN\(Lai et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib11)\)learns node\-specific propagation iterations\. These studies show that graph reasoning benefits from adapting where to collect information and how far to propagate\.

However, these methods are mainly optimized for a specific task, graph distribution or supervision objective and are not designed for cross\-graph transfer\. In contrast, AgentGFM learns a shared information\-flow policy from multiple source graphs and transfers it to unseen target graphs\. It jointly controls information sources, signal channels and propagation duration while using prediction–observation discrepancy for state correction\. A mechanism\-level comparison is provided in Appendix[A](https://arxiv.org/html/2607.26533#A1)\.

## 3\.Preliminaries

### 3\.1\.Problem Setup

Let𝒢=\(𝒱,ℰ,𝐗\)\\mathcal\{G\}=\(\\mathcal\{V\},\\mathcal\{E\},\\mathbf\{X\}\)denote a graph, where𝒱\\mathcal\{V\}is the node set,ℰ\\mathcal\{E\}is the edge set and𝐗∈ℝ\|𝒱\|×d\\mathbf\{X\}\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}\|\\times d\}is the node feature matrix\. We denote the adjacency matrix by𝐀\\mathbf\{A\}and the neighborhood of nodevvby𝒩​\(v\)\\mathcal\{N\}\(v\)\. Each nodev∈𝒱v\\in\\mathcal\{V\}is associated with a feature vector𝐱v∈ℝd\\mathbf\{x\}\_\{v\}\\in\\mathbb\{R\}^\{d\}\.

We consider a cross\-domain GFM setting in which the model is trained on a set of source graphs𝒢S\\mathcal\{G\}\_\{S\}and evaluated on a disjoint set of target graphs𝒢T\\mathcal\{G\}\_\{T\}, where𝒢S∩𝒢T=∅\\mathcal\{G\}\_\{S\}\\cap\\mathcal\{G\}\_\{T\}=\\varnothing\. The objective is to learn transferable graph knowledge from𝒢S\\mathcal\{G\}\_\{S\}and generalize to unseen graphs in𝒢T\\mathcal\{G\}\_\{T\}with limited supervision\.

### 3\.2\.Information\-Flow Control Formulation

Let𝐡v\(k\)\\mathbf\{h\}\_\{v\}^\{\(k\)\}denote the representation of nodevvat propagation stepkk\. Conventional message passing first aggregates neighboring representations:

\(1\)𝐦v\(k\)=AGG\(k\)⁡\(\{𝐡u\(k\)∣u∈𝒩​\(v\)\}\),\\mathbf\{m\}\_\{v\}^\{\(k\)\}=\\operatorname\{AGG\}^\{\(k\)\}\\left\(\\left\\\{\\mathbf\{h\}\_\{u\}^\{\(k\)\}\\mid u\\in\\mathcal\{N\}\(v\)\\right\\\}\\right\),and then updates the node representation:

\(2\)𝐡v\(k\+1\)=ϕ\(k\)​\(𝐡v\(k\),𝐦v\(k\)\),\\mathbf\{h\}\_\{v\}^\{\(k\+1\)\}=\\phi^\{\(k\)\}\\left\(\\mathbf\{h\}\_\{v\}^\{\(k\)\},\\mathbf\{m\}\_\{v\}^\{\(k\)\}\\right\),whereAGG\(k\)\\operatorname\{AGG\}^\{\(k\)\}andϕ\(k\)\\phi^\{\(k\)\}are specified by the model architecture\. Consequently, the same propagation mechanism is typically applied to different nodes and transferred to new graphs with limited adaptation\.

To support node\-specific propagation, we formulate information\-flow control as a policy\-learning problem\. At stepkk, nodevvconstructs a state

\(3\)𝐬v\(k\)=fs​\(𝐡v\(k\),𝐦v\(k\)\),\\mathbf\{s\}\_\{v\}^\{\(k\)\}=f\_\{s\}\\left\(\\mathbf\{h\}\_\{v\}^\{\(k\)\},\\mathbf\{m\}\_\{v\}^\{\(k\)\}\\right\),and a shared policy produces a node\-specific decision:

\(4\)𝐝v\(k\)=πθ​\(𝐬v\(k\)\)\.\\mathbf\{d\}\_\{v\}^\{\(k\)\}=\\pi\_\{\\theta\}\\left\(\\mathbf\{s\}\_\{v\}^\{\(k\)\}\\right\)\.The representation is then updated through a decision\-conditioned transition:

\(5\)𝐡v\(k\+1\)=ft​\(𝐡v\(k\),𝐦v\(k\),𝐝v\(k\)\)\.\\mathbf\{h\}\_\{v\}^\{\(k\+1\)\}=f\_\{t\}\\left\(\\mathbf\{h\}\_\{v\}^\{\(k\)\},\\mathbf\{m\}\_\{v\}^\{\(k\)\},\\mathbf\{d\}\_\{v\}^\{\(k\)\}\\right\)\.This formulation allows different nodes to execute different propagation decisions while sharing the same policy parameters\. AgentGFM provides a concrete realization of this formulation in Section[4](https://arxiv.org/html/2607.26533#S4)\.

## 4\.Method

### 4\.1\.Overview of AgentGFM

Given graphs from different domains, AgentGFM first aligns their node features into a shared space using truncated SVD\. The aligned features are then processed by a shared encoder in which each node acts as an agent for information\-flow control, as illustrated in Figure[2](https://arxiv.org/html/2607.26533#S4.F2)\. The encoder operates through a recurrent predict–act–observe–correct process\. The process is organized into a small number of outer interaction rounds\. Across rounds, each node updates its state using the feedback from the preceding prediction–observation comparison, while the act stage performs an inner information\-flow rollout over the graph\. At each round, a node first predicts the information expected from its current state\. It then makes propagation decisions, receives the resulting information from the graph and updates its state according to the discrepancy between the prediction and observation\. All nodes follow the same end\-to\-end trainable policy while making node\-specific decisions\. The following subsections introduce the node\-level actions, observation construction, state correction and training objectives\.

### 4\.2\.Node\-Agent Information\-Flow Control

AgentGFM realizes information\-flow control through interactions between node agents and the graph\. Each node acts as an agent, while the graph serves as the environment that returns contextual information in response to the node’s actions\.

A node agent is not an independent model assigned to a node\. Instead, it is a node\-specific execution of a shared information\-flow policy\. All nodes share the same policy parameters, while their states, actions, observations and halting decisions remain node\-specific\.

For nodeviv\_\{i\}, we define its agent state at rollout stepttas

\(6\)𝐬it=𝐜it,\\mathbf\{s\}\_\{i\}^\{t\}=\\mathbf\{c\}\_\{i\}^\{t\},where𝐜it∈ℝd\\mathbf\{c\}\_\{i\}^\{t\}\\in\\mathbb\{R\}^\{d\}is the carrier state,dddenotes the hidden representation dimension and𝐜i0=𝐡i\\mathbf\{c\}\_\{i\}^\{0\}=\\mathbf\{h\}\_\{i\}\. Given𝐬it\\mathbf\{s\}\_\{i\}^\{t\}, the shared policy produces

\(7\)𝐝it=\(𝜸it,𝝆it,zit\),\\mathbf\{d\}\_\{i\}^\{t\}=\\left\(\\boldsymbol\{\\gamma\}\_\{i\}^\{t\},\\boldsymbol\{\\rho\}\_\{i\}^\{t\},z\_\{i\}^\{t\}\\right\),where𝜸it\\boldsymbol\{\\gamma\}\_\{i\}^\{t\}and𝝆it\\boldsymbol\{\\rho\}\_\{i\}^\{t\}represent edge\-wise source\-reception and signal\-channel decisions, whilezitz\_\{i\}^\{t\}denotes node\-wise halting\.

The node agent repeatedly executes these decisions in the graph environment\. At each rollout step, its actions determine how information is received and processed from neighboring nodes\. The rollout terminates whenzitz\_\{i\}^\{t\}indicates halting or when the maximum rollout length is reached\.

After the rollout, nodeviv\_\{i\}obtains an observation𝐨i\\mathbf\{o\}\_\{i\}that summarizes the contextual information returned by the graph environment\. This observation is compared with the predicted observation𝐨^i\\widehat\{\\mathbf\{o\}\}\_\{i\}and their discrepancy is used to correct the node representation:

\(8\)𝐡i′=𝒯​\(𝐡i,𝐨^i,𝐨i\),\\mathbf\{h\}\_\{i\}^\{\\prime\}=\\mathcal\{T\}\\left\(\\mathbf\{h\}\_\{i\},\\widehat\{\\mathbf\{o\}\}\_\{i\},\\mathbf\{o\}\_\{i\}\\right\),where𝒯​\(⋅\)\\mathcal\{T\}\(\\cdot\)denotes the state\-correction function and𝐡i′\\mathbf\{h\}\_\{i\}^\{\\prime\}is the corrected representation of nodeviv\_\{i\}\.

![Refer to caption](https://arxiv.org/html/2607.26533v1/fig/method.jpg)

Overview of AgentGFM\. Each node acts as an agent that performs predictive observation, information\-flow decisions, graph interaction and feedback\-based state correction\.

Figure 2\.Overall architecture of AgentGFM\. Multi\-domain node attributes are first aligned into a shared feature space\. Each node then performs a predict–act–observe–correct interaction, where information\-flow actions control source reception, signal\-channel selection and halting\. The encoder is pretrained with masked reconstruction and prediction\-observation alignment objectives and transferred to downstream tasks\.
### 4\.3\.Predictive Observation Estimation

At the beginning of each interaction round, nodeviv\_\{i\}predicts the observation expected from the graph environment based on its current state\. This prediction provides a reference for evaluating the information returned after the node executes its actions\.

Given the current state𝐬i=𝐡i\\mathbf\{s\}\_\{i\}=\\mathbf\{h\}\_\{i\}, the predicted observation is computed as

\(9\)𝐨^i=fpred​\(𝐡i\),\\widehat\{\\mathbf\{o\}\}\_\{i\}=f\_\{\\mathrm\{pred\}\}\\left\(\\mathbf\{h\}\_\{i\}\\right\),wherefpred:ℝd→ℝdf\_\{\\mathrm\{pred\}\}:\\mathbb\{R\}^\{d\}\\rightarrow\\mathbb\{R\}^\{d\}is a trainable prediction function\.𝐨^i\\widehat\{\\mathbf\{o\}\}\_\{i\}denotes the observation that nodeviv\_\{i\}expects to receive from the graph environment\.

After the node acts, the graph environment returns an actual observation𝐨i\\mathbf\{o\}\_\{i\}\. AgentGFM compares𝐨^i\\widehat\{\\mathbf\{o\}\}\_\{i\}with𝐨i\\mathbf\{o\}\_\{i\}to measure their discrepancy\. This discrepancy reflects the mismatch between the node’s current expectation and the information returned by the graph\. It is used as feedback to correct the node state and guide the propagation decision in the next interaction round\.

### 4\.4\.Policy\-Conditioned Information\-Flow Rollout

After predictive observation estimation, each node interacts with the graph environment through a policy\-conditioned information\-flow rollout over the original topology\. At rollout steptt, nodeviv\_\{i\}decides which sources to receive information from and which signal channel to use for message transmission\.

Let𝐜it\\mathbf\{c\}\_\{i\}^\{t\}denote the carrier state of nodeviv\_\{i\}at steptt, with𝐜i0=𝐡i\\mathbf\{c\}\_\{i\}^\{0\}=\\mathbf\{h\}\_\{i\}\. Each node maintains a forwarding budgetbit∈\[0,1\]b\_\{i\}^\{t\}\\in\[0,1\]and an activity variableaita\_\{i\}^\{t\}\. The forwarding budget controls the information transmitted by the source node, whileaita\_\{i\}^\{t\}indicates whether nodeviv\_\{i\}remains active during the rollout\. The activity variable follows a binary halting trajectory in the forward pass and is optimized through the straight\-through estimator introduced in Section[4\.5](https://arxiv.org/html/2607.26533#S4.SS5)\.

For each original edge\(u,i\)∈ℰ\(u,i\)\\in\\mathcal\{E\}, AgentGFM computes a source\-reception score:

\(10\)γu→it=σ​\(gγ​\(𝐜ut,𝐜it,ϵu,ϵi,but\)\),\\gamma\_\{u\\to i\}^\{t\}=\\sigma\\left\(g\_\{\\gamma\}\\left\(\\mathbf\{c\}\_\{u\}^\{t\},\\mathbf\{c\}\_\{i\}^\{t\},\\epsilon\_\{u\},\\epsilon\_\{i\},b\_\{u\}^\{t\}\\right\)\\right\),whereϵu\\epsilon\_\{u\}andϵi\\epsilon\_\{i\}denote feedback cues from the preceding prediction–observation interaction\. The scoreγu→it\\gamma\_\{u\\to i\}^\{t\}determines how strongly nodeviv\_\{i\}receives information from source nodevuv\_\{u\}under the current rollout state\. The effective transmission strength along edge\(u,i\)\(u,i\)is defined as

\(11\)μu→it=ait​but​γu→it\.\\mu\_\{u\\to i\}^\{t\}=a\_\{i\}^\{t\}b\_\{u\}^\{t\}\\gamma\_\{u\\to i\}^\{t\}\.Accordingly,μu→it∈\[0,1\]\\mu\_\{u\\rightarrow i\}^\{t\}\\in\[0,1\]is a scalar effective transmission strength\. Thus, an active target node receives information according to the source\-reception decision, while a source node transmits information according to its forwarding budget\.

To capture complementary structural signals, AgentGFM constructs low\- and high\-frequency channels\. Letγu→it−1\\gamma\_\{u\\to i\}^\{t\-1\}denote the source\-reception score from the preceding rollout step\. Fort\>0t\>0, these scores are normalized over the incoming neighbors:

\(12\)αu→it=sg​\(γu→it−1\)∑v∈𝒩​\(i\)sg​\(γv→it−1\)\+ε,\\alpha\_\{u\\to i\}^\{t\}=\\frac\{\\mathrm\{sg\}\\left\(\\gamma\_\{u\\to i\}^\{t\-1\}\\right\)\}\{\\sum\_\{v\\in\\mathcal\{N\}\(i\)\}\\mathrm\{sg\}\\left\(\\gamma\_\{v\\to i\}^\{t\-1\}\\right\)\+\\varepsilon\},wheresg​\(⋅\)\\mathrm\{sg\}\(\\cdot\)denotes stop\-gradient andε\>0\\varepsilon\>0is a small constant for numerical stability\. At the first rollout step, the weights are initialized uniformly over neighboring nodes\.

Using these policy\-conditioned weights, the local low\- and high\-frequency components are computed as

\(13\)𝐜i,lowt=∑u∈𝒩​\(i\)αu→it​𝐜ut,𝐜i,hight=𝐜it−𝐜i,lowt\.\\mathbf\{c\}\_\{i,\\mathrm\{low\}\}^\{t\}=\\sum\_\{u\\in\\mathcal\{N\}\(i\)\}\\alpha\_\{u\\to i\}^\{t\}\\mathbf\{c\}\_\{u\}^\{t\},\\quad\\mathbf\{c\}\_\{i,\\mathrm\{high\}\}^\{t\}=\\mathbf\{c\}\_\{i\}^\{t\}\-\\mathbf\{c\}\_\{i,\\mathrm\{low\}\}^\{t\}\.The low\-frequency component summarizes contextual information from the selected sources, while the high\-frequency component preserves the deviation of the node state from its local context\.

For source nodevuv\_\{u\}, AgentGFM constructs two candidate messages:

\(14\)𝐦u,lowt=𝐖low​𝐜u,lowt,𝐦u,hight=𝐖high​𝐜u,hight\.\\mathbf\{m\}\_\{u,\\mathrm\{low\}\}^\{t\}=\\mathbf\{W\}\_\{\\mathrm\{low\}\}\\mathbf\{c\}\_\{u,\\mathrm\{low\}\}^\{t\},\\quad\\mathbf\{m\}\_\{u,\\mathrm\{high\}\}^\{t\}=\\mathbf\{W\}\_\{\\mathrm\{high\}\}\\mathbf\{c\}\_\{u,\\mathrm\{high\}\}^\{t\}\.
The edge\-specific channel gateρu→it\\rho\_\{u\\to i\}^\{t\}is computed from source–target compatibility, feedback cues and the source forwarding budget and is mapped into\(0,1\)\(0,1\)through a sigmoid function\. It combines the low\- and high\-frequency messages as

\(15\)𝐦u→it=ρu→it​𝐦u,lowt\+\(1−ρu→it\)​λhigh​𝐦u,hight,\\mathbf\{m\}\_\{u\\to i\}^\{t\}=\\rho\_\{u\\to i\}^\{t\}\\mathbf\{m\}\_\{u,\\mathrm\{low\}\}^\{t\}\+\\left\(1\-\\rho\_\{u\\to i\}^\{t\}\\right\)\\lambda\_\{\\mathrm\{high\}\}\\mathbf\{m\}\_\{u,\\mathrm\{high\}\}^\{t\},whereλhigh\\lambda\_\{\\mathrm\{high\}\}controls the contribution of the high\-frequency channel\.

Finally, the contextual information received by nodeviv\_\{i\}at rollout stepttis

\(16\)𝐫it=∑u∈𝒩​\(i\)μu→it​𝐦u→it∈ℝd\.\\mathbf\{r\}\_\{i\}^\{t\}=\\sum\_\{u\\in\\mathcal\{N\}\(i\)\}\\mu\_\{u\\rightarrow i\}^\{t\}\\mathbf\{m\}\_\{u\\rightarrow i\}^\{t\}\\in\\mathbb\{R\}^\{d\}\.

### 4\.5\.Gain\-Aware Halting and Observation Construction

After receiving the contextual signal𝐫it\\mathbf\{r\}\_\{i\}^\{t\}, nodeviv\_\{i\}constructs a candidate carrier state:

\(17\)𝐜~it\+1=Norm2⁡\(𝐜it\+fc​\(\[𝐜it∥𝐫it\]\)\)∈ℝd\.\\widetilde\{\\mathbf\{c\}\}\_\{i\}^\{t\+1\}=\\operatorname\{Norm\}\_\{2\}\\left\(\\mathbf\{c\}\_\{i\}^\{t\}\+f\_\{c\}\\left\(\\left\[\\mathbf\{c\}\_\{i\}^\{t\}\\\|\\mathbf\{r\}\_\{i\}^\{t\}\\right\]\\right\)\\right\)\\in\\mathbb\{R\}^\{d\}\.
To estimate whether the current information\-flow step provides useful additional information, we introduce a local gain predictorfgain​\(⋅\)f\_\{\\mathrm\{gain\}\}\(\\cdot\)\. It measures the discrepancy between the received context and the predicted context before and after state refinement:

\(18\)di,−t\\displaystyle d\_\{i,\-\}^\{t\}=1d​‖𝐫it−fgain​\(𝐜it\)‖1,\\displaystyle=\\frac\{1\}\{d\}\\left\\\|\\mathbf\{r\}\_\{i\}^\{t\}\-f\_\{\\mathrm\{gain\}\}\\left\(\\mathbf\{c\}\_\{i\}^\{t\}\\right\)\\right\\\|\_\{1\},di,\+t\\displaystyle d\_\{i,\+\}^\{t\}=1d​‖𝐫it−fgain​\(𝐜~it\+1\)‖1,\\displaystyle=\\frac\{1\}\{d\}\\left\\\|\\mathbf\{r\}\_\{i\}^\{t\}\-f\_\{\\mathrm\{gain\}\}\\left\(\\widetilde\{\\mathbf\{c\}\}\_\{i\}^\{t\+1\}\\right\)\\right\\\|\_\{1\},git\\displaystyle g\_\{i\}^\{t\}=di,−t−di,\+t\.\\displaystyle=d\_\{i,\-\}^\{t\}\-d\_\{i,\+\}^\{t\}\.
The resulting predictive gaingit∈ℝg\_\{i\}^\{t\}\\in\\mathbb\{R\}is node\-specific\. A positive gain indicates that the candidate update better captures the received context, while a small or negative gain suggests limited benefit from further information acquisition\.

Based on the current carrier state, received context, predictive gain, forwarding budget and rollout\-step embedding, a halting policy generates the halting score:

\(19\)κit=fhalt​\(𝐜it,𝐫it,𝐜~it\+1,git,bit,𝐞t\),\\kappa\_\{i\}^\{t\}=f\_\{\\mathrm\{halt\}\}\\left\(\\mathbf\{c\}\_\{i\}^\{t\},\\mathbf\{r\}\_\{i\}^\{t\},\\widetilde\{\\mathbf\{c\}\}\_\{i\}^\{t\+1\},g\_\{i\}^\{t\},b\_\{i\}^\{t\},\\mathbf\{e\}\_\{t\}\\right\),where𝐞t∈ℝ4\\mathbf\{e\}\_\{t\}\\in\\mathbb\{R\}^\{4\}denotes a learnable embedding of the current rollout step andκit∈\(0,1\)\\kappa\_\{i\}^\{t\}\\in\(0,1\)is the scalar halting score\.

To retain discrete node\-wise execution while enabling end\-to\-end optimization, we apply a straight\-through estimator:

\(20\)zi,ht\\displaystyle z\_\{i,\\mathrm\{h\}\}^\{t\}=𝕀​\(κit≥θ0\),\\displaystyle=\\mathbb\{I\}\\left\(\\kappa\_\{i\}^\{t\}\\geq\\theta\_\{0\}\\right\),zit\\displaystyle z\_\{i\}^\{t\}=zi,ht\+κit−sg⁡\(κit\),\\displaystyle=z\_\{i,\\mathrm\{h\}\}^\{t\}\+\\kappa\_\{i\}^\{t\}\-\\operatorname\{sg\}\\left\(\\kappa\_\{i\}^\{t\}\\right\),ait\+1\\displaystyle a\_\{i\}^\{t\+1\}=ait​\(1−zit\)\.\\displaystyle=a\_\{i\}^\{t\}\\left\(1\-z\_\{i\}^\{t\}\\right\)\.
The carrier state is updated according to the activity status at the current rollout step:

\(21\)𝐜it\+1=ait​𝐜~it\+1\+\(1−ait\)​𝐜it\.\\mathbf\{c\}\_\{i\}^\{t\+1\}=a\_\{i\}^\{t\}\\widetilde\{\\mathbf\{c\}\}\_\{i\}^\{t\+1\}\+\\left\(1\-a\_\{i\}^\{t\}\\right\)\\mathbf\{c\}\_\{i\}^\{t\}\.
The forward pass follows the binary halting decision, while gradients are propagated through the continuous halting score during optimization\. An active node incorporates the contextual information received at the current rollout step\. Once a node halts, its carrier state remains unchanged in subsequent rollout steps, while the frozen carrier can still provide source information for active nodes\.

Finally, the observation of nodeviv\_\{i\}is constructed by aggregating the contextual information collected throughout its node\-specific information\-flow rollout:

\(22\)𝐨i=ReLU⁡\(LN⁡\(∑t=0T−1𝐫it\)\)\.\\mathbf\{o\}\_\{i\}=\\operatorname\{ReLU\}\\left\(\\operatorname\{LN\}\\left\(\\sum\_\{t=0\}^\{T\-1\}\\mathbf\{r\}\_\{i\}^\{t\}\\right\)\\right\)\.

### 4\.6\.Feedback\-Gated State Correction

After obtaining the actual observation from the graph environment, each node compares it with the observation predicted from its current state\. Their discrepancy reflects how far the returned information deviates from the node’s expectation\. AgentGFM uses this discrepancy to regulate the contribution of the actual observation during state correction\.

We define the prediction–observation discrepancy as

\(23\)𝜹i=\|𝐨i−𝐨^i\|∈ℝd,\\boldsymbol\{\\delta\}\_\{i\}=\\left\|\\mathbf\{o\}\_\{i\}\-\\widehat\{\\mathbf\{o\}\}\_\{i\}\\right\|\\in\\mathbb\{R\}^\{d\},where𝐨^i\\widehat\{\\mathbf\{o\}\}\_\{i\}is the predicted observation and𝐨i\\mathbf\{o\}\_\{i\}is the actual observation returned by the graph environment\. This discrepancy is used for the current state correction and can also provide feedback for subsequent interactions\. Based on𝜹i\\boldsymbol\{\\delta\}\_\{i\}, AgentGFM computes a correction gate:

\(24\)𝐪i=σ​\(frel​\(𝐡i,𝐨^i,𝐨i,𝜹i\)\)∈\(0,1\)d,\\mathbf\{q\}\_\{i\}=\\sigma\\left\(f\_\{\\mathrm\{rel\}\}\(\\mathbf\{h\}\_\{i\},\\widehat\{\\mathbf\{o\}\}\_\{i\},\\mathbf\{o\}\_\{i\},\\boldsymbol\{\\delta\}\_\{i\}\)\\right\)\\in\(0,1\)^\{d\},where𝐪i\\mathbf\{q\}\_\{i\}is a feature\-wise correction gate that controls the contribution of each dimension of the actual graph observation\.

The corrected observation is obtained through gated fusion:

\(25\)𝐨~i=𝐪i⊙𝐨i\+\(1−𝐪i\)⊙𝐨^i\.\\widetilde\{\\mathbf\{o\}\}\_\{i\}=\\mathbf\{q\}\_\{i\}\\odot\\mathbf\{o\}\_\{i\}\+\\left\(1\-\\mathbf\{q\}\_\{i\}\\right\)\\odot\\widehat\{\\mathbf\{o\}\}\_\{i\}\.The node state is then updated through a residual correction:

\(26\)𝐡i′=LN⁡\(𝐡i\+Dropout⁡\(𝐨~i\)\),\\mathbf\{h\}^\{\\prime\}\_\{i\}=\\operatorname\{LN\}\\left\(\\mathbf\{h\}\_\{i\}\+\\operatorname\{Dropout\}\(\\widetilde\{\\mathbf\{o\}\}\_\{i\}\)\\right\),whereLN​\(⋅\)\\mathrm\{LN\}\(\\cdot\)denotes layer normalization\.

This feedback\-gated correction allows the node to balance the actual graph observation with its prediction before updating the state\. Unlike directly weighting messages by their transmission strength, the correction gate is conditioned on the discrepancy between predicted and observed information, making the state update explicitly feedback\-driven\.

Table 1\.Cross\-domain 1\-shot node classification accuracy \(%\)\. The best result on each dataset is highlighted in bold\. Avg\. Rank denotes the average ranking across all datasets, where a lower value is better\.A comparison of fourteen graph learning methods across ten node classification datasets under the cross\-domain 1\-shot protocol\. Methods are grouped into task\-supervised GNNs, self\-supervised pretraining methods and graph foundation models\. AgentGFM achieves the best average rank and the highest accuracy on nine datasets\.
### 4\.7\.Training Objective and Optimization

AgentGFM is pretrained on source graphs using self\-supervised objectives\. Given a masked node setℳ\\mathcal\{M\}, the masked attribute reconstruction objective is

\(27\)ℒmae=1\|ℳ\|​∑vi∈ℳ‖𝐱i−fdec​\(𝐡i′\)‖1\.\\mathcal\{L\}\_\{\\mathrm\{mae\}\}=\\frac\{1\}\{\|\\mathcal\{M\}\|\}\\sum\_\{v\_\{i\}\\in\\mathcal\{M\}\}\\left\\\|\\mathbf\{x\}\_\{i\}\-f\_\{\\mathrm\{dec\}\}\(\\mathbf\{h\}\_\{i\}^\{\\prime\}\)\\right\\\|\_\{1\}\.This objective encourages the model to recover masked attributes using contextual information from the graph rather than relying only on visible node features\.

To train predictive observation estimation, we align the predicted observation with the actual observation returned by the graph environment:

\(28\)ℒpred=1\|𝒱\|​∑vi∈𝒱\(1−𝐨^i⊤​sg⁡\(𝐨i\)‖𝐨^i‖2​‖𝐨i‖2\+ε\),\\mathcal\{L\}\_\{\\mathrm\{pred\}\}=\\frac\{1\}\{\|\\mathcal\{V\}\|\}\\sum\_\{v\_\{i\}\\in\\mathcal\{V\}\}\\left\(1\-\\frac\{\\widehat\{\\mathbf\{o\}\}\_\{i\}^\{\\top\}\\operatorname\{sg\}\(\\mathbf\{o\}\_\{i\}\)\}\{\\\|\\widehat\{\\mathbf\{o\}\}\_\{i\}\\\|\_\{2\}\\\|\\mathbf\{o\}\_\{i\}\\\|\_\{2\}\+\\varepsilon\}\\right\),wheresg⁡\(⋅\)\\operatorname\{sg\}\(\\cdot\)denotes stop\-gradient\.

To prevent excessive information propagation, we regularize source reception, effective transmission strength and rollout length:

\(29\)ℒγ=𝔼t,\(u,i\)∈ℰ​\[γu→it\],ℒμ=𝔼t,\(u,i\)∈ℰ​\[μu→it\],ℒa=𝔼t,i​\[ait\+1\]\.\\mathcal\{L\}\_\{\\gamma\}=\\mathbb\{E\}\_\{t,\(u,i\)\\in\\mathcal\{E\}\}\\left\[\\gamma\_\{u\\rightarrow i\}^\{t\}\\right\],\\quad\\mathcal\{L\}\_\{\\mu\}=\\mathbb\{E\}\_\{t,\(u,i\)\\in\\mathcal\{E\}\}\\left\[\\mu\_\{u\\rightarrow i\}^\{t\}\\right\],\\quad\\mathcal\{L\}\_\{a\}=\\mathbb\{E\}\_\{t,i\}\\left\[a\_\{i\}^\{t\+1\}\\right\]\.These terms discourage uniformly strong source reception, excessive information transmission and unnecessarily long rollouts, respectively\.

The final training objective is

\(30\)ℒ=λmae​ℒmae\+λpred​ℒpred\+λγ​ℒγ\+λμ​ℒμ\+λa​ℒa\.\\mathcal\{L\}=\\lambda\_\{\\mathrm\{mae\}\}\\mathcal\{L\}\_\{\\mathrm\{mae\}\}\+\\lambda\_\{\\mathrm\{pred\}\}\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\+\\lambda\_\{\\gamma\}\\mathcal\{L\}\_\{\\gamma\}\+\\lambda\_\{\\mu\}\\mathcal\{L\}\_\{\\mu\}\+\\lambda\_\{a\}\\mathcal\{L\}\_\{a\}\.
##### Implementation of learnable components\.

Observation prediction, carrier updating, gain estimation, gain\-aware halting and reliability gating are implemented using lightweight two\-layer MLPs, while the edge\-level source\-reception and low\-/high\-frequency channel gates combine source–target query–key projections with scalar MLPs\. The detailed parameterization, normalization operations and parameter\-sharing scheme are provided in Appendix[B\.3](https://arxiv.org/html/2607.26533#A2.SS3)\.

## 5\.Experiments

### 5\.1\.Experimental Setup

#### 5\.1\.1\.Datasets and Tasks

We evaluate AgentGFM on the cross\-domain 1\-shot setting\. For node classification, we use ten datasets, including Texas, Wisconsin, Cornell, Chameleon, Squirrel\(Pei et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib17)\), Cora, CiteSeer, PubMed\(Yang et al\.,[2016](https://arxiv.org/html/2607.26533#bib.bib31)\), Computers and Photo\(Shchur et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib19)\)\. These datasets cover diverse domains and topological patterns, including both homophilous and heterophilous graphs\. For graph classification, we evaluate on MUTAG, DD\(Shervashidze et al\.,[2011](https://arxiv.org/html/2607.26533#bib.bib20)\), IMDB\-BINARY\(Yanardag and Vishwanathan,[2015](https://arxiv.org/html/2607.26533#bib.bib30)\), ENZYMES\(Borgwardt et al\.,[2005](https://arxiv.org/html/2607.26533#bib.bib3)\)and PROTEINS\(Dobson and Doig,[2003](https://arxiv.org/html/2607.26533#bib.bib6)\)\. To further examine scalability, we conduct large\-scale node classification experiments on Physics\(Shchur et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib19)\), Ogbn\-Products and Ogbn\-Arxiv\(Hu et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib9)\)\.

#### 5\.1\.2\.Evaluation Protocol

We adopt a leave\-one\-dataset\-out cross\-domain 1\-shot protocol\. For each target dataset, the model is pretrained only on the remaining source datasets, ensuring that the target graph is unseen during pretraining\. For downstream evaluation, we randomly sample one labeled instance per class as the support set and use its representation as the class prototype\. Each remaining instance is assigned to the prototype with the highest cosine similarity\. We independently sample the support set 100 times and report the mean accuracy and standard deviation\. For graph classification, graph representations are obtained by mean pooling node representations and one labeled graph per class is used to construct the prototypes\. All methods are evaluated under identical source–target splits and 1\-shot settings\.

#### 5\.1\.3\.Baselines

We compare AgentGFM with three groups of methods\. The first group includes task\-supervised GNNs, such as GCN\(Kipf and Welling,[2017](https://arxiv.org/html/2607.26533#bib.bib10)\), GAT\(Velickovic et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib22)\), FAGCN\(Bo et al\.,[2021](https://arxiv.org/html/2607.26533#bib.bib2)\)and GPRGNN\(Chien et al\.,[2021](https://arxiv.org/html/2607.26533#bib.bib4)\)\. The second group includes self\-supervised graph pretraining methods, such as DGI\(Velickovic et al\.,[2019](https://arxiv.org/html/2607.26533#bib.bib23)\), GraphCL\(You et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib33)\)and GraphMAE\(Hou et al\.,[2022](https://arxiv.org/html/2607.26533#bib.bib8)\)\. The third group consists of GFMs, including SAMGPT\(Yu et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib34)\), BRIDGE\(Yuan et al\.,[2025](https://arxiv.org/html/2607.26533#bib.bib36)\), R\-GFM\(Liu et al\.,[2026](https://arxiv.org/html/2607.26533#bib.bib12)\), GCOPE\(Zhao et al\.,[2024](https://arxiv.org/html/2607.26533#bib.bib37)\)and GraphAny\(Zhao et al\.,[2025b](https://arxiv.org/html/2607.26533#bib.bib39)\)\. All methods are evaluated under the same protocol\.

### 5\.2\.Performance Analysis

#### 5\.2\.1\.Cross\-Domain 1\-Shot Node Classification

Table[1](https://arxiv.org/html/2607.26533#S4.T1)reports the cross\-domain 1\-shot node classification results\. AgentGFM achieves the best average rank of 1\.10 across ten target datasets\. It obtains the best performance on nine datasets and remains competitive on CiteSeer\. The improvement is especially clear on heterophilous graphs, where fixed propagation rules are more likely to absorb misleading neighboring signals\. Baselines exhibit unstable rankings across target datasets, especially under heterophilous settings\. In contrast, AgentGFM remains consistently competitive across both homophilous and heterophilous graphs, demonstrating the robustness of topology\-adaptive node\-agent information\-flow control under diverse structural patterns\.

Table 2\.Cross\-domain 1\-shot graph classification accuracy \(%\)\.Table 3\.Cross\-domain 1\-shot node classification accuracy \(%\) on large\-scale datasets\. The best result on each dataset is highlighted in bold\.Cross\-domain one\-shot node classification accuracy comparison across three large\-scale node classification datasets\.
#### 5\.2\.2\.Cross\-Domain 1\-Shot Graph Classification

Table[2](https://arxiv.org/html/2607.26533#S5.T2)reports the cross\-domain 1\-shot graph classification results\. AgentGFM ranks first on all five datasets and achieves the best average rank of 1\.00\. Although the margins over the strongest baselines are modest on some datasets, the consistent improvements across molecular, bioinformatics and social graph benchmarks demonstrate that the transferability of AgentGFM extends to graph\-level tasks\. We also observe that the strongest baseline varies across datasets, whereas AgentGFM maintains consistently strong performance, suggesting that node\-agent information\-flow control provides stable graph\-level transfer\.

#### 5\.2\.3\.Cross\-Domain 1\-Shot Large\-Scale Evaluation

Table[3](https://arxiv.org/html/2607.26533#S5.T3)reports the results on large\-scale node classification datasets\. AgentGFM is pretrained on eight source graphs, including Cora, CiteSeer, PubMed, Computers, Photo, Texas, Wisconsin and Cornell and then transferred to Physics, Ogbn\-Products and Ogbn\-Arxiv\. AgentGFM achieves the best results on Ogbn\-Products and Ogbn\-Arxiv and remains competitive on Physics\. Notably, R\-GFM runs out of memory on Ogbn\-Products and Ogbn\-Arxiv, whereas AgentGFM completes evaluation on both datasets\. AgentGFM performs its rollout over the original edge set without explicitly constructing dense higher\-order neighborhoods, which helps retain practical scalability on large graphs\. These results demonstrate strong cross\-domain transfer performance on large graphs\.

### 5\.3\.Ablation Study

![Refer to caption](https://arxiv.org/html/2607.26533v1/x2.png)Bar chart showing component\-wise ablation results of AgentGFM on representative cross\-domain node classification datasets\. The full model is compared with variants removing source reception, signal\-channel selection, feedback, or halting control\.

Figure 3\.Mechanism\-aligned ablation study on cross\-domain 1\-shot node classification\.![Refer to caption](https://arxiv.org/html/2607.26533v1/fig/node_stop_distribution.png)Distribution of node\-specific stopping steps under the maximum rollout horizon\. The figure shows the proportion of nodes terminating at different propagation steps across multiple datasets\.

Figure 4\.Distribution of node\-specific halting steps in AgentGFM\.#### 5\.3\.1\.Mechanism\-Aligned Ablation

Following the mechanism\-level comparison in Table[6](https://arxiv.org/html/2607.26533#A1.T6), we evaluate four ablations that isolate the key control mechanisms of AgentGFM:w/o Source\-Reception Control,w/o Channel Selection,w/o Feedbackandw/o Gain\-Aware Halting\. These variants remove node\-specific source reception, signal\-channel selection, prediction–observation feedback and adaptive node\-wise halting respectively\. Figure[3](https://arxiv.org/html/2607.26533#S5.F3)presents the results on four representative datasets, while the complete results on all ten node classification datasets are reported in Appendix[C](https://arxiv.org/html/2607.26533#A3)\.

As shown in Figure[3](https://arxiv.org/html/2607.26533#S5.F3), the full model achieves the best performance on all four datasets\. Removing Source\-Reception Control consistently reduces accuracy, supporting the contribution of node\-specific reception\. Removing Channel Selection also degrades performance, with clearer effects on Texas and Cornell, indicating the benefit of adapting signal channels across different graph structures\. Removing Feedback causes the largest drops on Cora, CiteSeer and Texas, highlighting the role of prediction–observation feedback in state correction\. Gain\-Aware Halting further improves performance by allowing nodes to use different rollout lengths\. These results empirically support the mechanism\-level distinctions summarized in Table[6](https://arxiv.org/html/2607.26533#A1.T6)\.

Table 4\.Hyperparameter analysis of the maximum rollout horizonTmaxT\_\{\\max\}\. The best result on each dataset is highlighted in bold\. Lower average rank indicates better overall performance\.![Refer to caption](https://arxiv.org/html/2607.26533v1/fig/case_study.jpg)Visualization of the learned information\-flow trajectory for target node 1403 from the Cora dataset\. The figure shows selected source nodes, propagation hops, edge reception strengths and channel preferences\.

Figure 5\.Node\-agent information\-flow trace for target node 1403 in Cora\. Edge width denotes source\-reception strengthγ\\gammaand color indicates channel preference\.
#### 5\.3\.2\.Node\-Specific Information\-Flow Depth

Figure[4](https://arxiv.org/html/2607.26533#S5.F4)shows the distribution of node\-specific halting steps under a maximum rollout horizon ofT=5T=5\. Most nodes halt before reaching the maximum horizon\. On average, 82\.42% of nodes halt by the second rollout step, showing that AgentGFM does not apply the full rollout to every node\.

The distributions also vary across datasets\. On Cornell, 96\.72% of nodes halt by the second step\. In contrast, Cora, Computers and Photo retain a larger proportion of nodes at later steps, with 12\.52%, 13\.69% and 13\.13% of nodes halting at steps 4 or 5 respectively\. These differences show that AgentGFM does not rely on a single global rollout depth\. Instead, different nodes terminate according to their states and estimated propagation gains\. Together with the ablation results, this analysis confirms that gain\-aware halting actively controls the duration of node\-level information acquisition\.

#### 5\.3\.3\.Node\-Agent Information\-Flow Case Study

Figure[5](https://arxiv.org/html/2607.26533#S5.F5)visualizes the learned information\-flow trace of target node 1403 in Cora\. Each edge represents a source\-reception decision, with edge width indicating reception strength and color denoting channel preference\. AgentGFM selectively collects information along multiple structural branches, reaching nodes up to five hops away\. The varying path lengths, reception strengths and channel preferences show that the shared policy produces a node\-specific propagation pattern rather than applying a uniform aggregation rule\.

#### 5\.3\.4\.Sensitivity to Rollout Horizon

Table[4](https://arxiv.org/html/2607.26533#S5.T4)reports the sensitivity of AgentGFM to the maximum rollout horizonTmaxT\_\{\\max\}\. SettingTmax=5T\_\{\\max\}=5achieves the best average rank and the best performance on seven of ten datasets, showing that a moderate horizon provides sufficient context for most node agents\. Smaller horizons may restrict information acquisition, whereas increasingTmaxT\_\{\\max\}to 6 brings no consistent improvement and may introduce less useful contextual signals\. These results support an adaptive but bounded information\-flow rollout and we useTmax=5T\_\{\\max\}=5as the default setting\.

## 6\.Efficiency and Complexity Analysis

AgentGFM performs recurrent information\-flow rollouts over the original edge set\. LetRRdenote the number of outer predict–act–observe–correct interaction rounds,TTthe maximum rollout horizon within each round andddthe hidden dimension\. Source reception, channel selection and message aggregation are computed along existing edges at each rollout step, resulting in a per\-step complexity ofO​\(\|E\|​d\)O\(\|E\|d\)and an overall information\-flow complexity ofO​\(R​T​\|E\|​d\)O\(RT\|E\|d\)\. During each rollout step, the node states and edge\-level control variables requireO​\(\|V\|​d\+\|E\|\)O\(\|V\|d\+\|E\|\)memory\. Since AgentGFM does not explicitly construct dense higher\-order neighborhoods or all\-pair structural contexts, it remains scalable for large graphs whenRRandTTare bounded by small constants\.

Table 5\.Efficiency comparison when pretraining on eight source datasets and evaluating on ogbn\-arxiv\.Table[5](https://arxiv.org/html/2607.26533#S6.T5)reports the empirical efficiency comparison when pretraining on eight source datasets and evaluating on ogbn\-arxiv\. AgentGFM takes 1\.14 seconds per pretraining epoch and 10\.21 GB of training memory, which is more memory\-efficient than BRIDGE while remaining practical in training time\. During target evaluation, AgentGFM takes 0\.89 seconds with 3\.24 GB of memory, making it faster than GraphAny and SAMGPT at inference\. Although AgentGFM is not the fastest or most memory\-efficient method in every aspect, it achieves a balanced efficiency profile while retaining adaptive node\-agent information\-flow control\.

## 7\.Conclusion

In this paper, we study topology generalization in GFMs through node\-agent information\-flow control\. Existing GFMs typically transfer predefined propagation schemes to new graphs with limited adaptation, which can be restrictive when nodes require different propagation behaviors under unseen topologies\. We therefore propose AgentGFM, which treats each node as an agent interacting with the graph environment under a shared transferable policy\. Through a predict–act–observe–correct process, AgentGFM enables node\-specific source reception, signal\-channel selection and gain\-aware halting, followed by feedback\-gated state correction\. Experiments on cross\-domain node classification, graph classification and large\-scale node classification demonstrate consistent transfer performance across diverse graph structures\.

AgentGFM currently incurs additional computation due to its recurrent information\-flow rollout\. Future work will investigate more efficient interaction and halting mechanisms while preserving node\-level adaptability and cross\-graph transferability\.

## References

- \(1\)
- Bo et al\.\(2021\)Deyu Bo, Xiao Wang, Chuan Shi, and Huawei Shen\. 2021\.Beyond Low\-frequency Information in Graph Convolutional Networks\. In*Thirty\-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty\-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2\-9, 2021*\. AAAI Press, 3950–3957\.[doi:10\.1609/AAAI\.V35I5\.16514](https://doi.org/10.1609/AAAI.V35I5.16514)
- Borgwardt et al\.\(2005\)Karsten M Borgwardt, Cheng Soon Ong, Stefan Schönauer, SVN Vishwanathan, Alex J Smola, and Hans\-Peter Kriegel\. 2005\.Protein function prediction via graph kernels\.*Bioinformatics*21, suppl\_1 \(2005\), i47–i56\.
- Chien et al\.\(2021\)Eli Chien, Jianhao Peng, Pan Li, and Olgica Milenkovic\. 2021\.Adaptive Universal Generalized PageRank Graph Neural Network\. In*9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3\-7, 2021*\. OpenReview\.net\.[https://openreview\.net/forum?id=n6jl7fLxrP](https://openreview.net/forum?id=n6jl7fLxrP)
- Das et al\.\(2018\)Rajarshi Das, Shehzaad Dhuliawala, Manzil Zaheer, Luke Vilnis, Ishan Durugkar, Akshay Krishnamurthy, Alex Smola, and Andrew McCallum\. 2018\.Go for a Walk and Arrive at the Answer: Reasoning Over Paths in Knowledge Bases using Reinforcement Learning\. In*6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 \- May 3, 2018, Conference Track Proceedings*\. OpenReview\.net\.[https://openreview\.net/forum?id=Syg\-YfWCW](https://openreview.net/forum?id=Syg-YfWCW)
- Dobson and Doig \(2003\)Paul D Dobson and Andrew J Doig\. 2003\.Distinguishing enzyme structures from non\-enzymes without alignments\.*Journal of molecular biology*330, 4 \(2003\), 771–783\.
- Fang et al\.\(2023\)Taoran Fang, Yunchao Zhang, Yang Yang, Chunping Wang, and Lei Chen\. 2023\.Universal Prompt Tuning for Graph Neural Networks\. In*Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 \- 16, 2023*, Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine \(Eds\.\)\.
- Hou et al\.\(2022\)Zhenyu Hou, Xiao Liu, Yukuo Cen, Yuxiao Dong, Hongxia Yang, Chunjie Wang, and Jie Tang\. 2022\.GraphMAE: Self\-Supervised Masked Graph Autoencoders\. In*KDD ’22: The 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, August 14 \- 18, 2022*\. ACM, 594–604\.
- Hu et al\.\(2020\)Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec\. 2020\.Open graph benchmark: Datasets for machine learning on graphs\.*Advances in neural information processing systems*33 \(2020\), 22118–22133\.
- Kipf and Welling \(2017\)Thomas N\. Kipf and Max Welling\. 2017\.Semi\-Supervised Classification with Graph Convolutional Networks\. In*5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24\-26, 2017, Conference Track Proceedings*\. OpenReview\.net\.
- Lai et al\.\(2020\)Kwei\-Herng Lai, Daochen Zha, Kaixiong Zhou, and Xia Hu\. 2020\.Policy\-GNN: Aggregation Optimization for Graph Neural Networks\. In*KDD ’20: The 26th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Virtual Event, CA, USA, August 23\-27, 2020*, Rajesh Gupta, Yan Liu, Jiliang Tang, and B\. Aditya Prakash \(Eds\.\)\. ACM, 461–471\.[doi:10\.1145/3394486\.3403088](https://doi.org/10.1145/3394486.3403088)
- Liu et al\.\(2026\)Haokun Liu, Zezhong Ding, and Xike Xie\. 2026\.Learning Graph Foundation Models on Riemannian Graph\-of\-Graphs\.*CoRR*abs/2605\.09993 \(2026\)\.arXiv:2605\.09993[doi:10\.48550/ARXIV\.2605\.09993](https://doi.org/10.48550/ARXIV.2605.09993)
- Liu et al\.\(2023a\)Jiawei Liu, Cheng Yang, Zhiyuan Lu, Junze Chen, Yibo Li, Mengmei Zhang, Ting Bai, Yuan Fang, Lichao Sun, Philip S Yu, et al\.2023a\.Towards graph foundation models: A survey and beyond\.*arXiv preprint arXiv:2310\.11829*\(2023\)\.
- Liu et al\.\(2025\)Lei Liu, Xingyu Xia, Qianqian Xie, Ben Liu, Wenjie Xu, and Min Peng\. 2025\.Enhanced Expert Merging for Mixture\-of\-Experts in Graph Foundation Models\. In*The Thirty\-ninth Annual Conference on Neural Information Processing Systems*\.
- Liu et al\.\(2023b\)Zemin Liu, Xingtong Yu, Yuan Fang, and Xinming Zhang\. 2023b\.GraphPrompt: Unifying Pre\-Training and Downstream Tasks for Graph Neural Networks\. In*Proceedings of the ACM Web Conference 2023, WWW 2023, Austin, TX, USA, 30 April 2023 \- 4 May 2023*, Ying Ding, Jie Tang, Juan F\. Sequeda, Lora Aroyo, Carlos Castillo, and Geert\-Jan Houben \(Eds\.\)\. ACM, 417–428\.[doi:10\.1145/3543507\.3583386](https://doi.org/10.1145/3543507.3583386)
- Palomares et al\.\(2021\)Iván Palomares, Carlos Porcel, Luiz Pizzato, Ido Guy, and Enrique Herrera\-Viedma\. 2021\.Reciprocal Recommender Systems: Analysis of state\-of\-art literature, challenges and opportunities towards social recommendation\.*Inf\. Fusion*69 \(2021\), 103–127\.[doi:10\.1016/j\.inffus\.2020\.12\.001](https://doi.org/10.1016/j.inffus.2020.12.001)
- Pei et al\.\(2020\)Hongbin Pei, Bingzhe Wei, Kevin Chen\-Chuan Chang, Yu Lei, and Bo Yang\. 2020\.Geom\-GCN: Geometric Graph Convolutional Networks\. In*8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26\-30, 2020*\. OpenReview\.net\.
- Schlichtkrull et al\.\(2018\)Michael Sejr Schlichtkrull, Thomas N\. Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling\. 2018\.Modeling Relational Data with Graph Convolutional Networks\. In*The Semantic Web \- 15th International Conference, ESWC 2018, Heraklion, Crete, Greece, June 3\-7, 2018, Proceedings**\(Lecture Notes in Computer Science, Vol\. 10843\)*, Aldo Gangemi, Roberto Navigli, Maria\-Esther Vidal, Pascal Hitzler, Raphaël Troncy, Laura Hollink, Anna Tordai, and Mehwish Alam \(Eds\.\)\. Springer, 593–607\.[doi:10\.1007/978\-3\-319\-93417\-4\_38](https://doi.org/10.1007/978-3-319-93417-4_38)
- Shchur et al\.\(2018\)Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Günnemann\. 2018\.Pitfalls of graph neural network evaluation\.*arXiv preprint arXiv:1811\.05868*\(2018\)\.
- Shervashidze et al\.\(2011\)Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M\. Borgwardt\. 2011\.Weisfeiler\-Lehman Graph Kernels\.*J\. Mach\. Learn\. Res\.*12 \(2011\), 2539–2561\.[doi:10\.5555/1953048\.2078187](https://doi.org/10.5555/1953048.2078187)
- Sun et al\.\(2025\)Li Sun, Zhenhao Huang, Suyang Zhou, Qiqi Wan, Hao Peng, and Philip S\. Yu\. 2025\.RiemannGFM: Learning a Graph Foundation Model from Riemannian Geometry\. In*Proceedings of the ACM on Web Conference 2025, WWW 2025, Sydney, NSW, Australia, 28 April 2025\- 2 May 2025*, Guodong Long, Michale Blumestein, Yi Chang, Liane Lewin\-Eytan, Zi Helen Huang, and Elad Yom\-Tov \(Eds\.\)\. ACM, 1154–1165\.[doi:10\.1145/3696410\.3714952](https://doi.org/10.1145/3696410.3714952)
- Velickovic et al\.\(2018\)Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio\. 2018\.Graph Attention Networks\. In*6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 \- May 3, 2018, Conference Track Proceedings*\. OpenReview\.net\.
- Velickovic et al\.\(2019\)Petar Velickovic, William Fedus, William L\. Hamilton, Pietro Liò, Yoshua Bengio, and R\. Devon Hjelm\. 2019\.Deep Graph Infomax\. In*7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6\-9, 2019*\. OpenReview\.net\.
- Vlaic et al\.\(2018\)Sebastian Vlaic, Theresia Conrad, Christian Tokarski\-Schnelle, Mika Gustafsson, Uta Dahmen, Reinhard Guthke, and Stefan Schuster\. 2018\.ModuleDiscoverer: Identification of regulatory modules in protein\-protein interaction networks\.*Scientific reports*8, 1 \(2018\), 433\.
- Wang et al\.\(2024\)Zehong Wang, Zheyuan Zhang, Nitesh V\. Chawla, Chuxu Zhang, and Yanfang Ye\. 2024\.GFT: Graph Foundation Model with Transferable Tree Vocabulary\. In*Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems 2024, NeurIPS 2024, Vancouver, BC, Canada, December 10 \- 15, 2024*, Amir Globersons, Lester Mackey, Danielle Belgrave, Angela Fan, Ulrich Paquet, Jakub M\. Tomczak, and Cheng Zhang \(Eds\.\)\.
- Wu et al\.\(2019\)Shu Wu, Yuyuan Tang, Yanqiao Zhu, Liang Wang, Xing Xie, and Tieniu Tan\. 2019\.Session\-Based Recommendation with Graph Neural Networks\. In*The Thirty\-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty\-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 \- February 1, 2019*\. AAAI Press, 346–353\.
- Xia et al\.\(2024\)Lianghao Xia, Ben Kao, and Chao Huang\. 2024\.OpenGraph: Towards Open Graph Foundation Models\. In*Findings of the Association for Computational Linguistics: EMNLP 2024, Miami, Florida, USA, November 12\-16, 2024**\(Findings of ACL, Vol\. EMNLP 2024\)*, Yaser Al\-Onaizan, Mohit Bansal, and Yun\-Nung Chen \(Eds\.\)\. Association for Computational Linguistics, 2365–2379\.[doi:10\.18653/V1/2024\.FINDINGS\-EMNLP\.132](https://doi.org/10.18653/V1/2024.FINDINGS-EMNLP.132)
- Xiong et al\.\(2017\)Wenhan Xiong, Thien Hoang, and William Yang Wang\. 2017\.DeepPath: A Reinforcement Learning Method for Knowledge Graph Reasoning\. In*Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, EMNLP 2017, Copenhagen, Denmark, September 9\-11, 2017*, Martha Palmer, Rebecca Hwa, and Sebastian Riedel \(Eds\.\)\. Association for Computational Linguistics, 564–573\.[doi:10\.18653/V1/D17\-1060](https://doi.org/10.18653/V1/D17-1060)
- Xue et al\.\(2021\)Fuzhao Xue, Aixin Sun, Hao Zhang, and Eng Siong Chng\. 2021\.GDPNet: Refining Latent Multi\-View Graph for Relation Extraction\. In*Thirty\-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty\-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2\-9, 2021*\. AAAI Press, 14194–14202\.[doi:10\.1609/AAAI\.V35I16\.17670](https://doi.org/10.1609/AAAI.V35I16.17670)
- Yanardag and Vishwanathan \(2015\)Pinar Yanardag and S\. V\. N\. Vishwanathan\. 2015\.Deep Graph Kernels\. In*Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Sydney, NSW, Australia, August 10\-13, 2015*, Longbing Cao, Chengqi Zhang, Thorsten Joachims, Geoffrey I\. Webb, Dragos D\. Margineantu, and Graham Williams \(Eds\.\)\. ACM, 1365–1374\.[doi:10\.1145/2783258\.2783417](https://doi.org/10.1145/2783258.2783417)
- Yang et al\.\(2016\)Zhilin Yang, William Cohen, and Ruslan Salakhudinov\. 2016\.Revisiting semi\-supervised learning with graph embeddings\. In*International conference on machine learning*\. PMLR, 40–48\.
- Yang et al\.\(2025\)Zhe\-Rui Yang, Jindong Han, Chang\-Dong Wang, and Hao Liu\. 2025\.GraphLoRA: Structure\-Aware Contrastive Low\-Rank Adaptation for Cross\-Graph Transfer Learning\. In*Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining, V\.1, KDD 2025, Toronto, ON, Canada, August 3\-7, 2025*, Yizhou Sun, Flavio Chierichetti, Hady W\. Lauw, Claudia Perlich, Wee Hyong Tok, and Andrew Tomkins \(Eds\.\)\. ACM, 1785–1796\.[doi:10\.1145/3690624\.3709186](https://doi.org/10.1145/3690624.3709186)
- You et al\.\(2020\)Yuning You, Tianlong Chen, Yongduo Sui, Ting Chen, Zhangyang Wang, and Yang Shen\. 2020\.Graph Contrastive Learning with Augmentations\. In*Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6\-12, 2020, virtual*, Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria\-Florina Balcan, and Hsuan\-Tien Lin \(Eds\.\)\.
- Yu et al\.\(2025\)Xingtong Yu, Zechuan Gong, Chang Zhou, Yuan Fang, and Hui Zhang\. 2025\.SAMGPT: Text\-free Graph Foundation Model for Multi\-domain Pre\-training and Cross\-domain Adaptation\. In*Proceedings of the ACM on Web Conference 2025, WWW 2025, Sydney, NSW, Australia, 28 April 2025\- 2 May 2025*, Guodong Long, Michale Blumestein, Yi Chang, Liane Lewin\-Eytan, Zi Helen Huang, and Elad Yom\-Tov \(Eds\.\)\. ACM, 1142–1153\.[doi:10\.1145/3696410\.3714828](https://doi.org/10.1145/3696410.3714828)
- Yu et al\.\(2024\)Xingtong Yu, Chang Zhou, Yuan Fang, and Xinming Zhang\. 2024\.Text\-free multi\-domain graph pre\-training: Toward graph foundation models\.*arXiv preprint arXiv:2405\.13934*\(2024\)\.
- Yuan et al\.\(2025\)Haonan Yuan, Qingyun Sun, Junhua Shi, Xingcheng Fu, Bryan Hooi, Jianxin Li, and Philip S\. Yu\. 2025\.How Much Can Transfer? BRIDGE: Bounded Multi\-Domain Graph Foundation Model with Generalization Guarantees\. In*Forty\-second International Conference on Machine Learning*\.
- Zhao et al\.\(2024\)Haihong Zhao, Aochuan Chen, Xiangguo Sun, Hong Cheng, and Jia Li\. 2024\.All in One and One for All: A Simple yet Effective Method towards Cross\-domain Graph Pretraining\. In*Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, KDD 2024, Barcelona, Spain, August 25\-29, 2024*, Ricardo Baeza\-Yates and Francesco Bonchi \(Eds\.\)\. ACM, 4443–4454\.[doi:10\.1145/3637528\.3671913](https://doi.org/10.1145/3637528.3671913)
- Zhao et al\.\(2026\)Jitao Zhao, Yi Wang, Yawen Li, Dongxiao He, Di Jin, Zhiyong Feng, and Weixiong Zhang\. 2026\.Towards Graph Foundation Model: Node Feature Transfer Invariant Modeling on General Graphs\. In*Proceedings of the ACM Web Conference 2026*\(United Arab Emirates\)*\(WWW ’26\)*\. Association for Computing Machinery, New York, NY, USA, 810–821\.[doi:10\.1145/3774904\.3792236](https://doi.org/10.1145/3774904.3792236)
- Zhao et al\.\(2025b\)Jianan Zhao, Zhaocheng Zhu, Mikhail Galkin, Hesham Mostafa, Michael M\. Bronstein, and Jian Tang\. 2025b\.Fully\-inductive Node Classification on Arbitrary Graphs\. In*The Thirteenth International Conference on Learning Representations, ICLR 2025, Singapore, April 24\-28, 2025*\. OpenReview\.net\.
- Zhao et al\.\(2025a\)Qifang Zhao, Weidong Ren, Tianyu Li, Hong Liu, Xingsheng He, and Xiaoxiao Xu\. 2025a\.GraphGPT: Generative Pre\-trained Graph Eulerian Transformer\. In*Forty\-second International Conference on Machine Learning, ICML 2025, Vancouver, BC, Canada, July 13\-19, 2025**\(Proceedings of Machine Learning Research, Vol\. 267\)*, Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste\-Julien, Felix Berkenkamp, Tegan Maharaj, Kiri Wagstaff, and Jerry Zhu \(Eds\.\)\. PMLR / OpenReview\.net\.[https://proceedings\.mlr\.press/v267/zhao25r\.html](https://proceedings.mlr.press/v267/zhao25r.html)
- Zhu et al\.\(2024\)Yun Zhu, Yaoke Wang, Haizhou Shi, Zhenshuo Zhang, Dian Jiao, and Siliang Tang\. 2024\.GraphControl: Adding Conditional Control to Universal Graph Pre\-trained Models for Graph Domain Transfer Learning\. In*Proceedings of the ACM Web Conference 2024**\(WWW ’24\)*\. ACM, 539–550\.[doi:10\.1145/3589334\.3645439](https://doi.org/10.1145/3589334.3645439)

## Appendix AMechanism\-Level Positioning

To further clarify the distinction between AgentGFM and existing graph propagation paradigms, we compare their primary control objects, optimization signals and interaction mechanisms in Table[6](https://arxiv.org/html/2607.26533#A1.T6)\. The comparison focuses on how information\-flow execution is determined rather than on specific model architectures\.

Table 6\.Mechanism\-level positioning of AgentGFM among graph propagation paradigms\. Opt\. Signal denotes the objective used for parameter learning, while Interaction Feedback denotes contextual signals used to condition subsequent information\-flow decisions\.A comparison of six graph propagation paradigms in terms of source control, signal control, depth control, optimization signal and interaction feedback\. AgentGFM differs from the other paradigms by combining node\- and edge\-level source control, low\- and high\-frequency signal selection, node\-wise halting, pretraining objectives and explicit prediction\-observation feedback\.
Table[6](https://arxiv.org/html/2607.26533#A1.T6)distinguishes existing paradigms according to the primary object controlled during information\-flow execution\. Fixed and multi\-hop message passing specify information sources and propagation ranges mainly through the model architecture\. Attention\-based methods adapt neighbor importance, while adaptive GNNs may further adjust neighborhood selection or propagation depth\. However, these methods generally optimize task\-specific propagation decisions without explicitly using contextual observations to correct node states and condition subsequent actions\.

Component\-adaptive GFMs mainly adapt prompts, experts, structural encodings, or model components for cross\-graph transfer, while the underlying information\-flow execution remains largely predetermined\. In contrast, AgentGFM treats the node\-specific information\-flow trajectory as the adaptive object\. Each node controls source reception, signal\-channel selection and node\-wise halting and uses prediction–observation feedback to correct its state and condition subsequent interactions\.

## Appendix BExperimental Settings

### B\.1\.Datasets

We evaluate node\-level performance on a diverse set of benchmark graphs\. Citation networks include Cora, CiteSeer and PubMed\(Yang et al\.,[2016](https://arxiv.org/html/2607.26533#bib.bib31)\)\. Web page networks consist of Texas, Cornell and Wisconsin\(Pei et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib17)\)\. Wikipedia topic graphs include Chameleon and Squirrel\(Pei et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib17)\)\. E\-commerce graphs include Photo and Computers\(Shchur et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib19)\)\. We additionally consider the large\-scale ogbn\-products and ogbn\-arxiv datasets from the Open Graph Benchmark\(Hu et al\.,[2020](https://arxiv.org/html/2607.26533#bib.bib9)\)and Physics\(Shchur et al\.,[2018](https://arxiv.org/html/2607.26533#bib.bib19)\)

Table 7\.Overview of node classification datasets used in the experiments\.Statistics of thirteen node classification datasets, including the numbers of nodes, edges, input features and classes\.For graph\-level tasks, we use molecular and biological datasets including MUTAG, DD\(Shervashidze et al\.,[2011](https://arxiv.org/html/2607.26533#bib.bib20)\), ENZYMES\(Borgwardt et al\.,[2005](https://arxiv.org/html/2607.26533#bib.bib3)\)and PROTEINS\(Dobson and Doig,[2003](https://arxiv.org/html/2607.26533#bib.bib6)\)\. We further include social network datasets IMDB\-BINARY\(Yanardag and Vishwanathan,[2015](https://arxiv.org/html/2607.26533#bib.bib30)\)\. All datasets are commonly used benchmarks in graph representation learning and cover a wide range of graph sizes, structures and homophily characteristics\. For graph classification datasets from TUDataset, if the original graphs do not contain node features, we construct node features as follows\. If node labels are available, we use one\-hot encodings of node labels as node features\. Otherwise, we use one\-hot encodings of node degrees, where the one\-hot dimension is determined by the maximum label value or maximum degree across the entire dataset\. If the original dataset provides node featuresxx, we directly use them\.

Table 8\.Overview of graph classification datasets used in the experiments\. Avg\. Nodes and Avg\. Edges denote the average numbers of nodes and edges per graph, respectively\.Statistics of five graph classification datasets, including the number of graphs, average graph size, input feature dimension and number of classes\. IMDB\-BINARY contains no original node attributes\.Table 9\.Full component ablation results on cross\-domain 1\-shot node classification\. Source\-Reception Control corresponds toγ\\gamma, Channel Selection corresponds toρ\\rho, Feedback denotes prediction–observation feedback and Gain\-Aware Halting denotes the node\-wise halting mechanism\. The best result on each dataset is highlighted in bold\. Avg\. Rank denotes the average rank across all datasets, where a lower value is better\.
### B\.2\.Implementation Details

Table 10\.Default hyperparameter settings of AgentGFM\.We implement AgentGFM in PyTorch and use the same default configuration across all datasets unless otherwise specified\. Node attributes are projected into a shared 50\-dimensional space and the hidden dimension is set to 128\. The model contains two predict–act–observe–correct interaction rounds, with the maximum rollout horizon set toTmax=5T\_\{\\max\}=5\. We train AgentGFM for 500 epochs using a learning rate of5×10−45\\times 10^\{\-4\}, a weight decay of5×10−45\\times 10^\{\-4\}, a dropout rate of 0\.1 and a masking ratio of 0\.35\. The complete implementation and hyperparameter settings are summarized in Table[10](https://arxiv.org/html/2607.26533#A2.T10)\. For 1\-shot evaluation, we repeat the sampling process 100 times and report the mean accuracy and standard deviation\.

### B\.3\.Information\-Flow Policy and Regularization Details

This section provides additional details of the node\-level information\-flow policy and the associated regularizers used in AgentGFM\. The policy is shared across nodes and graphs, while its source\-reception, channel\-selection, forwarding\-budget and halting decisions are computed for each local rollout trajectory\.

#### B\.3\.1\.Information\-Flow Policy

##### Source\-reception policy\.

For each original edge\(u,i\)∈ℰ\(u,i\)\\in\\mathcal\{E\}, the source\-reception strength is computed from carrier compatibility, prediction–observation feedback and the source forwarding budget:

\(31\)su→iγ,t=\(𝐖qγ​𝐜ut\)⊤​\(𝐖kγ​𝐜it\)r\+gγ​\(\[ϵu​‖ϵi‖​but\]\),s\_\{u\\rightarrow i\}^\{\\gamma,t\}=\\frac\{\\left\(\\mathbf\{W\}\_\{q\}^\{\\gamma\}\\mathbf\{c\}\_\{u\}^\{t\}\\right\)^\{\\top\}\\left\(\\mathbf\{W\}\_\{k\}^\{\\gamma\}\\mathbf\{c\}\_\{i\}^\{t\}\\right\)\}\{\\sqrt\{r\}\}\+g\_\{\\gamma\}\\left\(\\left\[\\epsilon\_\{u\}\\\|\\epsilon\_\{i\}\\\|b\_\{u\}^\{t\}\\right\]\\right\),\(32\)γu→it=σ​\(su→iγ,t\),\\gamma\_\{u\\rightarrow i\}^\{t\}=\\sigma\\left\(s\_\{u\\rightarrow i\}^\{\\gamma,t\}\\right\),whereϵu\\epsilon\_\{u\}andϵi\\epsilon\_\{i\}are prediction–observation feedback cues andbutb\_\{u\}^\{t\}is the source forwarding budget\. The resultingγu→it\\gamma\_\{u\\rightarrow i\}^\{t\}measures the source\-reception strength from nodevuv\_\{u\}to nodeviv\_\{i\}under the current rollout state\.

##### Channel\-selection policy\.

The channel\-selection gate determines how low\- and high\-frequency information are combined along each edge\. We compute

\(33\)ρu→it=σ​\(su→iρ,t\+βsmoothτρ\),\\rho\_\{u\\rightarrow i\}^\{t\}=\\sigma\\left\(\\frac\{s\_\{u\\rightarrow i\}^\{\\rho,t\}\+\\beta\_\{\\mathrm\{smooth\}\}\}\{\\tau\_\{\\rho\}\}\\right\),where

su→iρ,t=\\displaystyle s\_\{u\\rightarrow i\}^\{\\rho,t\}=\(𝐖qρ​𝐜ut\)⊤​\(𝐖kρ​𝐜it\)r\\displaystyle\\frac\{\\left\(\\mathbf\{W\}\_\{q\}^\{\\rho\}\\mathbf\{c\}\_\{u\}^\{t\}\\right\)^\{\\top\}\\left\(\\mathbf\{W\}\_\{k\}^\{\\rho\}\\mathbf\{c\}\_\{i\}^\{t\}\\right\)\}\{\\sqrt\{r\}\}\(34\)\+gρ​\(\[su​it​‖1−su​it‖​ϵu​‖ϵi‖​but\]\)\.\\displaystyle\+g\_\{\\rho\}\\left\(\\left\[s\_\{ui\}^\{t\}\\\|1\-s\_\{ui\}^\{t\}\\\|\\epsilon\_\{u\}\\\|\\epsilon\_\{i\}\\\|b\_\{u\}^\{t\}\\right\]\\right\)\.Here,su​its\_\{ui\}^\{t\}denotes the cosine\-similarity diagnostic between the source and target carrier states\. The smoothness biasβsmooth\\beta\_\{\\mathrm\{smooth\}\}and temperatureτρ\\tau\_\{\\rho\}control the preference and sharpness of the low\-/high\-frequency interpolation\. The final message is

\(35\)𝐦u→it=ρu→it​𝐦u,lowt\+\(1−ρu→it\)​λhigh​𝐦u,hight\.\\mathbf\{m\}\_\{u\\rightarrow i\}^\{t\}=\\rho\_\{u\\rightarrow i\}^\{t\}\\mathbf\{m\}\_\{u,\\mathrm\{low\}\}^\{t\}\+\\left\(1\-\\rho\_\{u\\rightarrow i\}^\{t\}\\right\)\\lambda\_\{\\mathrm\{high\}\}\\mathbf\{m\}\_\{u,\\mathrm\{high\}\}^\{t\}\.A largerρu→it\\rho\_\{u\\rightarrow i\}^\{t\}assigns greater weight to low\-frequency contextual information, whereas a smaller value assigns greater weight to high\-frequency residual information\.

##### Forwarding\-budget update\.

The forwarding budget represents the current capacity of a node to forward contextual information rather than a monotonically consumed resource\. It is updated according to the effective information mass received at the current rollout step:

\(36\)b~it\+1=𝒯bmin​\[clip⁡\(ηb​∑u∈𝒩​\(i\)μu→it,0,1\)\],\\widetilde\{b\}\_\{i\}^\{t\+1\}=\\mathcal\{T\}\_\{b\_\{\\min\}\}\\left\[\\operatorname\{clip\}\\left\(\\eta\_\{b\}\\sum\_\{u\\in\\mathcal\{N\}\(i\)\}\\mu\_\{u\\rightarrow i\}^\{t\},0,1\\right\)\\right\],whereηb\\eta\_\{b\}is the budget coefficient and𝒯bmin​\(⋅\)\\mathcal\{T\}\_\{b\_\{\\min\}\}\(\\cdot\)sets values belowbminb\_\{\\min\}to zero\. The budget state is updated according to the current activity status:

\(37\)bit\+1=ait​b~it\+1\+\(1−ait\)​bit\.b\_\{i\}^\{t\+1\}=a\_\{i\}^\{t\}\\widetilde\{b\}\_\{i\}^\{t\+1\}\+\\left\(1\-a\_\{i\}^\{t\}\\right\)b\_\{i\}^\{t\}\.An active node updates its forwarding budget using the newly received information mass, whereas a halted node retains its previous budget\. Stronger received information sustains later propagation, while weak information mass reduces subsequent forwarding capacity\.

##### Network parameterization\.

The observation predictorfpredf\_\{\\mathrm\{pred\}\}, carrier updaterfcf\_\{c\}, gain predictorfgainf\_\{\\mathrm\{gain\}\}, gain\-aware halting controllerfhaltf\_\{\\mathrm\{halt\}\}and reliability gatefrelf\_\{\\mathrm\{rel\}\}are implemented as lightweight two\-layer MLPs with PReLU activations and dropout\. Layer normalization is applied to representation\-valued outputs, whereas sigmoid functions are used for scalar or feature\-wise gates\. The source\-reception and channel\-selection policies combine source–target query–key compatibility scores with lightweight scalar MLPs\. Within each interaction round, these modules share parameters across all nodes, edges, graph domains and rollout steps, whereas successive interaction rounds use separate parameter sets\.

#### B\.3\.2\.Information\-Flow Regularization

The node\-level information\-flow policy contains three coupled decisions: which sources to receive from, how much effective information mass to transmit and how long each node should continue its rollout\. Without additional constraints, the policy may degenerate into uniformly receiving all neighboring information, transmitting excessive information mass, or keeping most nodes active until the maximum horizon\. We therefore regularize these three aspects separately\.

##### Source\-reception regularization\.

The first regularizer controls the average source\-reception strength:

\(38\)ℒγ=1T​\|ℰ\|​∑t=0T−1∑\(u,i\)∈ℰγu→it\.\\mathcal\{L\}\_\{\\gamma\}=\\frac\{1\}\{T\|\\mathcal\{E\}\|\}\\sum\_\{t=0\}^\{T\-1\}\\sum\_\{\(u,i\)\\in\\mathcal\{E\}\}\\gamma\_\{u\\rightarrow i\}^\{t\}\.This term discourages the policy from assigning uniformly high reception scores to all neighboring sources\. A smallerℒγ\\mathcal\{L\}\_\{\\gamma\}encourages the model to be selective about which neighbors contribute contextual information rather than reverting to indiscriminate neighbor aggregation\.

##### Propagation\-mass regularization\.

The second regularizer controls the effective information mass transmitted along graph edges\. Recall that

\(39\)μu→it=ait​but​γu→it,\\mu\_\{u\\rightarrow i\}^\{t\}=a\_\{i\}^\{t\}b\_\{u\}^\{t\}\\gamma\_\{u\\rightarrow i\}^\{t\},whereaita\_\{i\}^\{t\}indicates whether the target node is active,butb\_\{u\}^\{t\}is the source forwarding budget andγu→it\\gamma\_\{u\\rightarrow i\}^\{t\}is the source\-reception strength\. We regularize the average transmitted mass by

\(40\)ℒμ=1T​\|ℰ\|​∑t=0T−1∑\(u,i\)∈ℰμu→it\.\\mathcal\{L\}\_\{\\mu\}=\\frac\{1\}\{T\|\\mathcal\{E\}\|\}\\sum\_\{t=0\}^\{T\-1\}\\sum\_\{\(u,i\)\\in\\mathcal\{E\}\}\\mu\_\{u\\rightarrow i\}^\{t\}\.Unlikeℒγ\\mathcal\{L\}\_\{\\gamma\}, this term penalizes the realized information flow after accounting for target activity and source forwarding budget\. It discourages excessive propagation even when individual reception scores remain moderate\.

##### Continuation regularization\.

The third regularizer controls the node\-wise rollout length:

\(41\)ℒa=1T​\|𝒱\|​∑t=0T−1∑vi∈𝒱ait\+1\.\\mathcal\{L\}\_\{a\}=\\frac\{1\}\{T\|\\mathcal\{V\}\|\}\\sum\_\{t=0\}^\{T\-1\}\\sum\_\{v\_\{i\}\\in\\mathcal\{V\}\}a\_\{i\}^\{t\+1\}\.Sinceait\+1a\_\{i\}^\{t\+1\}indicates whether nodeviv\_\{i\}remains active after steptt, minimizingℒa\\mathcal\{L\}\_\{a\}discourages unnecessarily long rollouts\. This term prevents most nodes from remaining active until the maximum horizon while allowing nodes with positive predictive gain to continue collecting contextual information\.

Together, the three regularizers constrain node\-level information flow from complementary perspectives\.ℒγ\\mathcal\{L\}\_\{\\gamma\}encourages selective source reception,ℒμ\\mathcal\{L\}\_\{\\mu\}limits the effective information mass transmitted along active paths andℒa\\mathcal\{L\}\_\{a\}promotes adaptive early stopping\. In this way, the learned policy is discouraged from degenerating into uniform all\-neighbor aggregation, excessive message transmission, or fixed full\-horizon propagation\.

## Appendix CFull Component Ablation Results

Table[9](https://arxiv.org/html/2607.26533#A2.T9)reports the complete ablation results on all ten node\-level datasets\. Removing source receptionγ\\gammareduces the average accuracy from 0\.4540 to 0\.4388, showing that selective source acquisition helps suppress irrelevant neighboring evidence\. Removing signal\-channel selectionρ\\rhocauses a larger average drop to 0\.4200, with particularly clear degradation on Texas and Cornell\. This confirms the importance of adapting the signal type to local topology\. The largest performance decrease occurs when prediction–observation feedback is removed, reducing the average accuracy to 0\.3646\. The degradation is especially pronounced on Cora and Photo, indicating that feedback is central to assessing contextual reliability and correcting node states\. Removing gain\-aware stopping also lowers the average accuracy to 0\.4282\. Although the variant slightly improves results on Wisconsin and Chameleon, it degrades performance on most datasets, particularly PubMed, Computers and Photo\.