MA-DAR: Manifold-Aligned Dynamic Adaptive Routing for Continual Temporal Knowledge Graph Reasoning

# MA-DAR: Manifold-Aligned Dynamic Adaptive Routing for Continual Temporal Knowledge Graph Reasoning
Source: [https://arxiv.org/html/2607.21949](https://arxiv.org/html/2607.21949)
Xiangjun Shi1, Chong Mu1, Jinchuan Zhang1,\*, Lizong Zhang1, Yuefeng He1, Shang Liu1

###### Abstract

Continual temporal knowledge graph \(TKG\) reasoning aims to continuously incorporate newly emerging facts while preserving previously acquired knowledge\. Replay\-based continual learning has achieved promising performance by revisiting historical representations\. However, existing methods primarily focus on what to replay, while largely overlooking how replayed representations should be integrated with current ones\. Such direct integration often gives rise to two critical forms of representation conflict:norm dominationandsemantic blurring, ultimately degrading continual reasoning performance\. To address these challenges, we propose MA\-DAR \(Manifold\-Aligned Dynamic Adaptive Routing\), a lightweight plug\-and\-play framework for replay representation fusion\. MA\-DAR first aligns replayed and current representations onto a shared manifold to alleviate distribution discrepancies\. It then employs a dynamic gating mechanism to learn dimension\-wise fusion weights, adaptively determining the contribution of replayed and current representations to the fused representation\. Furthermore, a polarization regularizer encourages more decisive routing behaviors by discouraging ambiguous gating decisions, resulting in more stable and effective knowledge integration\. Extensive experiments on four public continual TKG benchmarks demonstrate that MA\-DAR consistently improves the performance of representative TKG encoders while remaining effective under different replay settings\. Comprehensive ablation studies and visualization analyses further verify the effectiveness of manifold alignment and dynamic adaptive routing in mitigating representation conflicts and improving continual reasoning\.

## Introduction

Temporal Knowledge Graphs \(TKGs\) have emerged as highly effective structures for capturing the evolutionary dynamics of facts across time, underpinning critical downstream applications such as event forecasting and intelligent risk analysis\(Huanget al\.[2024](https://arxiv.org/html/2607.21949#bib.bib14); Chenet al\.[2024a](https://arxiv.org/html/2607.21949#bib.bib13); Zheng and Cai[2020](https://arxiv.org/html/2607.21949#bib.bib43)\)\. A TKG represents facts as sequences of quadruples, comprising subject, relation, object, and timestamp\. A primary challenge in this domain is extrapolation reasoning: inferring missing facts at unseen future timestamps based on historical topological patterns\(Lacroixet al\.[2020](https://arxiv.org/html/2607.21949#bib.bib5); Goelet al\.[2020](https://arxiv.org/html/2607.21949#bib.bib7); Liet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib8); Jinet al\.[2020](https://arxiv.org/html/2607.21949#bib.bib6)\)\.

Real\-world TKG data typically arrives in continuous, streaming snapshots\. Sequential training on such streaming graphs natively induces catastrophic forgetting\(Kirkpatricket al\.[2017](https://arxiv.org/html/2607.21949#bib.bib15)\), where the model abruptly loses previously acquired historical dependencies\. To address this, considerable efforts have been focused on updating graph encoders to capture richer and more expressive structural representations\. For instance, RE\-GCN models local structural evolution\(Liet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib8)\), TiRGN\(Liet al\.[2022](https://arxiv.org/html/2607.21949#bib.bib9)\)incorporates local\-global historical patterns, and LogCL\(Menget al\.[2023](https://arxiv.org/html/2607.21949#bib.bib36)\)introduces contrastive learning to alleviate data sparsity\. Meanwhile, experience replay methods attempt to preserve historical knowledge by revisiting past information during incremental updates\. For instance, ER\(Rolnicket al\.[2019](https://arxiv.org/html/2607.21949#bib.bib18)\)mitigates forgetting by storing and rehearsing a raw subset of historical facts, while DGAR\(Zhanget al\.[2025b](https://arxiv.org/html/2607.21949#bib.bib1)\)explores generative replay by producing pseudo\-historical distributions through a diffusion model\.

However, existing replay\-based methods mainly focus on what to replay, while paying less attention to how replayed representations should be integrated with current representations\. Most baselines rely on global scalar weights or linear concatenation to merge the historical replay representation with the current topological update\(Wuet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib21); Cuiet al\.[2023](https://arxiv.org/html/2607.21949#bib.bib22)\)\. This coarse\-grained fusion strategy ignores the semantic complexity of high\-dimensional embeddings, leading to two critical representation conflicts:norm dominationandsemantic blurring\. Specifically, generative replay methods such as DGAR\(Zhanget al\.[2025b](https://arxiv.org/html/2607.21949#bib.bib1)\)may introduce scale discrepancies during the reconstruction of historical embeddings, causing the replayed representations to dominate the fusion process\. As depicted in Figure[1](https://arxiv.org/html/2607.21949#Sx1.F1)\(a\), such magnitude imbalance weakens the contribution of current structural updates during representation integration\. Meanwhile, without dimension\-aware constraints, conventional gating mechanisms tend to produce ambiguous weights around 0\.5, resulting in insufficient separation between historical information and current dynamics and ultimately blurring the synthesized semantic space\.

![Refer to caption](https://arxiv.org/html/2607.21949v1/x1.png)Figure 1:Comparison of feature fusion mechanisms in TKG reasoning\.In this paper, we propose the Manifold\-Aligned Dynamic Adaptive Routing \(MA\-DAR\)\. Diverging from standard end\-to\-end architectures, MA\-DAR acts as a generic, plug\-and\-play routing plugin that decouples replay representation fusion from current\-step structural encoding\. As depicted in Figure[1](https://arxiv.org/html/2607.21949#Sx1.F1)\(b\), MA\-DAR establishes dual data interfaces to ingest both the current temporal state from an arbitrary base encoder \(e\.g\., RE\-GCN, TiRGN, LogCL\) and the long\-term historical representations provided by an independent replay model, executing localized feature rectification within its own module\. To overcome fusion bottlenecks, the MA\-DAR plugin executes a compact sequence of operations: Manifold Alignment first normalizes historical and current representations to map them without norm domination; Dynamic Gating then aligns semantics by calculating distinct weights for individual dimensions; and finally, a Polarization Penalty regularizes the fusion process to force explicit, binary\-like retention decisions\.

Our contributions are summarized as follows:

- •We propose the MA\-DAR framework, a lightweight and plug\-and\-play replay representation fusion framework that identifies and alleviates two critical representation conflicts, namelynorm dominationandsemantic blurring, in continual TKG reasoning\.
- •We design a multi\-stage routing protocol inside the plugin, blending Manifold Alignment, Dynamic Gating, and a differentiable Polarization Penalty to eliminate norm discrepancy and achieve sharp, binary\-like feature decoupling\.
- •We conduct extensive evaluations across multiple continual TKG benchmarks, demonstrating that MA\-DAR achieves state\-of\-the\-art performance while remaining effective under different replay settings and TKG encoders\.

## Related Work

### Specialized Architectural Optimization

Traditional and early temporal knowledge graph reasoning approaches captured evolutionary dynamics via static time embeddings\(Bordeset al\.[2013](https://arxiv.org/html/2607.21949#bib.bib2); Leblay and Chekol[2018](https://arxiv.org/html/2607.21949#bib.bib3); Dasguptaet al\.[2018](https://arxiv.org/html/2607.21949#bib.bib4)\)or autoregressive sequence layers\(Jinet al\.[2020](https://arxiv.org/html/2607.21949#bib.bib6)\)\. Subsequently, message\-passing schemes via structural GNNs became prevalent, with RE\-GCN capturing local evolution\(Liet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib8)\)and TiRGN modeling local\-global periodic patterns\(Liet al\.[2022](https://arxiv.org/html/2607.21949#bib.bib9)\)\. More recently, deep\-tuning paradigms have introduced advanced multi\-granularity and contrastive regularizers to alleviate topological sparsity\(Xuet al\.[2023](https://arxiv.org/html/2607.21949#bib.bib11); Menget al\.[2023](https://arxiv.org/html/2607.21949#bib.bib36); Chenet al\.[2024b](https://arxiv.org/html/2607.21949#bib.bib12); Wanget al\.[2024b](https://arxiv.org/html/2607.21949#bib.bib42)\), while others leverage dual correspondence or latent relational associations to mine periodic patterns\(Lianget al\.[2023](https://arxiv.org/html/2607.21949#bib.bib38); Zhanget al\.[2023b](https://arxiv.org/html/2607.21949#bib.bib39)\)\. To capture unobserved topologies, contemporary architectures incorporate multi\-graph convolutions or historically relevant event structures to dynamically prune dependency paths\(Zhanget al\.[2024](https://arxiv.org/html/2607.21949#bib.bib41),[2025a](https://arxiv.org/html/2607.21949#bib.bib37)\)\. Despite their expressive structural encoding capabilities, these models generally lack explicit memory retention mechanisms; continuous streaming updates may gradually overwrite previously captured patterns, leading to catastrophic forgetting\.

### Regularization and Distillation

Regularization\-based paradigms penalize parameter or representation deviation to preserve historical weight locations\(Kirkpatricket al\.[2017](https://arxiv.org/html/2607.21949#bib.bib15); Zenkeet al\.[2017](https://arxiv.org/html/2607.21949#bib.bib17); Aljundiet al\.[2018](https://arxiv.org/html/2607.21949#bib.bib34)\)\. Early alignment methods enforced consistency via embedding coordinate tracking\(Wanget al\.[2019](https://arxiv.org/html/2607.21949#bib.bib19)\)\. In graph domains, these constraints manifest as temporal consistency metrics\(Wuet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib21)\),L2L\_\{2\}\-norm penalties over expanding structures\(Cuiet al\.[2023](https://arxiv.org/html/2607.21949#bib.bib22)\), or specialized debiasing paths to counter spurious forgetting\(Zhuet al\.[2025a](https://arxiv.org/html/2607.21949#bib.bib26)\)\.

Beyond weight penalties, knowledge distillation and evolutionary tokens act as structural regularizers, inheriting the foundational principle of learning without forgetting\(Li and Hoiem[2017](https://arxiv.org/html/2607.21949#bib.bib33)\)\. Modern variants align incremental graphs via incremental distillation from offline teachers\(Liuet al\.[2024](https://arxiv.org/html/2607.21949#bib.bib23)\)or handle entity growth through scale\-aware gradual evolution\(Liet al\.[2025](https://arxiv.org/html/2607.21949#bib.bib24)\)\. To optimize parameter efficiency during streaming, task\-driven tokens\(Zhuet al\.[2025b](https://arxiv.org/html/2607.21949#bib.bib25)\)and selective incremental subgraph training\(Jiaet al\.[2025](https://arxiv.org/html/2607.21949#bib.bib27)\)have been explored\. Local\-global structural distillation also serves to simultaneously constrain multi\-granularity graph semantics\(Shiet al\.[2025](https://arxiv.org/html/2607.21949#bib.bib35)\)\. Nevertheless, these continuous scalar penalties inherently force representations into a compromised intermediate state, failing to cleanly separate historical stability from current plasticity\.

### Experience Replay

Experience replay mitigates forgetting by interleaving stored or generated historical distributions with current streaming updates\(Lopez\-Paz and Ranzato[2017](https://arxiv.org/html/2607.21949#bib.bib16); Rolnicket al\.[2019](https://arxiv.org/html/2607.21949#bib.bib18); Darunaet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib20)\)\. To accommodate continuous relational drift, advanced frameworks integrate external guidance from large language models\(Wanget al\.[2024a](https://arxiv.org/html/2607.21949#bib.bib40)\)or utilize dynamic state tracking to update representations\(Zhanget al\.[2023a](https://arxiv.org/html/2607.21949#bib.bib10)\)\. Concurrently, generative replay based on conditional score\-matching and denoising diffusion probabilistic models has emerged to synthesize high\-fidelity historical contexts resilient to long\-term drift\(Hoet al\.[2020](https://arxiv.org/html/2607.21949#bib.bib29); Austinet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib30); Longet al\.[2024](https://arxiv.org/html/2607.21949#bib.bib31); Caiet al\.[2024](https://arxiv.org/html/2607.21949#bib.bib32); Zhanget al\.[2025b](https://arxiv.org/html/2607.21949#bib.bib1); Chenet al\.[2026](https://arxiv.org/html/2607.21949#bib.bib28)\)\. However, a critical bottleneck remains: many existing replay\-based TKG methods rely on coarse global scalar weights or vector concatenation to fuse replayed distributions with current states\. This ignores high\-dimensional scale disparities, inducing severenorm dominationandsemantic blurring, which motivates the need for a more adaptive replay representation fusion mechanism\.

## Problem Formulation

A Temporal Knowledge Graph𝒢\\mathcal\{G\}is formalized as a sequence of snapshots𝒢=\{G1,G2,…,G\|𝒯\|\}\\mathcal\{G\}=\\\{G\_\{1\},G\_\{2\},\\dots,G\_\{\|\\mathcal\{T\}\|\}\\\}\. Each snapshotGtG\_\{t\}contains a set of quadruples\(s,r,o,t\)\(s,r,o,t\)\. In the streaming TKG reasoning setting, the model is tasked with answering queries\(s,r,?,t\)\(s,r,?,t\)or\(?,r,o,t\)\(?,r,o,t\)given the historical graphs up tot−1t\-1\. The model is continuously trained on sequentially arriving snapshots and evaluated on its ability to perform continual reasoning while preserving previously acquired knowledge\.

## Methodology

The overall architecture of the proposed Manifold\-Aligned Dynamic Adaptive Routing \(MA\-DAR\) framework is illustrated in Figure[2](https://arxiv.org/html/2607.21949#Sx4.F2)\. Architecturally, MA\-DAR is designed as a standalone modular plugin that establishes dual data interfaces to seamlessly couple with two independent external input streams: Current Encoding and Historical Replay Representation\. Upon receiving these heterogeneous feature representations, the MA\-DAR plugin routes them through its core components to systematically alleviatenorm dominationandsemantic blurring\.

![Refer to caption](https://arxiv.org/html/2607.21949v1/x2.png)Figure 2:Overall architecture of the MA\-DAR plugin\. Operating as a plug\-and\-play extension, MA\-DAR establishes dual data interfaces to ingest external current temporal features and replayed historical representations\. These heterogeneous context representations are sequentially aligned and routed within the plugin through Manifold Alignment, Dynamic Gating, and a Polarization Penalty\.### Current Encoding

To capture the short\-term evolutionary dynamics of the current TKG snapshot, MA\-DAR interfaces with a base structural encoderℰ\\mathcal\{E\}\. Let𝒢t−m:t−1\\mathcal\{G\}\_\{t\-m:t\-1\}denote the sequence of recent historical subgraphs\. At each timestamp, the base encoder typically employs a multi\-relational Graph Convolutional Network \(GCN\) to aggregate local neighborhood information\. For a given entityee, its structural representation at layerllis updated via message passing:

he\(l\)=σ​\(∑r∈ℛ∑v∈𝒩r​\(e\)Wr\(l\)​hv\(l−1\)\)h\_\{e\}^\{\(l\)\}=\\sigma\\left\(\\sum\_\{r\\in\\mathcal\{R\}\}\\sum\_\{v\\in\\mathcal\{N\}\_\{r\}\(e\)\}W\_\{r\}^\{\(l\)\}h\_\{v\}^\{\(l\-1\)\}\\right\)\(1\)where𝒩r​\(e\)\\mathcal\{N\}\_\{r\}\(e\)is the set of neighbors connected by relationrr, andWr\(l\)W\_\{r\}^\{\(l\)\}is the relation\-specific weight matrix\.

Subsequently, a Gated Recurrent Unit \(GRU\) evolves the entity state over the temporal window to capture dynamic trends, producing the current\-step representationHc​u​r​r​e​n​t\(t\)H\_\{current\}^\{\(t\)\}:

Hc​u​r​r​e​n​t\(t\)=GRU​\(HG​C​N\(t\),Hc​u​r​r​e​n​t\(t−1\)\)H\_\{current\}^\{\(t\)\}=\\text\{GRU\}\(H\_\{GCN\}^\{\(t\)\},H\_\{current\}^\{\(t\-1\)\}\)\(2\)The mathematical independence of MA\-DAR from the internal mechanics ofℰ\\mathcal\{E\}empowers it to seamlessly interface with diverse representative TKG encoders, treating their final output identically asHc​u​r​r​e​n​t\(t\)H\_\{current\}^\{\(t\)\}\.

### Historical Replay Representation

Relying solely on recent topological updates leads to the catastrophic forgetting of long\-term patterns\. To provide a robust historical counterpart toHc​u​r​r​e​n​t\(t\)H\_\{current\}^\{\(t\)\}, MA\-DAR receives historical replay representationsHr​e​p​l​a​y\(t\)H\_\{replay\}^\{\(t\)\}provided by external replay mechanisms\. In our implementation, we instantiate the replay mechanism with the diffusion\-based generative replay strategy adopted in DGAR as a representative source of historical representations\.

In our diffusion\-based instantiation, we follow DGAR to obtain the historical replay representation through a guided reverse\-denoising process conditioned on the temporal query\. The resulting representationHreplay\(t\)H\_\{\\mathrm\{replay\}\}^\{\(t\)\}is treated as an external input to MA\-DAR\. Importantly, MA\-DAR is agnostic to the internal form of the replay mechanism and can also operate with directly sampled experience replay, as evaluated in RQ3\.

### Components of MA\-DAR

Upon receivingHcurrent\(t\)H\_\{\\text\{current\}\}^\{\(t\)\}andHreplay\(t\)H\_\{\\text\{replay\}\}^\{\(t\)\}via its dual entry interfaces, the MA\-DAR plugin processes these representations through three sequentially cascaded alignment and gating steps\.

#### Manifold Alignment

Vectors of disparate scales skew the fusion result, causing the magnitude\-heavy representation to overshadow the other\. To eliminate thisnorm dominationprior to integration, MA\-DAR projects both the current and replay representations onto a uniform hypersphere manifold:

H~c​u​r​r​e​n​t\\displaystyle\\tilde\{H\}\_\{current\}=Hc​u​r​r​e​n​t‖Hc​u​r​r​e​n​t‖2\+ϵ\\displaystyle=\\frac\{H\_\{current\}\}\{\\\|H\_\{current\}\\\|\_\{2\}\+\\epsilon\}\(3\)H~r​e​p​l​a​y\\displaystyle\\tilde\{H\}\_\{replay\}=Hr​e​p​l​a​y‖Hr​e​p​l​a​y‖2\+ϵ\\displaystyle=\\frac\{H\_\{replay\}\}\{\\\|H\_\{replay\}\\\|\_\{2\}\+\\epsilon\}\(4\)whereϵ\\epsilonis a small constant for numerical stability\. This normalization removes norm\-based scale advantages before the subsequent routing operation\. The normalization is applied independently to each entity representation\.

#### Dynamic Gating

To alleviate thesemantic blurringcaused by ambiguous global scalar weights, MA\-DAR computes a fine\-grained, dimension\-level gating vectorz∈\[0,1\]dz\\in\[0,1\]^\{d\}using a shallow multilayer perceptron \(MLP\):

z=σ​\(MLP​\(H~r​e​p​l​a​y⊕H~c​u​r​r​e​n​t\)\)z=\\sigma\(\\text\{MLP\}\(\\tilde\{H\}\_\{replay\}\\oplus\\tilde\{H\}\_\{current\}\)\)\(5\)where⊕\\oplusdenotes concatenation andσ\\sigmais the sigmoid activation function\. This vector evaluates information at the feature\-wise level, allowing the model to adaptively route specific semantic dimensions from either the historical distribution or the current context\. Finally, the adaptive semantic synthesis is performed using this gating vector:

Hf​i​n​a​l=z⊙H~r​e​p​l​a​y\+\(1−z\)⊙H~c​u​r​r​e​n​tH\_\{final\}=z\\odot\\tilde\{H\}\_\{replay\}\+\(1\-z\)\\odot\\tilde\{H\}\_\{current\}\(6\)where⊙\\odotrepresents the element\-wise Hadamard product\.

#### Polarization Penalty

Without additional structural constraints, the routing vectorzzcalculated in the gating stage may remain in ambiguous intermediate states \(e\.g\., values near 0\.5\), leading to less decisive retention behaviors\. To force definitive choices and prevent features from blurring into a noisy superposition state, we introduce a strict Polarization Penaltyℒp​o​l​a​r\\mathcal\{L\}\_\{polar\}during the optimization phase:

ℒp​o​l​a​r=1\|ℬ\|​d​∑e∈ℬ∑j=1dze,j​\(1−ze,j\),\\mathcal\{L\}\_\{polar\}=\\frac\{1\}\{\|\\mathcal\{B\}\|d\}\\sum\_\{e\\in\\mathcal\{B\}\}\\sum\_\{j=1\}^\{d\}z\_\{e,j\}\(1\-z\_\{e,j\}\),\(7\)whereℬ\\mathcal\{B\}denotes the set of entities in the current mini\-batch andze,jz\_\{e,j\}is the gate value of entityeeat dimensionjj\. This regularizer penalizes ambiguous intermediate gate values and encourages each dimension toward more decisive values near0or11, without imposing non\-differentiable hard routing decisions\.

### Joint Training Objective

After obtaining the final polarized representationHf​i​n​a​lH\_\{final\}, the decoder evaluates scores for candidate entities\. We treat entity prediction as a multi\-class classification task\. For a streaming snapshot at tasktt, the task\-specific training lossℒt​a​s​k\\mathcal\{L\}\_\{task\}is optimized via cross\-entropy:

ℒt​a​s​k=−∑q∈𝒟t​r​a​i​n\(t\)∑e∈ℰyq,e​log⁡pθ​\(e∣q\),\\mathcal\{L\}\_\{task\}=\-\\sum\_\{q\\in\\mathcal\{D\}\_\{train\}^\{\(t\)\}\}\\sum\_\{e\\in\\mathcal\{E\}\}y\_\{q,e\}\\log p\_\{\\theta\}\(e\\mid q\),\(8\)whereqqdenotes an entity prediction query,yq,ey\_\{q,e\}is its one\-hot ground\-truth label, andpθ​\(e∣q\)p\_\{\\theta\}\(e\\mid q\)is the predicted probability of candidate entityee\.

To counteract the historical information loss caused by overfitting to current temporal facts, we incorporate historical replay facts as an auxiliary replay objective\. The loss associated with historical replay,ℒr​e​p​l​a​y\\mathcal\{L\}\_\{replay\}, is calculated identically to Equation[8](https://arxiv.org/html/2607.21949#Sx4.E8), with the difference that it computes classification errors over historical facts sampled from𝒫replay\\mathcal\{P\}\_\{\\text\{replay\}\}instead of current snapshots\.

By incorporating the proposed dimension\-level polarization regularizer while maintaining the original optimization objective of the backbone encoder, the joint training objectiveℒ\\mathcal\{L\}for the entire framework is formalized as follows:

ℒ=ℒt​a​s​k\+μ​ℒr​e​p​l​a​y\+λ​ℒp​o​l​a​r\+α​ℒo​r​i​g\\mathcal\{L\}=\\mathcal\{L\}\_\{task\}\+\\mu\\mathcal\{L\}\_\{replay\}\+\\lambda\\mathcal\{L\}\_\{polar\}\+\\alpha\\mathcal\{L\}\_\{orig\}\(9\)whereμ\\mu,λ\\lambda, andα\\alphacontrol the contributions of historical replay, dimensional polarization, and the original optimization objective of the backbone encoder, respectively\.

The additional computational overhead introduced by MA\-DAR is negligible, as it only involves lightweight manifold alignment, dynamic gating, and polarization operations\. Empirical profiling shows that the MA\-DAR routing module accounts for less than 1% of the total training time across both ER and diffusion\-based replay settings\. In the profiled diffusion\-based configurations, it further accounts for only 0\.18%\-0\.22% of the total inference time\. Detailed efficiency analysis and hardware profiling are provided in the supplementary material\.

Table 1:Performance \(in percentage\) comparisons on ICEWS14s, ICEWS18, GDELT, and ICEWS05\-15 datasets under the time\-aware filter setting\. Bold values indicate the best performance in each base encoder group\.

## Experiments

To systematically evaluate the proposed framework, our experiments are designed to answer the following five Research Questions \(RQs\):

- •RQ1 \(Superiority\):Does MA\-DAR consistently enhance representative base TKG encoders in streaming scenarios?
- •RQ2 \(Ablation\):Do the core components of Manifold Alignment, Dynamic Gating, and Polarization Penalty contribute synergistically to performance improvements?
- •RQ3 \(Replay Compatibility\):Can MA\-DAR effectively integrate historical representations generated by different replay mechanisms?
- •RQ4 \(Norm Analysis\):How does Manifold Alignment alleviate scale imbalance and resolvenorm domination?
- •RQ5 \(Gating Analysis\):How does the Polarization Penalty mitigatesemantic blurringand encourage decisive dimension\-wise routing?

### Experimental Setup

#### Datasets

We evaluate the proposed MA\-DAR framework on four widely used Temporal Knowledge Graph benchmark datasets: ICEWS14s, ICEWS18, ICEWS05\-15, and GDELT\. The detailed statistics of these temporal knowledge graph benchmarks, including their chronological snapshots and temporal granularities, are provided in the supplementary material\.

#### Baselines

We compare MA\-DAR against representative state\-of\-the\-art baselines from different paradigms: \(1\) continual knowledge graph reasoning methods, including FT \(Fine\-Tuning\), ER\(Rolnicket al\.[2019](https://arxiv.org/html/2607.21949#bib.bib18)\), TIE\(Wuet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib21)\), LKGE\(Cuiet al\.[2023](https://arxiv.org/html/2607.21949#bib.bib22)\), IncDE\(Liuet al\.[2024](https://arxiv.org/html/2607.21949#bib.bib23)\), and DGAR\(Zhanget al\.[2025b](https://arxiv.org/html/2607.21949#bib.bib1)\); and \(2\) representative TKG reasoning methods with specialized modeling strategies, including RPC\(Lianget al\.[2023](https://arxiv.org/html/2607.21949#bib.bib38)\), L2TKG\(Zhanget al\.[2023b](https://arxiv.org/html/2607.21949#bib.bib39)\), and HisRES\(Zhanget al\.[2025a](https://arxiv.org/html/2607.21949#bib.bib37)\)\. To demonstrate the plug\-and\-play capability of MA\-DAR, we integrate it with diverse representative base encoders, including RE\-GCN\(Liet al\.[2021](https://arxiv.org/html/2607.21949#bib.bib8)\), TiRGN\(Liet al\.[2022](https://arxiv.org/html/2607.21949#bib.bib9)\), and LogCL\(Menget al\.[2023](https://arxiv.org/html/2607.21949#bib.bib36)\)\.

### Implementation Details

Following common settings in continual TKG reasoning, the embedding dimensionddis set to 200, the learning rate is set to 0\.001, and optimization is performed using Adam\. The historical replay sample sizekkis configured to 35, 25, 40, and 32 for ICEWS14s, ICEWS18, ICEWS05\-15, and GDELT, respectively, withμ=1\\mu=1\.

For the internal routing operations within MA\-DAR, the stability constantϵ\\epsilonin Manifold Alignment is fixed at1×10−51\\times 10^\{\-5\}\. The Dynamic Gating network is implemented as a shallow MLP with sigmoid activation\. Its hidden dimension, number of layers, and dropout rate are selected according to the adopted backbone encoder\. The polarization penalty coefficient is set toλ=0\.2\\lambda=0\.2for RE\-GCN andλ=0\.5\\lambda=0\.5for TiRGN and LogCL\. Detailed backbone\-specific configurations are provided in the supplementary material\. To ensure fair evaluation, the auxiliary loss coefficientα\\alphafollows the default structural objectives of different base encoders:α=0\.1\\alpha=0\.1for RE\-GCN’s static constraints,α=1\.0\\alpha=1\.0for TiRGN’s evolution loss, andα=0\.5\\alpha=0\.5for LogCL’s graph contrastive objective\. Detailed hyperparameter analysis is provided in the supplementary material\. To assess robustness to random initialization, we repeat the representative RE\-GCN\+MA\-DAR configuration three times with different random seeds on all four datasets\. The mean results are reported in the main paper, while the corresponding mean±\\pmstandard deviation results are provided in the supplementary material\.

Additionally, the replay mechanism employed in the main experiments is the diffusion‑based generative replay\. Under this setting, MA\-DAR can be regarded as diffusion \+ RE\-GCN\.

### Main Results \(RQ1\)

The continual reasoning performance across four benchmark datasets under the strict time\-aware filter setting is presented in Table[1](https://arxiv.org/html/2607.21949#Sx4.T1)\. The upper section of the table reports the results of continual learning baselines and representative TKG reasoning methods with specialized modeling strategies, while the lower section presents the performance of representative base encoders equipped with our MA\-DAR plugin to demonstrate the improvements brought by MA\-DAR\.

Table 2:Ablation study of the proposed MA\-DAR on three datasets across representative base encoders\. “w/o” denotes the removal of a specific component\.Based on the quantitative results, several observations can be established\. First, equipping diverse base encoders with MA\-DAR consistently improves performance across different datasets\. Regardless of whether the underlying model relies on local structural evolution modeling \(RE\-GCN\), local\-global periodic pattern modeling \(TiRGN\), or structural contrastive learning \(LogCL\), the consistent improvements from eachBaselinetoBaseline\+MA\-DAR counterpart in Table[1](https://arxiv.org/html/2607.21949#Sx4.T1)demonstrate the general applicability of MA\-DAR as a plug\-and\-play enhancement module\. For instance, integrating MA\-DAR with the basic RE\-GCN model achieves a relative MRR improvement of26\.53%on ICEWS14s and a relative H@10 improvement of56\.80%on the high\-frequency GDELT dataset under the same filter evaluation protocol\.

Second, combining LogCL with MA\-DAR achieves the strongest overall performance among all evaluated methods\. Compared with competitive baselines such as HisRES and DGAR, LogCL\+MA\-DAR achieves an absolute MRR of66\.20%on ICEWS14s, outperforming HisRES by an absolute margin of15\.72%\. It also obtains relative MRR improvements of22\.37%on ICEWS18 and9\.85%on ICEWS05\-15 over HisRES\. These results demonstrate the effectiveness of fine\-grained dimension\-level routing in MA\-DAR for integrating historical and current representations\.

### Ablation Study \(RQ2\)

To evaluate the contribution of the core components within the plugin, we examine three variants of MA\-DAR across all representative encoders: \(1\) w/o MA \(Manifold Alignment\), which skips hypersphere projection and directly feeds raw embeddings into the routing module; \(2\) w/o DG \(Dynamic Gating\), which disables the dimension\-level MLP router and replaces it with simple scalar averaging \(Hf​i​n​a​l=0\.5​H~r​e​p​l​a​y\+0\.5​H~c​u​r​r​e​n​tH\_\{final\}=0\.5\\tilde\{H\}\_\{replay\}\+0\.5\\tilde\{H\}\_\{current\}\); and \(3\) w/o PP \(Polarization Penalty\), which optimizes without theℒp​o​l​a​r\\mathcal\{L\}\_\{polar\}regularizer\.

Quantitative results in Table[2](https://arxiv.org/html/2607.21949#Sx5.T2)show that removing any individual component leads to performance degradation across all backbones, demonstrating the contribution of each component to the overall effectiveness of MA\-DAR\. Specifically, the w/o DG variant suffers the largest performance degradation \(e\.g\., LogCL’s MRR decreases by 9\.80% on ICEWS14s\), indicating that scalar averaging is insufficient for effective representation integration\. Meanwhile, the degradation of the w/o MA variant demonstrates that removing hypersphere alignment makes the fusion process more sensitive to scale discrepancies between historical and current representations\. Finally, the performance decrease of the w/o PP variant indicates that removing polarization regularization weakens the ability of the gating module to make decisive dimension\-wise routing decisions\. Overall, these results demonstrate that Manifold Alignment, Dynamic Gating, and Polarization Penalty provide complementary benefits, and their combination enables MA\-DAR to effectively integrate historical and current representations, answering RQ2\.

### Replay Compatibility Analysis \(RQ3\)

To investigate whether MA\-DAR can effectively utilize historical representations from different replay mechanisms, we further evaluate its compatibility with both real experience replay \(ER\) and generative replay settings\. Specifically, we combine MA\-DAR with LogCL under different historical representation sources and compare the resulting performance\.

Table 3:Replay compatibility analysis of MA\-DAR on ICEWS14s\.The results are summarized in Table[3](https://arxiv.org/html/2607.21949#Sx5.T3)\. Under the real experience replay setting, MA\-DAR improves LogCL from 48\.87% to 51\.70% MRR, demonstrating that the proposed alignment and routing mechanism remains effective when historical representations are obtained through direct sampling\. Furthermore, under the generative replay setting, integrating MA\-DAR with LogCL further improves the MRR from 51\.58% to 66\.20%\. The consistent improvements under different replay sources indicate that MA\-DAR focuses on resolving representation integration conflicts rather than relying on a specific replay generation mechanism\.

### Norm Domination Analysis \(RQ4\)

![Refer to caption](https://arxiv.org/html/2607.21949v1/x3.png)Figure 3:Analysis of feature norms\. \(a\) L2 norm distributions of raw current and replay representations\. \(b\) Projection onto the unit hypersphere manifold via Manifold Alignment\.To answer RQ4, we visualize the L2 norms of the hidden representations before and after Manifold Alignment\. As depicted in Figure[3](https://arxiv.org/html/2607.21949#Sx5.F3)\(a\), the raw feature spaces exhibit noticeablenorm domination\. Specifically, the historical replay representations \(Hr​e​p​l​a​yH\_\{replay\}\) exhibit a different magnitude distribution compared with the current representations \(Hc​u​r​r​e​n​tH\_\{current\}\), resulting in an imbalance during direct feature fusion\. This magnitude discrepancy indicates that directly combining the raw representations may cause one representation source to dominate the fusion process\. If these raw representations are directly fused, the magnitude\-dominant representation may bias the optimization process and weaken the contribution of the other representation source\.

As shown in Figure[3](https://arxiv.org/html/2607.21949#Sx5.F3)\(b\), Manifold Alignment projects these heterogeneous representations onto a uniform hypersphere manifold \(‖H‖2=1\|\|H\|\|\_\{2\}=1\)\. This normalization reduces the scale discrepancy between historical and current representations, alleviatingnorm dominationbefore subsequent semantic routing\.

### Regularized Gating Analysis \(RQ5\)

![Refer to caption](https://arxiv.org/html/2607.21949v1/x4.png)Figure 4:Distribution of dimension\-level gating valueszzon ICEWS14s using RE\-GCN equipped with MA\-DAR under the diffusion replay setting\. \(a\) Standard soft\-gating without the polarization penalty\. \(b\) Gating distribution with the polarization penalty, encouraging more decisive dimension\-wise routing decisions\.![Refer to caption](https://arxiv.org/html/2607.21949v1/gate_trajectory_ICEWS14s.png)Figure 5:Case study of temporal gating evolution for representative entities on ICEWS14s using RE\-GCN equipped with MA\-DAR under the diffusion replay setting\.To address RQ5, we extract and visualize the test\-phase distribution of dimension\-level gating weightsz∈\[0,1\]dz\\in\[0,1\]^\{d\}on ICEWS14s using RE\-GCN equipped with MA\-DAR under the diffusion replay setting\. As illustrated in Figure[4](https://arxiv.org/html/2607.21949#Sx5.F4)\(a\), soft gating withoutℒp​o​l​a​r\\mathcal\{L\}\_\{polar\}regularization leaves a large proportion of gating values \(41\.5%\) in the ambiguous middle zone \(0\.4≤z≤0\.60\.4\\leq z\\leq 0\.6\), which may lead tosemantic blurringduring representation fusion\.

With the proposed Polarization Penalty, the proportion of ambiguous intermediate values is reduced to 19\.7% \(Figure[4](https://arxiv.org/html/2607.21949#Sx5.F4)\(b\)\), encouraging gating values toward clearer binary\-like decisions\. This distribution indicates that MA\-DAR produces more decisive dimension\-wise routing patterns, reducing uncertain intermediate allocations during representation fusion\. Unlike a non\-differentiable hard threshold, the proposed penalty encourages decisive routing while preserving end\-to\-end gradient\-based optimization\.

Beyond the statistical distribution, we further provide a case study to illustrate how MA\-DAR dynamically adjusts gating behaviors during temporal evolution\. As shown in Figure[5](https://arxiv.org/html/2607.21949#Sx5.F5), we track the gating trajectories of representative entities on ICEWS14s using the same RE\-GCN\-based MA\-DAR configuration under diffusion replay\. The trajectories show that gating values vary across timestamps rather than remaining fixed, indicating that MA\-DAR dynamically adjusts the contributions of historical and current representations according to evolving temporal contexts\.

## Conclusion

In this paper, we introduced MA\-DAR, a general plug\-and\-play routing for replay representation fusion in continual TKG reasoning\. By decoupling historical representation integration from backbone temporal encoding, MA\-DAR can be flexibly combined with diverse architectures, including RE\-GCN, TiRGN, and LogCL\. Through Manifold Alignment, Dynamic Gating, and Polarization Penalty, MA\-DAR effectively resolves representation conflicts between historical and current features by alleviatingnorm dominationandsemantic blurring\. Extensive experiments under the strict time\-aware filter protocol demonstrate that MA\-DAR consistently improves different backbone encoders and establishes new state\-of\-the\-art performance across multiple benchmarks\.

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