Dynamic Link Prediction with Temporally Enhanced Signed Graph Neural Networks

# Dynamic Link Prediction with Temporally Enhanced Signed Graph Neural Networks
Source: [https://arxiv.org/html/2605.26290](https://arxiv.org/html/2605.26290)
###### Abstract

Temporal signed networks \(TSNs\) model the time evolution of cooperative and adversarial relationships that arise in applications such as social media analysis, trust and reputation systems, and financial transaction networks\. While graph neural networks \(GNNs\) perform well for static or unsigned link prediction, effective learning in temporal signed graphs remains challenging due to the interaction of signed relations, evolving structure, and balance\-theoretic constraints\. To address this gap, we propose a*modular*temporal enhancement framework for signed GNNs that integrates historical context into otherwise static architectures\. The framework introduces a Historical Context Integration Module \(HCIM\) that combines learnable recency\-aware temporal weighting, LSTM\-based embedding trajectory modeling, and multi\-head temporal attention to capture both short\- and long\-term signed interaction dynamics\. Historical information is fused with current node representations using either global or node\-adaptive weighting, allowing the architecture\-agnostic framework to accommodate heterogeneous temporal behaviors\. We instantiate the approach on the Self\-Explainable Signed Graph Transformer \(SE\-SGformer\), preserving interpretability while extending it with temporal awareness\. Experiments on real\-world and synthetic TSNs, including Bitcoin OTC, Bitcoin Alpha, Reddit, and small\-world network models, demonstrate consistent and statistically significant improvements over the static baseline\.

## IIntroduction

Many real\-world networked systems operate in environments where interactions evolve over time\. Social platforms, trust and reputation systems, and financial transaction networks continuously change as entities form, reinforce, or sever relationships\. Modeling such systems therefore requires representations that capture not only connectivity, but also how interaction patterns evolve in response to behavioral and systemic dynamics\. This naturally leads to the formulation of*temporal graphs*, where network structure and interactions are treated as time\-varying rather than static\.

Recent advances in temporal graph neural networks \(GNNs\) have significantly improved the modeling of such evolving systems\. Methods such as temporal graph networks \(TGNs\)\[[1](https://arxiv.org/html/2605.26290#bib.bib1)\], DyRep\[[2](https://arxiv.org/html/2605.26290#bib.bib2)\], and JODIE\[[3](https://arxiv.org/html/2605.26290#bib.bib3)\]explicitly incorporate temporal dependencies and have demonstrated substantial gains over static approaches in dynamic link prediction tasks\. However, these models have been developed almost exclusively for*unsigned*temporal graphs, where edges encode only the presence or intensity of interactions\. This focus does not account for the rich semantics introduced by*signed*relationships, in which edges carry positive or negative polarity corresponding to trust versus distrust, endorsement versus opposition, or cooperation versus antagonism\.

Signed interactions are fundamental to understanding the dynamics of many real\-world networks\. In cryptocurrency trust networks, such as Bitcoin OTC and Bitcoin Alpha, positive edges reflect trustworthiness while negative edges often indicate fraudulent behavior or risky interactions\[[4](https://arxiv.org/html/2605.26290#bib.bib4),[5](https://arxiv.org/html/2605.26290#bib.bib5)\]\. Similarly, Reddit hyperlink networks encode editorial stance through signed references that capture supportive or contradictory relations between communities\[[3](https://arxiv.org/html/2605.26290#bib.bib3),[6](https://arxiv.org/html/2605.26290#bib.bib6)\]\. Accurately predicting not only whether a future link will occur, but also its polarity, referred to as*signed link prediction*, is therefore critical for applications ranging from fraud detection to community curation and recommendation systems\.

### I\-AThe Temporal Signed Graph Challenge

Classical signed network analysis relies on structural balance theory and motif analysis to explain network stability and hierarchy through signed triadic structures\[[7](https://arxiv.org/html/2605.26290#bib.bib7),[8](https://arxiv.org/html/2605.26290#bib.bib8)\]\. Building on these foundations, recent signed graph neural networks \(SGNNs\), particularly the Self\-Explainable Signed Graph Transformer \(SE\-SGformer\)\[[9](https://arxiv.org/html/2605.26290#bib.bib9)\], achieve strong performance in static signed link prediction through multi\-hop signed neighborhoods and structural attention biases while preserving interpretability\.

Despite these advances, most SGNNs still treat networks as static snapshots by either aggregating interactions across time or ignoring temporal evolution entirely\. Explicit temporal modeling remains largely absent in signed graph learning, limiting applicability in dynamic environments such as cryptocurrency trust networks and online social platforms, where historical interactions evolve in importance over time\. Integrating temporal dynamics into signed graph learning therefore introduces several challenges that existing methods do not adequately address:

- •Temporal information decay:The influence of historical signed interactions naturally diminishes as time passes, necessitating systematic mechanisms for weighting recent versus older evidence\. As such, static SGNNs lack adaptive models of temporal decay\.
- •Evolving balance structures:Structural balance patterns are not fixed, i\.e\., as networks grow and change, local signed motifs evolve, requiring continual recalibration of balance\-based representations\.
- •Heterogeneous temporal dynamics:Different nodes and network regions exhibit distinct temporal behaviors\. Highly central or influential nodes may retain long\-term relevance, whereas peripheral nodes may display rapidly decaying influence, motivating adaptive rather than uniform temporal weighting\[[10](https://arxiv.org/html/2605.26290#bib.bib10)\]\.
- •Interpretability under temporal complexity:While interpretability is essential in trust\-critical applications, incorporating temporal mechanisms often reduces transparency\. Existing temporal attention models provide limited insight into how temporal factors influence signed predictions\.

### I\-BMain Contributions

To address these limitations, we propose a temporal enhancement framework that extends static SGNNs with historical context integration while preserving their structural foundations and interpretability\. Rather than introducing a new temporal architecture, the proposed approach augments existing SGNNs with a modular temporal component\. The main contributions of this work are summarized as follows:

- •Historical Context Integration Module \(HCIM\):We introduce a systematic three\-stage temporal framework that combines learnable recency\-aware weighting, LSTM\-based embedding trajectory modeling, and multi\-head temporal attention\. As shown in Figure[1](https://arxiv.org/html/2605.26290#S1.F1), this design enables fine\-grained, data\-driven control over the influence of historical signed interactions\.
- •Dual temporal weighting strategies:We propose both complementary mechanisms for temporal information integration: \(i\) global parameterized weights for homogeneous temporal patterns in synthetic networks, and \(ii\) node\-adaptive Multilayer Perception \(MLP\)\-based weighting that accommodates heterogeneous temporal dynamics in real\-world data\.
- •Interpretable temporal integration:The framework preserves model explainability through explicit learning parameters for recency bias and temporal combination weights, thus enabling practitioners to understand and validate temporal decision factors in signed graph learning applications\.
- •Comprehensive empirical validation:Extensive experiments on real\-world and synthetic temporal signed networks \(TSNs\) demonstrate consistent and statistically significant improvements over static baselines, including relative AUC gains of 3\.79% on Bitcoin OTC, 5\.76% on Bitcoin Alpha, and over 13% on Barabási\-Albert \(BA\) and Watts\-Strogatz \(WS\) networks, with corresponding error reductions of up to 41\.77%\.

![Refer to caption](https://arxiv.org/html/2605.26290v1/x1.png)Figure 1:Our proposed module for integrating historical context into SGNN models\.The proposed framework is architecture\-agnostic and can be integrated with a wide range of SGNNs\. We validate its effectiveness using SE\-SGformer due to its strong performance and interpretability, while emphasizing that the approach generalizes beyond this specific backbone\.

### I\-CSignificance, Impact, and Technical Innovation

This work addresses a critical gap between temporal graph learning and SGNNs by introducing a mechanism for modeling evolving signed interactions while preserving the structural foundations and interpretability of existing SGNNs\. Unlike prior approaches that either ignore temporal dynamics or redesign signed architectures entirely, the proposed framework extends static SGNNs with explicit temporal awareness for accurate signed link prediction in dynamic and adversarial environments\.

The core technical innovation is the proposed HCIM, a modular component that injects temporal information into signed graph representations without modifying the backbone model’s signed\-structural encodings\. HCIM introduces two temporal fusion strategies, namelyglobalandnode\-adaptive weighting, to capture both homogeneous temporal behavior in synthetic networks and heterogeneous dynamics in real\-world systems\. Temporal influence is modeled through explicit, learnable parameters, including recency bias and fusion coefficients, enabling post\-hoc analysis of how historical context contributes to signed predictions\.

From an application perspective, the proposed framework advances temporal signed link prediction in domains such as financial fraud detection and online community moderation, where understanding the evolution of cooperative and adversarial relationships is essential\. By building upon balance theory\[[7](https://arxiv.org/html/2605.26290#bib.bib7)\]and status theory\[[11](https://arxiv.org/html/2605.26290#bib.bib11)\]while extending SGNNs to dynamic settings, this work establishes a scalable, interpretable, and architecture\-agnostic foundation for temporal signed graph learning\.

### I\-DPaper Organization

The remainder of this paper is organized as follows\. Section[II](https://arxiv.org/html/2605.26290#S2)reviews related work in signed and temporal graph learning\. Section[III](https://arxiv.org/html/2605.26290#S3)introduces the notation and mathematical preliminaries for temporal signed graphs together with the HCIM formulation\. Section[IV](https://arxiv.org/html/2605.26290#S4)presents the experimental setup and evaluation protocol\. Section[V](https://arxiv.org/html/2605.26290#S5)presents empirical results together with analytical insights and practical implications\. Section[VI](https://arxiv.org/html/2605.26290#S6)outlines limitations and future research directions\. Finally, Section[VII](https://arxiv.org/html/2605.26290#S7)concludes the paper\.

## IIRelated Works

Research on temporal signed graph learning spans dynamic GNNs, attention\-based temporal modeling, signed network representation learning, and temporal decay mechanisms\[[12](https://arxiv.org/html/2605.26290#bib.bib12)\]\. This section reviews the foundational literature and positions our contributions within the broader research landscape\.

### II\-ATheoretical Foundations for Signed Networks

Signed network evolution is governed by well\-established social theories that underpin modern computational approaches\.*Structural balance theory*\(\[[13](https://arxiv.org/html/2605.26290#bib.bib13),[7](https://arxiv.org/html/2605.26290#bib.bib7)\]\) states that a cycle is stable when the product of its edge signs is positive, reflecting the social principle that “the enemy of my enemy is my friend”\. This concept directly motivates the signed random walk encodings and structural attention biases used in SGNNs such as SE\-SGformer\.

Complementing balance theory,*status theory*\[[11](https://arxiv.org/html/2605.26290#bib.bib11)\]assumes that nodes possess latent status values, with edge signs determined by status differences\. This framework often provides stronger explanatory power in directed signed networks\. SE\-SGformer incorporates these hierarchical principles through centrality\-based encodings that bias attention toward status\-consistent signed relationships, enabling the model to learn implicit node hierarchies that drive relationship formation\.

### II\-BDynamic GNNs and Recurrent Architectures

Early research on temporal GNNs established the foundation for modeling evolving graph topologies through recurrent and sequential architectures\. E\-LSTM\-D introduced an encoder\-LSTM\-decoder framework for end\-to\-end dynamic link prediction\[[14](https://arxiv.org/html/2605.26290#bib.bib14)\], while GC\-LSTM incorporated graph convolution operations directly within LSTM cells to better capture structural evolution across graph snapshots\[[15](https://arxiv.org/html/2605.26290#bib.bib15)\]\. These pioneering efforts demonstrated the effectiveness of combining graph convolutions with sequential modeling\.

Subsequent works extended this paradigm through sophisticated temporal embedding strategies\. For instance, dynnode2vec incorporated historical initialization into node2vec\-based embeddings to improve temporal consistency across snapshots\[[16](https://arxiv.org/html/2605.26290#bib.bib16)\], while LSTM\-history tracking methods explicitly modeled embedding evolution through recurrent memory mechanisms\[[17](https://arxiv.org/html/2605.26290#bib.bib17)\]\. Bi\-DNE further advanced temporal representation learning by introducing bilayer architectures capable of jointly modeling micro\- and macro\-level network evolution patterns\[[18](https://arxiv.org/html/2605.26290#bib.bib18)\]\. More recently, JODIE demonstrated the effectiveness of trajectory\-based embedding evolution for temporal interaction networks, showing that dynamic representations derived from sequential dependencies substantially improve predictive performance\[[3](https://arxiv.org/html/2605.26290#bib.bib3)\]\.

### II\-CAttention\-Based Temporal Modeling and Recency Learning

Attention mechanisms and time\-aware weighting strategies have become central components of modern temporal graph learning frameworks\. Self\-attention architectures are particularly effective for capturing fine\-grained temporal dependencies and selectively emphasizing informative historical interactions\. TGAT pioneered the use of temporal encoding with attention for inductive learning on dynamic graphs, establishing a foundation for attention\-based temporal graph architectures\[[19](https://arxiv.org/html/2605.26290#bib.bib19)\]\. TGN further advanced this direction by integrating message passing with memory modules for continuous\-time graph representation learning\[[1](https://arxiv.org/html/2605.26290#bib.bib1)\], while DySAT showed the effectiveness of jointly modeling structural and temporal dependencies through dual self\-attention mechanisms\[[20](https://arxiv.org/html/2605.26290#bib.bib20)\]\. Recent efforts such as L\-STEP introduced learnable spatial\-temporal positional encodings, further refining attention\-based temporal modeling\[[21](https://arxiv.org/html/2605.26290#bib.bib21)\]\.

Complementing attention\-based approaches, several studies have explored explicit temporal decay and recency\-aware weighting mechanisms to model the diminishing relevance of historical interactions\. TPNet introduced a learnable time\-decay mechanism that assigns greater importance to recent interactions, revealing the effectiveness of adaptive temporal weighting in link prediction tasks\[[22](https://arxiv.org/html/2605.26290#bib.bib22)\]\. TREND modeled temporal significance through event\-node dynamics\[[23](https://arxiv.org/html/2605.26290#bib.bib23)\], while causal anonymous walks incorporated recency information through timestamp\-aware sequential updates\[[24](https://arxiv.org/html/2605.26290#bib.bib24)\]\. Collectively, these approaches emphasize that temporal influence in evolving graphs is inherently non\-uniform and should be learned rather than manually prescribed\. Together, these studies suggest that temporal influence in evolving graphs varies across time and should be learned rather than manually prescribed\. Our approach extends these concepts by combining exponential decay and recency amplification into a unified, learnable, and differentiable preprocessing step that can be seamlessly integrated with existing static architectures\.

### II\-DSigned and Temporal GNNs

Static SGNNs established the computational foundation for learning from signed networks\. The Signed Graph Convolutional Network \(SGCN\) was the first to incorporate balance and status theory into graph message passing, enabling effective aggregation of signed neighborhood information while preserving structural constraints\[[25](https://arxiv.org/html/2605.26290#bib.bib25)\]\. Building on SGCN, the Signed Graph Attention Network \(SiGAT\) introduced attention mechanisms tailored to signed graphs, allowing adaptive weighting of positive and negative neighbors\[[26](https://arxiv.org/html/2605.26290#bib.bib26)\]\. Subsequent models such as SIGNet\[[27](https://arxiv.org/html/2605.26290#bib.bib27)\], SiNE\[[28](https://arxiv.org/html/2605.26290#bib.bib28)\], and SDGNN\[[29](https://arxiv.org/html/2605.26290#bib.bib29)\]further improved signed representation learning through higher\-order structural modeling, embedding\-based formulations, and directed signed graph encoding\. In\[[9](https://arxiv.org/html/2605.26290#bib.bib9)\], SE\-SGformer extended transformer architectures to signed graph learning through signed random walk encodings and structural attention biases, achieving strong performance while providing interpretable predictions\. However, like most existing SGNNs, SE\-SGformer operates on static graph snapshots and does not explicitly model temporal evolution\.

Temporal dynamics have also begun to receive attention in signed graph learning\. DynamiSE\[[30](https://arxiv.org/html/2605.26290#bib.bib30)\]and PolarDSN\[[31](https://arxiv.org/html/2605.26290#bib.bib31)\]extend signed representation learning to dynamic settings, while SEMBA\[[32](https://arxiv.org/html/2605.26290#bib.bib32)\]introduces memory\-augmented architectures for continuous\-time signed network evolution\. Although these approaches improve temporal signed representation learning, they generally lack explicit recency\-aware weighting and interpretable sequential\-attention fusion mechanisms\.

Our framework builds upon foundational graph learning architectures, including graph convolutional networks \(GCNs\)\[[33](https://arxiv.org/html/2605.26290#bib.bib33)\], graph attention networks \(GATs\)\[[34](https://arxiv.org/html/2605.26290#bib.bib34)\], and GraphSAGE\[[35](https://arxiv.org/html/2605.26290#bib.bib35)\]\. Transformer architectures\[[36](https://arxiv.org/html/2605.26290#bib.bib36)\]and neural machine translation attention mechanisms\[[37](https://arxiv.org/html/2605.26290#bib.bib37)\]further motivate the multi\-head temporal attention component of HCIM\.

## IIIMathematical Framework and Notation

This section introduces the mathematical notation and methodological foundations used throughout the paper\.

AsignedgraphG=\(V,E\+,E−\)G=\(V,E^\{\+\},E^\{\-\}\)consists of a node setVVwith\|V\|=n\|V\|=n, positive edgesE\+⊆V×VE^\{\+\}\\subseteq V\\times V, and negative edgesE−⊆V×VE^\{\-\}\\subseteq V\\times V, whereE\+∩E−=∅E^\{\+\}\\cap E^\{\-\}=\\emptyset\. The edge sign functionϕ:E→\{\+1,−1\}\\phi:E\\rightarrow\\\{\+1,\-1\\\}assigns polarity to each edge\. Depending on the application domain, signed edges may be directed to capture asymmetric trust, reputation, or adversarial relationships\. For allu,v∈Vu,v\\in V, the signed adjacency matrix𝐀∈ℝn×n\\mathbf\{A\}\\in\\mathbb\{R\}^\{n\\times n\}is defined as:

𝐀u​v=\{\+1,if​\(u,v\)∈E\+−1,if​\(u,v\)∈E−0,otherwise\\mathbf\{A\}\_\{uv\}=\\begin\{cases\}\+1,&\\text\{if \}\(u,v\)\\in E^\{\+\}\\\\ \-1,&\\text\{if \}\(u,v\)\\in E^\{\-\}\\\\ 0,&\\text\{otherwise\}\\end\{cases\}\(1\)
Atemporalsigned graph is represented as a sequence𝒢=\{G\(1\),G\(2\),…,G\(T\)\}\\mathcal\{G\}=\\\{G^\{\(1\)\},G^\{\(2\)\},\\ldots,G^\{\(T\)\}\\\}, where each snapshotG\(t\)=\(V,E\+\(t\),E−\(t\)\)G^\{\(t\)\}=\(V,E^\{\+\(t\)\},E^\{\-\(t\)\}\)captures the signed network state at timett\. Each snapshot is independently processed by the backbone SGNN to generate node embeddings, which are subsequently integrated through the proposed HCIM to capture temporal dependencies across graph states\.

### III\-ASE\-SGformer Baseline Overview

Our framework builds upon the SE\-SGformer, a transformer\-based architecture designed for signed link prediction with interpretable structural reasoning\. SE\-SGformer combines spectral feature extraction, signed structural encoding, and transformer\-based representation learning to capture both local and higher\-order signed relationships\. Specifically, the model consists of three primary components:

1. 1\.Spectral feature generation:Singular value decomposition \(SVD\) is applied to the signed adjacency matrix to generate low\-dimensional spectral node representations that preserve global structural information\.
2. 2\.Structural encoding:Multiple signed structural descriptors, including centrality features, adjacency information, and signed random walk encodings, are incorporated to capture balance\- and status\-aware relational patterns\.
3. 3\.Transformer\-based representation learning:Multi\-head self\-attention layers equipped with structural bias terms learn contextualized signed node embeddings while preserving graph structural dependencies\.

The resulting node embeddings are processed through a three\-class discriminator for signed link prediction using geometric triplet\-based objectives\. Although SE\-SGformer achieves strong predictive performance and interpretability in static signed graphs, it does not explicitly model temporal evolution across graph snapshots\.

### III\-BHistorical Context Integration Module \(HCIM\)

Although SE\-SGformer effectively captures structural patterns in signed graphs, it operates on static graph snapshots and therefore cannot explicitly model the temporal evolution of signed interactions\. To address this limitation, we introduce the proposed HCIM, which extracts temporally informed representations from historical graph embeddings through three sequential stages:adaptive temporal weighting,sequential pattern modeling, andattention\-based context integration\.

Given a sequence of historical node embeddings\{𝐙\(t−T\),𝐙\(t−T\+1\),…,𝐙\(t−1\)\}\\\{\\mathbf\{Z\}^\{\(t\-T\)\},\\mathbf\{Z\}^\{\(t\-T\+1\)\},\\ldots,\\mathbf\{Z\}^\{\(t\-1\)\}\\\}, the HCIM given in Algorithm[1](https://arxiv.org/html/2605.26290#alg1)produces a temporally enriched representation𝐇context\\mathbf\{H\}\_\{\\text\{context\}\}that captures both short\- and long\-term signed interaction dynamics as detailed below\.

Algorithm 1Historical Context Integration Module \(HCIM\)0:Historical embeddings

\{𝐙\(0\),…,𝐙\(T−1\)\}\\\{\\mathbf\{Z\}^\{\(0\)\},\\dots,\\mathbf\{Z\}^\{\(T\-1\)\}\\\}, decay factor

λ\\lambda, recency parameter

γ\\gamma
0:Historical context representation

𝐇context∈ℝN×d\\mathbf\{H\}\_\{\\text\{context\}\}\\in\\mathbb\{R\}^\{N\\times d\}
1:Stage 1: Adaptive Temporal Weighting

2:for

τ=0\\tau=0to

T−1T\-1do

3:

wτ←λT−1−τ⋅eτ/γ∑i=0T−1λT−1−i⋅ei/γ\+ϵw\_\{\\tau\}\\leftarrow\\frac\{\\lambda^\{T\-1\-\\tau\}\\cdot e^\{\\tau/\\gamma\}\}\{\\sum\_\{i=0\}^\{T\-1\}\\lambda^\{T\-1\-i\}\\cdot e^\{i/\\gamma\}\+\\epsilon\}
4:

𝐙~\(τ\)←wτ⋅𝐙\(τ\)\\tilde\{\\mathbf\{Z\}\}^\{\(\\tau\)\}\\leftarrow w\_\{\\tau\}\\cdot\\mathbf\{Z\}^\{\(\\tau\)\}
5:endfor

6:Stage 2: Sequential Pattern Modeling

7:

𝐇seq←stack​\(𝐙~\(0\),…,𝐙~\(T−1\)\)\\mathbf\{H\}\_\{\\text\{seq\}\}\\leftarrow\\text\{stack\}\(\\tilde\{\\mathbf\{Z\}\}^\{\(0\)\},\\dots,\\tilde\{\\mathbf\{Z\}\}^\{\(T\-1\)\}\)
8:

𝐇lstm,\(𝐡T,𝐜T\)←LSTM​\(𝐇seq\)\\mathbf\{H\}\_\{\\text\{lstm\}\},\(\\mathbf\{h\}\_\{T\},\\mathbf\{c\}\_\{T\}\)\\leftarrow\\text\{LSTM\}\(\\mathbf\{H\}\_\{\\text\{seq\}\}\)
9:Stage 3: Attention\-Based Context Integration

10:

𝐐,𝐊,𝐕←𝐇lstm\\mathbf\{Q\},\\mathbf\{K\},\\mathbf\{V\}\\leftarrow\\mathbf\{H\}\_\{\\text\{lstm\}\}
11:

𝐇attn←MultiHeadAttn​\(𝐐,𝐊,𝐕\)\\mathbf\{H\}\_\{\\text\{attn\}\}\\leftarrow\\text\{MultiHeadAttn\}\(\\mathbf\{Q\},\\mathbf\{K\},\\mathbf\{V\}\)
12:

𝐇context←Proj​\(𝐇attn​\[:,T−1,:\]\)\\mathbf\{H\}\_\{\\text\{context\}\}\\leftarrow\\text\{Proj\}\(\\mathbf\{H\}\_\{\\text\{attn\}\}\[:,T\-1,:\]\)
13:return

𝐇context\\mathbf\{H\}\_\{\\text\{context\}\}

#### III\-B1Stage 1: Adaptive Temporal Weighting

The first stage applies learnable recency\-aware weighting to historical node embeddings in order to model the diminishing influence of older interactions\. Given historical embeddings\{𝐙\(t−T\),…,𝐙\(t−1\)\}\\\{\\mathbf\{Z\}^\{\(t\-T\)\},\\dots,\\mathbf\{Z\}^\{\(t\-1\)\}\\\}, the temporal weighting coefficient for timestepτ\\tauis defined as:

wτ=λT−1−τ⋅eτ/γ∑i=0T−1λT−1−i⋅ei/γ\+ϵ,w\_\{\\tau\}=\\frac\{\\lambda^\{T\-1\-\\tau\}\\cdot e^\{\\tau/\\gamma\}\}\{\\sum\_\{i=0\}^\{T\-1\}\\lambda^\{T\-1\-i\}\\cdot e^\{i/\\gamma\}\+\\epsilon\},\(2\)whereλ∈\(0,1\)\\lambda\\in\(0,1\)controls exponential temporal decay,γ\>0\\gamma\>0determines recency emphasis, andϵ\\epsilonis a numerical stability constant\. Larger values ofγ\\gammaamplify the influence of more recent graph snapshots, while smaller values encourage smoother temporal weighting across historical embeddings\. The weighted embeddings are then computed as𝐙~\(τ\)=wτ⋅𝐙\(τ\)\\tilde\{\\mathbf\{Z\}\}^\{\(\\tau\)\}\\\!=\\\!w\_\{\\tau\}\\\!\\cdot\\\!\\mathbf\{Z\}^\{\(\\tau\)\}\. This weighting strategy enables the framework to adaptively prioritize informative historical interactions rather than treating all past snapshots equally\.

#### III\-B2Stage 2: Sequential Pattern Modeling

The weighted historical embeddings are stacked into a temporal sequence and processed using an LSTM network:

𝐇seq\\displaystyle\\mathbf\{H\}\_\{\\text\{seq\}\}=stack​\(𝐙~\(0\),…,𝐙~\(T−1\)\),\\displaystyle=\\text\{stack\}\(\\tilde\{\\mathbf\{Z\}\}^\{\(0\)\},\\dots,\\tilde\{\\mathbf\{Z\}\}^\{\(T\-1\)\}\),\(3\)𝐇lstm\\displaystyle\\mathbf\{H\}\_\{\\text\{lstm\}\}=LSTM​\(𝐇seq\)\.\\displaystyle=\\text\{LSTM\}\(\\mathbf\{H\}\_\{\\text\{seq\}\}\)\.\(4\)The LSTM captures sequential dependencies in the evolution of node embeddings across graph snapshots, enabling the framework to model temporal interaction trajectories and long\-range temporal dependencies in signed relationships\.

#### III\-B3Stage 3: Attention\-Based Context Integration

Following sequential modeling, multi\-head self\-attention is applied to selectively emphasize historically informative temporal patterns:

𝐇attn=MultiHeadAttn​\(𝐇lstm,𝐇lstm,𝐇lstm\)\.\\mathbf\{H\}\_\{\\text\{attn\}\}=\\text\{MultiHeadAttn\}\(\\mathbf\{H\}\_\{\\text\{lstm\}\},\\mathbf\{H\}\_\{\\text\{lstm\}\},\\mathbf\{H\}\_\{\\text\{lstm\}\}\)\.\(5\)
The representation corresponding to the final timestep is subsequently projected to obtain the historical context representation:

𝐇context=Proj​\(𝐇attn​\[:,T−1,:\]\)\.\\mathbf\{H\}\_\{\\text\{context\}\}=\\text\{Proj\}\(\\mathbf\{H\}\_\{\\text\{attn\}\}\[:,T\-1,:\]\)\.\(6\)This stage enables the model to focus on temporally salient historical information beyond what can be captured through sequential recurrence alone\.

### III\-CTemporal Integration Strategies

The extracted historical context representation𝐇context\\mathbf\{H\}\_\{\\text\{context\}\}is integrated with the current node embeddings𝐙current\\mathbf\{Z\}\_\{\\text\{current\}\}using one of two temporal fusion strategies\.

#### III\-C1Global Weighting

The global weighting strategy applies a single learnable fusion parameterα∈\[0,1\]\\alpha\\in\[0,1\]uniformly across all nodes:

𝐙enhanced=\(1−α\)​𝐙current\+α​𝐇context\.\\mathbf\{Z\}\_\{\\text\{enhanced\}\}=\(1\-\\alpha\)\\mathbf\{Z\}\_\{\\text\{current\}\}\+\\alpha\\mathbf\{H\}\_\{\\text\{context\}\}\.\(7\)This formulation assumes relatively homogeneous temporal behavior across the graph and is particularly effective for synthetic or structurally regular networks\.

#### III\-C2Node\-Adaptive Weighting

To accommodate heterogeneous temporal behavior across nodes, we further introduce an adaptive fusion mechanism in which node\-specific fusion weights are generated using a multilayer perceptron \(MLP\):

𝜶node=σ​\(MLP​\(\[𝐙current∥𝐇context\]\)\),\\bm\{\\alpha\}\_\{\\text\{node\}\}=\\sigma\\left\(\\text\{MLP\}\(\[\\mathbf\{Z\}\_\{\\text\{current\}\}\\\|\\mathbf\{H\}\_\{\\text\{context\}\}\]\)\\right\),\(8\)whereσ​\(⋅\)\\sigma\(\\cdot\)denotes the sigmoid activation function and∥\\\|represents vector concatenation\. The enhanced embeddings are then computed as follows, where⊙\\odotdenotes element\-wise multiplication:

𝐙enhanced=\(1−𝜶node\)⊙𝐙current\+𝜶node⊙𝐇context\.\\mathbf\{Z\}\_\{\\text\{enhanced\}\}=\(1\-\\bm\{\\alpha\}\_\{\\text\{node\}\}\)\\odot\\mathbf\{Z\}\_\{\\text\{current\}\}\+\\bm\{\\alpha\}\_\{\\text\{node\}\}\\odot\\mathbf\{H\}\_\{\\text\{context\}\}\.\(9\)
Unlike global fusion, this strategy enables different nodes to dynamically adjust the relative importance of historical and current information according to their temporal interaction patterns\. The temporally enhanced embeddings𝐙enhanced\\mathbf\{Z\}\_\{\\text\{enhanced\}\}replace the original node embeddings in the final prediction layers of SE\-SGformer\. Importantly, the original loss formulation and signed prediction objectives remain unchanged, ensuring full compatibility with the baseline architecture while augmenting it with temporal modeling capabilities\. As a result, the proposed HCIM framework extends SE\-SGformer with temporal awareness without sacrificing its interpretability or signed structural reasoning mechanisms\. Table[I](https://arxiv.org/html/2605.26290#S3.T1)summarizes the key notation used in the paper\.

TABLE I:Key notation used throughout the paper\.

## IVExperimental Setup

This section presents the experimental environment, datasets, evaluation metrics, and computational considerations used to evaluate the proposed framework\.

### IV\-AImplementation Details

All models were implemented using PyTorch and PyTorch Geometric\. The framework supports automatic device detection with a hardware fallback hierarchy consisting of CUDA\-enabled GPUs, Apple Metal Performance Shaders \(MPS\), and CPU execution\. Numerical computations were performed using NumPy and SciPy, while Matplotlib was used for visualization and statistical analysis\. Dataset\-specific hyperparameter configurations are reported within their corresponding results sections, as different network types require distinct architectural and optimization settings depending on their structural and temporal characteristics\.

To ensure statistical reliability, all experiments were conducted across 30 independent runs using different random seeds\. Statistical significance was evaluated using paired t\-tests across runs, with confidence intervals computed for all reported metrics\. Improvements are considered statistically significant whenp<0\.001p<0\.001, while non\-significant differences are reported asp\>0\.05p\>0\.05\. In addition to mean performance, standard errors are reported to quantify result stability and experimental variance\. Gradient clipping with a maximum norm of1\.01\.0was employed across all experiments to improve training stability, particularly for temporal models processing sequential embeddings\. Batch processing with configurable batch sizes was used during evaluation to manage GPU memory consumption on large graphs\. To reduce redundant computation, historical node embeddings were cached during temporal training\. Furthermore, all temporal weighting and fusion operations were implemented using differentiable functions to preserve stable gradient flow throughout the HCIM architecture\.

### IV\-BDatasets and Characteristics

To evaluate the proposed framework, we use both real\-world and synthetic temporal signed networks with diverse structural and temporal properties\. Bitcoin OTC, Bitcoin Alpha, and Reddit Hyperlink provide empirical signed interactions with evolving trust or community dynamics, while synthetic BA and WS networks enable evaluation under scale\-free and small\-world structures\.

#### IV\-B1Bitcoin OTC

The Bitcoin OTC dataset \(\[[4](https://arxiv.org/html/2605.26290#bib.bib4),[5](https://arxiv.org/html/2605.26290#bib.bib5)\]\) is a directed*who\-trusts\-whom*network from a peer\-to\-peer Bitcoin trading platform\. Nodes represent users, while directed edges correspond to signed trust ratings between users\. The network contains 5,881 nodes and 35,592 timestamped edges with ratings ranging from−10\-10to\+10\+10, of which approximately 89% are positive\. For our experiments, ratings are binarized by polarity, mapping positive values to\+1\+1and negative values to−1\-1\. Temporal snapshots are constructed by discretizing timestamps into intervals that preserve sufficient edge density for training and evaluation, while the node set remains fixed across snapshots for consistent temporal tracking\.

#### IV\-B2Bitcoin Alpha

The Bitcoin Alpha dataset \(\[[4](https://arxiv.org/html/2605.26290#bib.bib4),[5](https://arxiv.org/html/2605.26290#bib.bib5)\]\) is a directed signed trust network from a peer\-to\-peer Bitcoin reputation platform\. It contains 3,783 nodes and 24,186 directed signed edges with ratings ranging from−10\-10to\+10\+10\. Compared with Bitcoin OTC, Bitcoin Alpha exhibits a higher proportion of positive edges \(approximately 93%\), enabling evaluation of the proposed temporal enhancement under stronger sign imbalance\. The same preprocessing procedure is applied, where ratings are binarized by polarity and temporal snapshots are constructed from timestamped interactions\.

#### IV\-B3Reddit Hyperlink Subset

The Reddit Hyperlink dataset \(\[[3](https://arxiv.org/html/2605.26290#bib.bib3),[6](https://arxiv.org/html/2605.26290#bib.bib6)\]\) is a directed, signed, temporal network of hyperlinks between subreddit communities\. Nodes are subreddits, and directed edges indicate hyperlinks from one subreddit to another\. Edge signs are derived from sentiment analysis of linking posts, producing positive and negative inter\-community relationships\.

Unlike the Bitcoin datasets, which represent user\-level trust, Reddit captures community\-level interaction dynamics and evolving social conflict\. The full dataset spans nearly 2\.5 years of activity and contains 54,075 nodes connected by 571,927 edges\. However, the network exhibits extreme degree heterogeneity and hub dominance\. Although the median node degree is only 2, the mean degree reaches 21\.2, with a maximum degree of 25,425\. Most critically, just 540 nodes \(approximately 1% of all subreddits\) account for nearly 79% of all edges, creating a dense core\-sparse periphery structure that poses significant challenges for GNN\-based models prone to over\-smoothing\. Furthermore, the top 10 most connected subreddits alone account for approximately 22% of all network edges, while the top 1,000 nodes dominate nearly 86% of all interactions\. Such extreme concentration creates substantial imbalance between highly connected hub communities and sparse peripheral nodes, making stable temporal representation learning particularly challenging\.

To enable tractable temporal modeling while preserving structural complexity, we construct a temporally persistent subset retaining representative community interactions and nodes appearing across multiple snapshots, ensuring sufficient historical context for temporal signed link prediction\.

#### IV\-B4Watts\-Strogatz \(WS\) Networks

We generate temporal WS networks \(\[[38](https://arxiv.org/html/2605.26290#bib.bib38),[39](https://arxiv.org/html/2605.26290#bib.bib39)\]\) to evaluate the proposed framework under small\-world structural conditions\. These networks are characterized by high clustering coefficients and short average path lengths\. Small rewiring probabilities preserve strong local clustering while introducing occasional long\-range connections that reduce path lengths\. In our setup, each node is initially connected to itsk=6k=6nearest neighbors on either side, producing a moderate\-density small\-world topology consistent with realistic social network sparsity\[[40](https://arxiv.org/html/2605.26290#bib.bib40)\]\.

To preserve clustering over time, triangle\-closing edges are assigned higher persistence probabilities so that clustered neighborhoods remain stable across temporal snapshots\. Low\-probability sign changes \(approximately 2% per timestep\) simulate occasional transitions between cooperative and adversarial interactions while preserving the dominance of positive trust relationships commonly observed in real signed networks\. New edge formation is additionally biased toward triangle closure and sign\-consistent interactions, reflecting principles of balance theory and homophily\.

#### IV\-B5Barabási\-Albert \(BA\) Networks

We generate temporal BA networks\[[41](https://arxiv.org/html/2605.26290#bib.bib41)\]to evaluate the proposed framework under scale\-free structural dynamics\. BA networks are formed through preferential attachment, where new edges connect to existing nodes with probability proportional to their degree\. This process produces heavy\-tailed degree distributions and persistent hub nodes, properties widely observed in real\-world social and information networks\.

In our temporal BA setting, nodes with higher degree are more likely to attract new connections, ensuring that hub nodes remain dominant over time\. Compared with the WS setting, higher edge persistence probabilities are assigned to preserve the stability of hub\-centric structures characteristic of scale\-free networks\. Edge\-sign dynamics are additionally biased to reflect reputation effects, where highly connected nodes are more likely to maintain positive relationships\. To preserve the power\-law structure, only low\-degree nodes are subjected to limited edge pruning during temporal evolution\.

![Refer to caption](https://arxiv.org/html/2605.26290v1/Degree_Distribution2.png)Figure 2:Degree distribution statistics across evaluated datasets\. The logarithmic scale highlights differences in structural heterogeneity and hub dominance\.
#### IV\-B6Dataset Comparison

The evaluated datasets exhibit substantially different structural characteristics, as illustrated in Figure[2](https://arxiv.org/html/2605.26290#S4.F2)\. The Reddit network shows the strongest degree heterogeneity, with extreme hub dominance and large disparities between median, mean, and maximum node degree \(with maximum\-to\-median ratios exceeding 4,000\)\. This dense core\-sparse periphery structure can significantly increase the risk of over\-smoothing and unstable message propagation in GNN\-based models\[[42](https://arxiv.org/html/2605.26290#bib.bib42)\]\. In contrast, the synthetic WS networks exhibit relatively uniform degree distributions with strong local clustering and limited hub formation \(ratio≈\\approx6\.8\), whereas the BA networks produce moderate heterogeneity through preferential attachment and persistent hub nodes \(ratio≈\\approx15\.6\)\. The Bitcoin OTC and Bitcoin Alpha trust networks occupy an intermediate regime, combining realistic interaction heterogeneity with manageable degree concentration and meaningful signed relationships\.

Consequently, these datasets provide complementary evaluation environments spanning clustered small\-world structures, hub\-dominated scale\-free networks, realistic trust systems, and large\-scale community interaction graphs\. This diversity enables comprehensive assessment of the proposed HCIM across diverse temporal and structural conditions relevant to signed graph learning\. Our implementation is publicly available online\[[43](https://arxiv.org/html/2605.26290#bib.bib43)\]\.

### IV\-CComputational Complexity Analysis

The proposed HCIM adds temporal processing on top of the static SE\-SGformer backbone\. Since the backbone model is unchanged, the additional computational cost comes only from the HCIM operations: temporal weighting, LSTM\-based sequence modeling, temporal attention, projection, and fusion\. Letnndenote the number of nodes,TTthe number of historical snapshots,ddthe node embedding dimension, andhhthe LSTM hidden dimension\.

First, the adaptive temporal weighting stage assigns a scalar weight to each historical embedding and rescales the corresponding node representation\. Since this operation is applied to all nodes, all timesteps, and all embedding dimensions, its complexity isO​\(T​n​d\)O\(Tnd\)\. Second, the weighted sequence is processed by an LSTM\. For each timestep and each node, the LSTM maps an input of dimensionddto a hidden representation of dimensionhh\. Ignoring constant factors associated with the gates, the dominant cost isO​\(T​n​d​h\)O\(Tndh\)\. Third, multi\-head temporal attention is applied along the temporal dimension of the LSTM outputs\. Since attention compares each timestep with every other timestep for each node, the temporal attention cost scales quadratically with the number of snapshots, i\.e\.,O​\(T2​n​h\)O\(T^\{2\}nh\)\. This term remains manageable in our experiments becauseTTis small relative to the number of nodes\. Fourth, the attention output is projected back to the embedding dimension and fused with the current node embeddings\. The projection from dimensionhhtoddrequiresO​\(n​h​d\)O\(nhd\), while the global fusion strategy requires onlyO​\(n​d\)O\(nd\)\. For the node\-adaptive fusion strategy, the MLP\-based computation of node\-specific weights introduces an additional cost of approximatelyO​\(n​d2\)O\(nd^\{2\}\), depending on the hidden size of the MLP\. Since the lower\-order terms are dominated by the LSTM, attention, and projection operations, the complexity of HCIM can be written as:

O​\(T​n​d​h\+T2​n​h\+n​h​d\)\.O\(Tndh\+T^\{2\}nh\+nhd\)\.\(10\)
![Refer to caption](https://arxiv.org/html/2605.26290v1/Runtime_comparison.png)Figure 3:Runtime comparison between the baseline and temporally enhanced SE\-SGformer across evaluated datasets under their respective training configurations\.With node\-adaptive fusion, the MLP introduces an additionalO​\(n​d2\)O\(nd^\{2\}\)term, givingO​\(T​n​d​h\+T2​n​h\+n​h​d\+n​d2\)O\(Tndh\+T^\{2\}nh\+nhd\+nd^\{2\}\)\. Thus, HCIM introduces a controlled overhead that scales linearly with the number of nodes and embedding dimension, and quadratically only with respect to the number of temporal snapshots\. Empirical runtime measurements in Figure[3](https://arxiv.org/html/2605.26290#S4.F3)show that, despite the additional temporal processing stages, the practical GPU\-accelerated overhead remains moderate\. Across datasets, the observed runtime increase ranges from approximately 1\.48% to 24\.80%, depending on graph size, embedding dimension, and temporal fusion strategy\.

Dataset\-specific architectural configurations introduce moderate variations in practical computational cost\. For the Bitcoin OTC and Bitcoin Alpha networks, the model employs two transformer layers, four attention heads, and node\-adaptive temporal fusion, resulting in an additional adaptive MLP cost of approximatelyO​\(n​d2\)O\(nd^\{2\}\)\. In contrast, the synthetic BA and WS networks utilize larger embedding dimensions \(d=256d=256\) together with global temporal fusion, eliminating the adaptive weighting overhead while increasing projection and sequential modeling costs\. The Reddit subset similarly adopts global weighting but retains multi\-layer temporal processing due to its higher structural heterogeneity\. Across all datasets, the dominant computational terms remain the LSTM sequential modeling componentO​\(T​n​d​h\)O\(Tndh\)and the temporal attention componentO​\(T2​n​h\)O\(T^\{2\}nh\)\.

## VResults and Discussions

This section presents and analyzes the experimental results of the proposed framework\.

### V\-APerformance Metrics

To capture the nuances of imbalanced, signed, and temporal network data, we employ Area Under the ROC Curve \(AUC\), binary F1\-score, and Precision@100 \(P@100\)\. In addition to raw performance values, we report relative improvement, error reduction, and standard error\. Relative improvement measures scale\-invariant gains over the baseline, while error reduction quantifies the decrease in prediction error achieved by the temporal enhancement\. Standard error is included to evaluate the statistical reliability and consistency of observed improvements across repeated runs\.

#### V\-A1Area Under the ROC Curve \(AUC\)

AUC measures the ability of a model to distinguish between positive and negative classes across all classification thresholds\[[44](https://arxiv.org/html/2605.26290#bib.bib44)\]\. It is particularly suitable for signed link prediction because it is threshold\-independent and robust to class imbalance in sparse signed networks\. Formally,

AUC=∫01TPR​\(t\)​𝑑FPR​\(t\),\\mathrm\{AUC\}=\\int\_\{0\}^\{1\}\\mathrm\{TPR\}\(t\)\\,d\\mathrm\{FPR\}\(t\),\(11\)whereTPR\\mathrm\{TPR\}andFPR\\mathrm\{FPR\}denote the true positive rate and false positive rate, respectively\. An AUC value of 0\.5 corresponds to random prediction, whereas values closer to 1 indicate stronger discriminative performance\.

TABLE II:Performance comparison between the baseline and temporally enhanced SE\-SGformer models\.
#### V\-A2Binary F1\-Score

The F1\-score combines precision and recall to provide a balanced measure of classification accuracy, particularly useful for imbalanced datasets\[[45](https://arxiv.org/html/2605.26290#bib.bib45)\]:

F1=2×Precision×RecallPrecision\+Recall,\\mathrm\{F1\}=2\\times\\frac\{\\mathrm\{Precision\}\\times\\mathrm\{Recall\}\}\{\\mathrm\{Precision\}\+\\mathrm\{Recall\}\},\(12\)wherePrecision=TP/\(TP\+FP\)\\text\{Precision\}=\\text\{TP\}/\(\\text\{TP\}\+\\text\{FP\}\),Recall=TP/\(TP\+FN\)\\text\{Recall\}=\\text\{TP\}/\(\\text\{TP\}\+\\text\{FN\}\), and true positives \(TP\), false positives \(FP\), and false negatives \(FN\) pertain to the positive edge class\.

Following prior signed link prediction studies, evaluation is formulated as a binary classification problem in which positive edges constitute the positive class and negative edges constitute the negative class, while non\-edges are excluded from evaluation\.

#### V\-A3Precision@100 \(P@100\)

P@100 evaluates the proportion of correctly predicted positive edges among the top 100 highest\-confidence predictions\[[46](https://arxiv.org/html/2605.26290#bib.bib46)\]\. Formally, given a ranked list of predicted positive edges, Precision@K is computed as:

P​@​100=\#​true positive edges in top 100100\.\\mathrm\{P@100\}=\\frac\{\\\#\\text\{ true positive edges in top 100\}\}\{100\}\.\(13\)
This metric is especially relevant for practical applications such as trust recommendation and fraud detection, where only the highest\-confidence predictions are actionable\.

### V\-BPerformance Comparison

Table[II](https://arxiv.org/html/2605.26290#S5.T2)summarizes the performance comparison between the baseline SE\-SGformer and the proposed temporally enhanced framework across all evaluated datasets\. Overall, the HCIM enhancement consistently improves AUC and F1\-score across all five datasets with statistically significant gains\. In contrast, P@100 exhibits mixed statistical significance, with synthetic networks showing significant improvements while real\-world datasets exhibit marginal changes\.

#### V\-B1Bitcoin OTC Network

On the Bitcoin OTC network, the temporally enhanced framework achieved a relative AUC improvement of 3\.787%, corresponding to a 30\.321% error reduction compared with the static baseline\. The F1\-score further improved by 1\.860%, with a corresponding error reduction of 23\.416%\. The evaluation employed 2 transformer layers with 4 attention heads and adaptive temporal weighting\. The network was partitioned into six temporal snapshots, where the second\-to\-last snapshot was selected as the prediction target to balance edge density with sufficient historical context\. Node embedding dimensions were automatically scaled according to graph size\. Although P@100 exhibited a slight decrease, the change was not statistically significant\. This behavior likely arises from the adaptive weighting mechanism occasionally emphasizing historical edges that are no longer active in later timesteps\. Nevertheless, the substantial improvements in AUC and F1\-score indicate a clear net benefit from temporal enhancement\.

#### V\-B2Bitcoin Alpha Network

On the Bitcoin Alpha dataset, the proposed framework achieved a relative AUC improvement of 5\.761% together with a 40\.058% reduction in prediction error\. Similarly, the F1\-score improved by 3\.167%, corresponding to a 33\.501% error reduction\. Experimental settings remained consistent with the Bitcoin OTC evaluation, including 2 transformer layers, 4 attention heads, adaptive temporal weighting, and automatically scaled node embedding dimensions\. The second\-to\-last temporal snapshot was again selected as the prediction target\. The stronger improvements suggest that historical interactions in this network contain highly informative temporal patterns\. As in Bitcoin OTC, P@100 exhibited a slight decrease, although the substantial gains in AUC and F1\-score clearly demonstrate the effectiveness of temporal modeling for trust prediction in temporal blockchain networks\.

#### V\-B3Reddit Hyperlink Network

The Reddit Hyperlink subset presented unique challenges due to its extreme structural heterogeneity and hub\-dominated topology\. The evaluation employed the same general configuration as the cryptocurrency datasets, with two key modifications: AdamW\[[47](https://arxiv.org/html/2605.26290#bib.bib47)\]optimization replaced Adam, and global temporal weighting was used instead of adaptive weighting\. AdamW regularization improved training stability under severe degree imbalance by mitigating dominant gradient updates from highly connected hub nodes, while global weighting provided more consistent temporal aggregation across structurally diverse network regions\. The final configuration employed 2 transformer layers, 4 attention heads, 1000 training epochs, and automatically scaled node embedding dimensions\.

The model achieved a relative AUC improvement of 1\.371%, corresponding to a 4\.551% error reduction\. The F1\-score improved by 2\.922%, with an associated error reduction of 11\.260%\. P@100 showed a slight improvement for the temporal model, indicating that temporal information remains beneficial even in large\-scale community interaction networks with highly irregular topology\.

#### V\-B4BA Scale\-Free Network

The BA network provided a controlled evaluation setting for scale\-free temporal graph dynamics\. The proposed framework achieved one of its strongest improvements on this dataset, with a relative AUC gain of 13\.943% and an error reduction of 41\.132%\. The F1\-score similarly improved by 10\.782%, corresponding to a 39\.934% reduction in error\.

The evaluation employed a configuration optimized for scale\-free structures, including a single transformer layer, 8 attention heads, 256\-dimensional node embeddings, AdamW optimization, and global temporal weighting\. The single\-layer design mitigated oversmoothing in hub\-dominated regions, while the larger embedding dimension improved representation capacity across heterogeneous node roles\. Unlike the cryptocurrency datasets, the BA network also exhibited improved P@100 performance, suggesting that temporal enhancement is particularly effective in preferential\-attachment environments with persistent hub structures\. Although variance increased slightly for AUC and F1\-score, the gains remained statistically significant across all runs\.

#### V\-B5WS Small\-World Network

The WS evaluation demonstrates the effectiveness of the proposed framework under small\-world structural conditions characterized by strong local clustering and sparse long\-range connectivity\. Similar to the BA evaluation, the WS experiments employed a single transformer layer, 8 attention heads, 256\-dimensional embeddings, AdamW optimization, and global temporal weighting\.

The framework achieved a relative AUC improvement of 13\.483%, corresponding to a 41\.949% reduction in prediction error\. The F1\-score improved by 11\.382%, with a 41\.820% error reduction\. These results indicate that historical context plays a particularly important role in small\-world graph evolution\. P@100 also improved significantly, further confirming the effectiveness of temporal aggregation in clustered graph structures\. The strong and consistent gains across all metrics demonstrate that the proposed HCIM effectively captures temporal dependencies in networks characterized by localized community structure and short path lengths\.

#### V\-B6Analysis and Discussion

Across all datasets, the temporally enhanced SE\-SGformer consistently outperformed the static baseline\. The largest relative improvements were observed on the synthetic BA and WS networks, indicating that HCIM effectively captures temporal dependencies in structured graph evolution processes such as preferential attachment and small\-world rewiring\. The cryptocurrency trust networks further showed that temporal modeling improves prediction performance even in already high\-performing settings, suggesting that historical interactions contain additional predictive information beyond static graph structure alone\. Although improvements on the Reddit Hyperlink subset were comparatively smaller, our framework remained effective despite the network’s extreme structural heterogeneity and hub dominance\.

Our findings suggest that effective temporal graph learning benefits from separating historical and structural representations before adaptive fusion\. The consistent gains across all datasets further indicate potential applicability beyond signed link prediction to broader temporal graph learning tasks\.

## VILimitations and Future Work

Despite the promising results shown by the HCIM, several limitations remain that motivate further investigation\.

### VI\-ALimitations

A fundamental limitation of the HCIM is its dependence on the underlying SGNN backbone\. Consequently, HCIM inherits certain representational and scalability limitations of the base architecture, although its modular design allows integration with improved SGNN models as they emerge\.

Secondly, HCIM introduces additional computational and memory overhead compared with static SGNNs\. Historical embedding sequences must be processed through recurrent and attention\-based temporal modules, increasing both training and inference complexity\. Although tractable for the evaluated datasets, scalability may become challenging for very large graphs or long temporal horizons\. The Reddit Hyperlink dataset, comprising of 55,863 nodes and 858,490 edges, highlights an additional limitation associated with highly heterogeneous and hub\-dominated graph structures\. As depicted in Figure[4](https://arxiv.org/html/2605.26290#S6.F4), about 1% of nodes account for nearly 79% of all edges, producing an extreme core\-periphery structure\. Such imbalance exacerbates oversmoothing effects in GNNs, reducing representation diversity in densely connected regions and limiting temporal modeling effectiveness\.

![Refer to caption](https://arxiv.org/html/2605.26290v1/Reddit_Edge_Concentration2.png)Figure 4:Edge concentration in the full Reddit Hyperlink dataset\. A small fraction of nodes controls the majority of edges, producing severe hub dominance and increasing oversmoothing effects in GNN\-based models\.
### VI\-BFuture Research Directions

An important direction for future research is the development of architecture\-general temporal enhancement frameworks for signed graph learning\. In particular, integrating HCIM\-like temporal modules with diverse SGNN backbones, including SGCN, SiGAT, and SIGNet, would enable systematic analysis of how different signed message\-passing mechanisms interact with temporal aggregation\. Beyond direct integration, future work should investigate modular graph learning architectures in which temporal, structural, semantic, and spatial components are decoupled and independently optimized before adaptive fusion\. Such a design paradigm could improve architectural flexibility, scalability, and interpretability while providing deeper insight into the role of temporal information in signed graph representation learning\.

Another research direction concerns the limited availability of diverse, high\-quality benchmark datasets for temporal signed graph learning\. Existing real\-world datasets are largely restricted to cryptocurrency and social network domains, and often suffer from privacy constraints, inconsistent preprocessing, and limited temporal granularity\. Future research should focus on developing standardized benchmark datasets, preprocessing pipelines, temporal partitioning strategies, and evaluation protocols to support fair comparison and reproducible research\. In parallel, more realistic synthetic temporal signed graph generators are needed to model trust evolution, antagonistic interactions, sign\-flip dynamics, and temporal correlation patterns observed in real\-world networks\.

A third promising avenue involves extending temporal signed graph learning toward deployment\-oriented and domain\-specific applications\. Future efforts could investigate incorporating application\-dependent signals, such as market dynamics in financial networks, evolving interaction patterns in social platforms, and trust\-aware behavior in recommendation systems\. In addition, developing temporal SGNN frameworks capable of supporting streaming graph updates, incremental learning, and large\-scale real\-time inference would further improve practical applicability\.

## VIIConclusion

This paper introduced the Historical Context Integration Module \(HCIM\), a modular temporal enhancement framework for signed graph neural networks\. By combining recency\-aware temporal weighting, LSTM\-based embedding trajectory modeling, and multi\-head temporal attention, HCIM enables static SGNNs to incorporate historical context while preserving structural interpretability\. Integrated with SE\-SGformer, the proposed framework achieved consistent and statistically significant improvements across both real\-world and synthetic temporal signed networks, including Bitcoin OTC, Bitcoin Alpha, Reddit Hyperlink, BA, and WS datasets\. The experimental results demonstrate that temporal signed interactions contain predictive information beyond static graph structure alone, highlighting the effectiveness of temporal enhancement for signed link prediction\. Overall, HCIM provides an interpretable and architecture\-agnostic foundation for future temporal SGNN research and broader temporal graph learning applications\.

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