IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
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Decoupled and Divergence-Conditioned Prompt for Multi-domain Dynamic Graph Foundation Models
arXiv:2605.13540v1 [cs.LG] 13 May 2026
Haonan Yuan, Qingyun Sun, Member, IEEE, Junhua Shi, Xingcheng Fu, Member, IEEE, Jianxin Li, Senior Member, IEEE, and Philip S. Yu, Life Fellow, IEEE
Recent advances in DGNNs have further enabled effective learning on continuous-time dynamic graphs, with applications in user behavior prediction [25]–[27], fraud detection [28]–[30], and molecular interaction modeling [31]–[33], etc. Despite their success, existing DGNNs are typically developed under task- or domain-specific training protocols, and thus often struggle to generalize across domains and tasks [34], [35]. This limitation resembles the early development of language [36] and vision [37] models before the emergence of foundation models [38], [39]. In the graph domain, Graph Foundation Models (GFMs) have recently shown that largescale pre-training on diverse static graphs can learn transferable representations that generalize well across various downstream tasks [40]–[42]. Instead of training a separate model for each task, an ideal GFM is pre-trained once on multi-domain graphs and then adapted to different tasks with limited supervision [43], [44], following the widely adopted “pretrain-then-finetune” paradigm [45]–[47]. Inspired by these advances, researchers have begun to explore dynamic graph foundation models, which aim to extend this paradigm from static graphs to time-evolving Index Terms—Graph foundation models, dynamic graph repre- graphs and support multi-domain, multi-task adaptation [35]. sentation learning, multi-domain graph pre-training, and graph However, building a dynamic graph foundation model over prompt learning. multiple domains remains highly challenging. Unlike static graphs, which can often be integrated through structural I. I NTRODUCTION unification or feature-space alignment, dynamic graphs carry Graphs provide a powerful abstraction for modeling complex domain-specific semantics, incompatible temporal granularities, relationships among entities, and have been widely adopted and divergent evolution patterns. These discrepancies make it in diverse domains, including social networks [1]–[3], rec- difficult to learn unified representations from multiple dynamic ommendation systems [4]–[6], knowledge graphs [7]–[9], and graphs during pre-training. Moreover, when the pre-trained biological systems [10]–[12], etc. In many real-world scenarios, model is adapted to downstream domains, mismatched semantic node interactions are inherently temporal, with graph structures or temporal patterns may introduce irrelevant source-domain and node features evolving continuously over time. Such knowledge and lead to negative transfer. Specifically: Challenge I: Misaligned temporal semantics across dynamic graphs are commonly characterized by asynchronous domains make multi-domain pre-training difficult. Unlike edge events [13]–[15], time-sensitive node states [16]–[18], static graphs that can be integrated through structural unificaand evolving topological patterns [19]–[21]. To capture these tion, dynamic graphs encode temporal order, relative intervals, temporal dependencies, Dynamic Graph Neural Networks and event causality as essential parts of their semantics. For (DGNNs) have emerged as a major paradigm, enabling temporal example, user-item interactions in an e-commerce platform encoding, event-aware message propagation, and inductive occur at millisecond-level timestamps, whereas citation netgeneralization over time-evolving data streams [22]–[24]. works typically evolve over years. Since most dynamic graph H. Yuan, Q. Sun, J. Shi, and J. Li are with the School of Computer Science models rely on relative time encodings or continuous-time and Engineering, Beihang University, Beijing 100191, China (Email: {yuanhn, propagation mechanisms, directly combining domains with sunqy, shijunhua, lijx}@buaa.edu.cn). different temporal granularities may introduce ambiguity and X. Fu is with the Key Lab of Education Blockchain and Intelligent Technology, Ministry of Education, Guangxi Normal University, Guilin 541004, distort the original semantics. Without a shared or aligned China (E-mail: [email protected]). notion of time, heterogeneous dynamic graphs cannot be simply P. S. Yu is with the Department of Computer Science, University of Illinois fused into a unified pre-training corpus. Consequently, the idea at Chicago, Chicago 60607, USA (E-mail: [email protected]). Manuscript received May 11, 2026; revised June 18, 2026. of scaling via data aggregation, which is central to the success Abstract—Dynamic graphs are ubiquitous in real-world systems, and building generalizable dynamic Graph Foundation Models has become a frontier in graph learning. However, dynamic graphs from different domains pose fundamental challenges to unified modeling, as their semantic and temporal patterns are inherently inconsistent, making the multi-domain pre-training difficult. Consequently, the widely used “pretrain-then-finetune” paradigm often suffers from severe negative knowledge transfer. To the best of our knowledge, there exists no multi-domain dynamic GFM. In this work, we propose DyGFM, a Dynamic Graph Foundation Model over multiple domains based on decoupled and divergenceconditioned prompting. To disentangle transferable semantics from the domain-specific dynamics, we introduce a dual-branch pretraining strategy with semantic-temporal decoupling. To alleviate negative transfer during domain adaptation, we further develop a cross-domain routing mechanism with divergence-aware expert selection. To enable efficient downstream fine-tuning, we design a divergence-conditioned prompt generator that injects lightweight, learnable graph prompts tailored to semantic and temporal traits. Extensive experiments on continuous dynamic graph benchmarks demonstrate that DyGFM consistently outperforms 12 state-ofthe-art baselines on both node classification and link prediction tasks, achieving superior effectiveness and efficiency.
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
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How to construct a dynamic GFM from multi-domain pre-training and cross-domain / -task adaptation? Fig. 1. Challenges of constructing a dynamic GFM.
cross-domain routing mechanism with divergence-aware expert selection, which selectively aggregates relevant source-domain experts according to semantic and temporal discrepancies. To enable efficient downstream fine-tuning, we further introduce a divergence-conditioned prompt generator that injects lightweight, learnable graph prompts tailored to domain-specific semantic and temporal patterns. Our contributions are: • We propose a multi-domain dynamic graph foundation model named DyGFM. To the best of our knowledge, this is the first work that successfully builds a multi-domain dynamic GFM by explicitly addressing both multi-domain pre-training and negative knowledge transfer. • Its architecture consists of semantic-temporal decoupled pretraining, divergence-aware expert routing, and divergenceconditioned prompting. This design enables DyGFM to handle temporal semantic incompatibility during pre-training and mitigate negative transfer during domain adaptation. • Extensive experiments on continuous dynamic graph benchmarks demonstrate that DyGFM consistently outperforms 12 state-of-the-art baselines in node classification and link prediction, achieving superior effectiveness and efficiency.
of static GFMs, becomes non-trivial in dynamic settings. Challenge II: Semantic-temporal divergence across domains leads to negative transfer during adaptation. Compared with the static settings, dynamic graphs differ not only in interaction patterns and feature distributions, but also in their temporal evolution patterns. For instance, a user churn dataset may reflect short-term behavioral bursts, while a co-authorship II. R ELATED W ORK network usually captures long-term collaboration trends. When the widely adopted “pretrain-then-finetune” paradigm is applied A. Dynamic Graph Learning to such heterogeneous dynamic domains, the model may Dynamic graph learning aims to learn representations that transfer temporal priors or semantic cues from irrelevant source evolve over time. Unlike static graphs, dynamic graphs require domains, thereby degrading downstream performance on the models to capture both structural and temporal dependencies target domain. This issue becomes particularly pronounced induced by timestamped interactions, event intervals, and node under limited supervision, where the model has insufficient states. Existing works can be grouped into structure-oriented, target-domain labels to correct misleading inductive biases. temporal-oriented, and application-oriented methods. Without explicitly modeling domain divergence, a unified Structure-oriented methods. Structure-oriented methods dynamic graph foundation model may generalize poorly and mainly focus on preserving evolving graph topology and even negatively affect domain-specific adaptation. capturing time-dependent structural patterns. One representative Several recent works have attempted to improve gener- line is the temporal random walk-based methods. CTDNE [48] alization in dynamic graph learning through prompt-based extends random walks to continuous-time dynamic graphs adaptation towards establishing DyGFMs. For instance, DyG- by enforcing chronological constraints during walk sampling. Prompt [35] introduces a node- and timestamp-conditioned CAWs [49] further introduces causal anonymous walks to imprompt generator, allowing fine-tuning of pre-trained temporal prove inductive temporal representation learning. NeurTW [20] graph models on downstream tasks. Similarly, TIGPrompt [34] enhances temporal walks with neural message passing and designs discrete temporal prompts for multi-task generalization. motif-aware temporal structures. PINT [50] studies the exWhile effective within single-domain settings, these methods pressiveness of temporal graph networks from the perspective generally assume that time semantics remain consistent across of temporal walks, while TPNet [51] improves temporal link all training and testing data. As a result, they do not account prediction through temporal walk matrix projection. Another for the structural and temporal divergence that naturally representative line is the temporal neighborhood-based methods. arises across multiple heterogeneous domains. Moreover, most TGAT [52] introduces time encoding and attention-based existing approaches rely on end-to-end fine-tuning of model temporal neighborhood aggregation. GraphMixer [53] uses backbones or restrict prompting to shallow embeddings, thereby simple MLP-based architectures to aggregate temporal neighlimiting scalability and flexibility, which remain major bottle- borhood information efficiently. DyGFormer [54] tokenizes necks in building practical dynamic GFMs. historical interactions and models them with transformer-style To tackle the aforementioned challenges, we propose sequence encoding. CNE-N [55] designs co-neighbor encoding DyGFM, a Dynamic Graph Foundation Model over multiple for efficient dynamic link prediction, while SEAN [56] and domains by decoupled and divergence-conditioned prompting. RepeatMixer [57] further improve temporal neighborhood DyGFM is built upon three key modules. To disentangle modeling through adaptive neighborhood selection and repeattransferable semantics from domain-specific dynamics, we aware sampling. These methods are effective in modeling local propose a dual-branch pre-training strategy with semantic- temporal structures and dynamic neighborhood dependencies, temporal decoupling, which separately captures static feature but they usually rely on sampled walks, historical neighborsemantics and fine-grained temporal patterns. To alleviate hoods, or local interaction windows, which may limit their negative transfer during domain adaptation, we design a scalability and long-range temporal modeling ability.
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
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Temporal-oriented methods. Temporal-oriented methods Early prompt-based methods, such as GraphPrompt [84], emphasize the modeling of temporal dependency itself, espe- HGPrompt [85], MultiGPrompt [86], and UniPrompt [87], cially under continuous-time and irregular interaction settings. adapt pre-trained GNN encoders to downstream tasks through RNN-based methods such as JODIE [58] and DyGNN [59] learnable prompts or unified task templates. Subsequent multiupdate node representations sequentially according to streaming domain methods, such as SAMGPT [88], GCOPE [89], interactions, thereby capturing the evolving states of nodes over MDGPT [90], MDGCL [91], BRIDGE [92], and MDGFM [93], time. RTRGN [60] further revises temporal information during further improve cross-domain transfer by introducing conrecurrent aggregation to better incorporate historical neigh- trastive alignment, domain tokens, source-domain selection, or bor states. Temporal point process-based methods, including spectral regularization. More recent methods explore specialHTNE [61], DyRep [62], M2DNE [63], TREND [14], and Easy- ized graph-native architectures. For example, GMoPE [94] introDGL [64], model the occurrence of future interactions through duces expert prompts for adaptive transfer, while GRAVER [95] conditional intensity functions, which enable them to capture constructs transferable graph vocabularies through hierarchical asynchronous events and temporal decay effects. Memory- routing. In addition, GraphMoRE [96], RiemannGFM [97], based methods such as TGN [65], NAT [66], TIGER [67], GraphGlue [98], and CRGFM [99] incorporate geometric or PRES [68], MemMap [69], and MSPipe [70] maintain node- Riemannian modeling to better capture complex structural level memory states and update them through event-driven patterns. These methods provide effective graph-native transfer message passing, supporting online representation learning over mechanisms, but they are mainly designed for static graphs streaming graphs. More recently, frequency-domain methods and do not explicitly model temporal order, time granularity, such as FreeDyG [71] and BandRank [72] transform temporal or dynamic evolution across domains. signals into the frequency space to capture periodic patterns, LLM-based GFMs. LLM-based GFMs reformulate graph long-term trends, and multi-scale temporal variations. These learning into language-compatible formats and use large methods provide powerful tools for modeling temporal evolu- language models as the primary backbone. Representative tion, but the learned temporal patterns are often tied to specific methods include LangGFM [100] and PromptGFM [101], datasets or domains. As a result, directly transferring with which textualize graph structures, node attributes, or graph tasks different temporal resolutions, event generation mechanisms, into language-readable inputs and perform graph reasoning or interaction patterns remains non-trivial. or prediction in the language space. Other methods further Application-oriented methods. Application-oriented meth- convert graphs into structured formats such as JSON, XML, ods adapt dynamic graph learning to specific downstream or tables, enabling graph data to reuse the general reasoning scenarios. For sequential recommendation, methods such and instruction-following abilities of pretrained foundation as TGSRec [73], DGSR [24], PTGCN [74], TCGC [75], models [102]. This line benefits from the semantic knowledge NeuFilter [76], and TGCL4SR [77] model dynamic user-item and flexible task interfaces of LLMs, especially for textinteractions and evolving preferences. For anomaly detection attributed graphs or graph-language tasks. However, LLMand community discovery, methods such as TagGen [78], based GFMs often rely on textualized graph descriptions SAD [79], GeneralDyG [80], CDGP [81], TGC [82], and DyG- and may lose fine-grained structural or temporal information MF [83] capture abnormal temporal behaviors or evolving during the transformation process. Therefore, they are not community structures. Although these methods achieve strong naturally suited for continuous-time dynamic graphs, where performance in their target applications, their architectures and irregular event intervals and temporal dependencies are central objectives are usually task-specific, offering limited support to representation learning. for unified pre-training and cross-domain adaptation. GNN-LLM Hybrid GFMs. Hybrid GFMs combine graph Overall, existing dynamic graph learning methods have encoders with language encoders or LLMs, aiming to exploit advanced the modeling of temporal interactions and evolving both graph-native structural modeling and language-level sestructures. However, most of them are developed for single- mantic reasoning. OFA [103] and UniGraph [46] provide early domain or task-specific scenarios, with limited consideration attempts to unify graph tasks through graph-language interfaces. of multi-domain pre-training, temporal scale inconsistency, GraphGPT [45] and LLaGA [104] connect graph encoders with and negative transfer during cross-domain adaptation. These LLMs through projection modules, while GraphCLIP [105] and limitations motivate our study on dynamic graph foundation BooG [106] align graph representations with textual supervision modeling over multiple domains. or class-level semantic information. UniGraph2 [107] further extends this paradigm to multimodal graph settings, and B. Static Graph Foundation Models. GOFA [108] integrates graph computation with frozen language Graph Foundation Models (GFMs) aim to learn transferable models for graph-aware generation. Compared with purely knowledge from large-scale graph data and generalize to diverse GNN-based or LLM-based methods, hybrid GFMs offer a downstream tasks and domains. According to the backbone more flexible interface for combining structural and semantic architecture, existing static GFMs can be broadly grouped into knowledge. Nevertheless, most hybrid GFMs still focus on static graphs or text-attributed graphs, and their pre-training and GNN-based GFMs, LLM-based GFMs, and hybrid GFMs. GNN-based GFMs. GNN-based GFMs use graph-native adaptation mechanisms do not explicitly address temporal scale encoders as the main backbone, where transferable knowledge inconsistency, domain-specific evolution patterns, or negative is learned through message passing, graph transformers, struc- transfer in multi-domain dynamic graph learning. tural tokenization, or domain-aware representation alignment. Overall, static GFMs have significantly advanced transferable
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
TABLE I C OMPARISON WITH REPRESENTATIVE G RAPH F OUNDATION M ODELS . Category
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GraphPrompt [84], SAMGPT [88], GNN-based Static GFMs GCOPE [89], BRIDGE [92], MDGFM [93], GRAVER [95] LLM-based Static GFMs LangGFM [100], PromptGFM [101] Hybrid Static GFMs
OFA [103], GraphGPT [45], LLaGA [104], GraphCLIP [105], GOFA [108] TIGPrompt [34], DDGPrompt [109]
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III. N OTATIONS AND P RELIMINARIES graph representation learning through graph-native pre-training, language-based reformulation, and graph-language hybrid A. Notations architectures. However, they generally lack explicit mechanisms Given a continuous-time dynamic graph G = {V, E, T }, for modeling continuous temporal evolution and aligning where V denotes the node set, T denotes the continuous time heterogeneous time semantics across domains. As a result, domain, and E = {(u, v, t)} denotes the temporal edge set. directly extending static GFMs to dynamic graph scenarios Each temporal edge (u, v, t) ∈ E indicates that node u interacts may lead to temporal semantic distortion during pre-training with node v at timestamp t ∈ T . The node feature matrix and negative transfer during cross-domain adaptation. This is denoted by X ∈ R|V|×d0 , where d0 is the input feature motivates the development of dynamic graph foundation models dimension. Let ztv ∈ Rd denote the embedding of node v at that can jointly support multi-domain pre-training, semantictime t, where d is the hidden dimension. temporal decoupling, and efficient downstream fine-tuning. Dynamic Graph Neural Networks (DGNNs) serve as the backbone architecture for dynamic graph foundation modeling. In general, DGNNs update node embeddings by aggregating C. Dynamic Graph Foundation Models. messages from historical temporal neighbors. Specifically, for Compared with static GFMs, dynamic graph foundation mod- node v at time t, the aggregation is performed over previous els remain much less explored. Existing related studies mainly interactions (u, v, t′ ) ∈ E with t′ < t: n ′ o focus on prompt-based adaptation for pre-trained dynamic t t ′ ′ ′ graph models rather than building foundation models over zv = AGG ϕ zu , T (t−t ), euv | (u, v, t ) ∈ E, t < t , (1) multiple dynamic graph domains. TIGPrompt [34] introduces ′ ztu denotes the historical embedding of node u at time a temporal prompt generator to bridge the temporal and where ′ semantic gaps between pre-training and downstream prediction t , T ′(·) is a relative time encoder that maps the time interval on temporal interaction graphs, and supports lightweight prompt t − t into a temporal representation, euv denotes the edge tuning over frozen TIG backbones. DyGPrompt [35] further feature between u and v when available, ϕ(·) is a temporal develops a pre-training and prompt learning framework for message function, and AGG(·) is a temporal neighborhood dynamic graphs, where node and time prompts are used to aggregator, e.g., the attention-based aggregator in TGAT [52]. alleviate the discrepancy between link-prediction pre-training and downstream tasks, and dual condition-nets are designed B. Pre-training and Few-shotFine-tuning n domains to capture evolving node-time patterns. More recently, DDG- Given n dynamic graphs GiS i=1 from source n n Prompt [109] proposes a data-centric prompt tuning strategy DiS i=1 with their corresponding labels YiS i=1 , the prethat refines pre-trained node embeddings through temporal bias, training stage aims to learn a dynamic graph learner h = g ◦ f , edge weight, and feature mask prompts, aiming to improve where f denotes the dynamic graph encoder that extracts timefew-shot adaptability across downstream tasks. Although these aware node representations, and g denotes the task head. After studies improve the adaptability of dynamic graph models, convergence, the learned parameters θ ⋆ are frozen and reused they are still mainly centered on single-domain prompt tuning for downstream adaptation. or task-level transfer. They do not explicitly address multiDuring fine-tuning, given a target dynamic graph G T from domain pre-training over dynamic graphs with incompatible a target domain DT (seen or unseen), only m labeled samples temporal semantics, nor do they model semantic-temporal are available Pn as the support set under the m-shot setting, where divergence between source and target domains for negative m ≪ i=1 |Vi |. The goal is to adapt the frozen f to the target transfer mitigation. Therefore, to the best of our knowledge, domain and predict labels for the unlabeled query set. In this multi-domain dynamic graph foundation modeling remains work, adaptation is achieved by injecting learnable prompts largely unexplored. into f , while keeping θ ⋆ unchanged.
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YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
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Fig. 2. The framework of DyGFM. (1) Dual-Branch Pre-training. DyGFM decouples transferable semantics and domain-specific temporal dynamics. (2) Cross-Domain Adaptation. Divergence-aware routing selects relevant source-domain experts according to semantic and temporal discrepancies. (3) Conditioned Fine-tuning. Divergence-conditioned graph prompts enable efficient target-domain adaptation with the frozen pre-trained encoder.
IV. M ETHOD
semantics while filtering domain-specific noise. Following the Information Bottleneck (IB) principle [112], [113], we formuIn this section, we elaborate on the proposed DyGFM. late a self-supervised objective that enforces both sufficiency, which preserves useful semantic information, and minimality, A. Pre-training with Semantic-Temporal Decoupling which filters out redundant variations: Multi-domain dynamic graphs contain two entangled factors: S bS transferable domain semantics and domain-specific temporal LSS-IB = −I ZSi ; ZS+ + βI Z ; X , (3) i i i dynamics. Across domains, features may have different dimenb S , AS , ZSi = fs X (4) i i sions, distributions, and semantics, while temporal interactions may evolve at incompatible time scales. Directly merging such heterogeneous dynamic graphs for pre-training may therefore where fs is the GNN encoder. The first term (prediction term) introduce feature-level semantic noise and temporal semantic maximizes the mutual information between ZSi and its positive S+ distortion. To address this issue, we propose semantic-temporal sample Zi drawn from its neighbors. The second term decoupled pre-training. The semantic branch learns transferable (compression term) penalizes the mutual information between b S. semantic representations from timestamp-removed graphs, ZSi and the aligned input X i while the temporal branch preserves domain-specific dynamics Since the two mutual information terms in Eq. (3) cannot through relative-time modeling and lightweight adapters. be directly computed in closed form, we instantiate the self1) Semantic Branch: The semantic branch aims to learn supervised IB objective through two tractable approximations. transferable domain semantics. Since source graphs may Specifically, the prediction term is lower-bounded by a conhave heterogeneous feature spaces and domain-specific noise, trastive objective, while the compression term is upper-bounded directly applying a shared encoder to raw node features is by a variational KL regularizer. ineffective. Therefore, we first project each source domain S+ S Proposition 1 (Lower Bound of I Z ; Z ). The prediction i i into a shared latent semantic space, and then constrain the S+ S is intractable but can be lower-bounded via learned representations to preserve informative semantics while term I Zi ; Zi InfoNCE [114]. For anchor node u and its positive neighbor suppressing redundant variations. For a source dynamic graph + u with negatives {v}, we have: S Gi , we remove timestamps and merge edges into a static graph S |Vi |×di with feature matrix Xi ∈ R . Removing timestamps I ZSi ; ZS+ ⩾ −LInfoNCE i allows the semantic branch to focus on features and structures N+ without being affected by incompatible temporal scales. exp zSu , zS+ /τ 1 X u = + log PN − , (5) Self-supervised Feature Alignment. Instead of aligning S S N u=1 v=0 exp (⟨zu , zv ⟩ /τ ) dimensions with trivial techniques like SVD [110], we propose a set of domain-specific aligners {Ai }ni=1 corresponding to where ⟨·, ·⟩ denotes the inner product (cosine similarity), τ each source domain. These aligners project heterogeneous node is a temperature parameter, and negatives are sampled across features into a shared latent semantic space: domains to avoid trivial alignment. b S ∈ R|Vi |×d = Ai XS , for each i ⩽ n, X (2) i i Proof. Fix a source domain Di . Define random variables for where the aligner is implemented by a two-layer MLP [111]. an anchor-positive pair sampled from domain Di : Given the aligned node features, our goal is to learn compressed yet expressive semantic tokens ZS that retain transferable ZSi , ZS+ ∼ pi z, z+ , (6) i
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
where drawing an anchor node u induces z = zSu and a positive u+ induces z+ = zS+ u . The mutual information is: pi (z | z+ ) + I ZSi ; ZS+ = E log . (7) pi (z,z ) i pi (z)
S bS by KL divergence [115] between the posterior qϕ z | x S and prior p z :
For each u (N + ) anchors in a batch, form a candidate set: S Cu = zS0 , · · · , zSN − , zS0 = zS+ (8) u , zv ∼ pneg (z) ,
Proof. The compression term penalizes how much information b S. the latent representation ZSi preserves about the input X i Formally, the mutual information can be written as: b) b S = Ep(bx,z) log p (z | x I ZSi ; X . (16) i p (z)
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where pneg is the marginal distribution over embeddings used for negative sampling. It can be the within-domain marginal pi ZSi or a pooled marginal across domains. Both approx- This is equivalent to: imate p (z) and keep the derivation valid. Let K = N − + 1 b S = Ep(bx) [KL (p (z | x b) ∥ p (z))] , (17) I ZSi ; X be the candidate count. Define the softmax posterior over the i index v ∈ {0, · · · , N − } of the positive within Cu : which indicates that compression measures the average discrep S S ancy between the posterior and the prior. The true posterior exp z , z /τ u v (9) p (z | x q v | zSu , Cu = PN − . b) is generally intractable. We approximate it with a S zSu , zk /τ k=0 exp b). By the non-negativity of the variational distribution qϕ (z | x Let v ⋆ = 0 be the true index of the positive. The standard KL divergence: InfoNCE argument, namely contrastive predictive coding, gives: b) ∥ p (z)) ⩽ KL (qϕ (z | x b) ∥ p (z)) . KL (p (z | x (18) S+ ⋆ S S E log p v = 0 | zu , Cu = I Zi ; Zi − log K, (10) Taking expectation over aligned inputs and replacing it with an empirical average over nodes u = 1, · · · , N , we obtain: where the expectation is over the joint sampling of nodes for N anchors, positives, and negatives. By the non-negativity of the 1 X S bS bSu p zSu . (19) KL qϕ zSu | x I Z ; X ⩽ i i KL divergence: N u=1 We conclude the proof. E log q 0 | zSu , Cu S S S = E log p 0 | zu , Cu − KL p · | zu , Cu q · | zu , Cu Proposition 2 constrains semantic representations with a ⩽ E log p 0 | zSu , Cu . (11) variational KL regularizer, encouraging them to remain close to a prior distribution and thereby reducing redundant domainAveraging over anchors u = 1, · · · , N + yields: specific variations for more stable cross-domain pre-training. Pre-training Objectives. After obtaining semantic tokens, I ZSi ; ZS+ ⩾ log K+ i we optimize the semantic branch using a self-supervised link" # N+ S+ S based objective on the timestamp-removed graph. This objective X exp zu , zu /τ 1 E log PN − . (12) encourages connected nodes to have consistent semantic + S S N u=1 v=0 exp (⟨zu , zv ⟩ /τ ) representations, while separating them from unconnected nodes across the merged training graphs. Inspired by prior studies [84], Replacing the expectation with the empirical batch average: each anchor node u is paired with a positive neighbor v + + N sampled from its observed adjacency and a negative neighbor exp zSu , zS+ /τ 1 X u LInfoNCE = − + log PN − . (13) v − sampled from unconnected nodes. Concretely, a similarity N u=1 zSu , zSu,v /τ v=0 exp discriminator g = MLP (⟨·, ·⟩) maps a pair of embeddings to a scalar score, where ⟨·, ·⟩ denotes the inner product. For It follows that: each non-redundant training quadruple (u, v + , v − , yu ), where S+ S − I Zi ; Zi ⩾ log N + 1 − LInfoNCE ⩾ −LInfoNCE , (14) yu ∈ {0, 1} indicates link existence, the objective is: Lpre θs , ZS where the additive log (N − + 1) is a batch-size constant that X does not affect optimization. We conclude the proof. exp g zSu , zSv+ /τ . (20) =− log exp g zSu , zSv+ /τ +exp g zSu , zSv− /τ (u,v + ,v − ) Proposition 1 makes the sufficiency constraint tractable by replacing the mutual information maximization with an InfoNCE- Finally, we combine the link-based pre-training loss with the based contrastive objective. This encourages semantic tokens self-supervised IB regularization. The overall objective for the to retain neighborhood-consistent predictive information, while semantic branch is: cross-domain negative samples prevent the representations from Lpre-sem θs , ZS = Lpre θs , ZS + λ1 · LSS-IB . (21) collapsing into trivial domain-specific alignments. b S ). The compres- Here, λ1 controls the strength of the information bottleneck regProposition 2 (Upper Bound of I ZSi ; X i ularization, balancing semantic discrimination and redundancy b S is intractable but can be upper-bounded suppression during pre-training. sion term I ZSi ; X i
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
2) Temporal Branch: While the semantic branch learns transferable feature semantics in a shared latent space, temporal dynamics require a different treatment. The timestamps of different domains usually do not share the same physical semantics: a short interval may indicate an important behavioral change in transaction networks, but may be negligible in citation or collaboration networks. Therefore, directly sharing a single time encoder across all domains may force incompatible temporal scales into the same representation space and distort domain-specific evolution patterns. To preserve temporal specificity while still learning reusable temporal aggregation knowledge, the temporal branch follows a shared-specific design. Specifically, domain-specific timers encode relative time within each source domain, a shared temporal backbone captures general temporal dependency patterns, and lightweight adapters absorb domain-specific residual dynamics. Relative-time Encoding. For each edge (u, v, t) ∈ Ei , a domain-specific timer T i maps t to a time embedding: rSi (t) = T i (t) ,
T i = MLPi (·) ◦ PE (·) ,
(22)
where PE (·) is a sinusoidal positional encoding similar to that used in Transformers [116], which maps the relative timestamp into a vector as: PE (t) = sin (ω1 t) , cos (ω1 t) , · · ·, sin ωdt /2 t , cos ωdt /2 t , (23)
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compensate for domain-specific temporal dynamics that cannot be captured by a universal temporal model alone. Alternating Training and Freezing Strategy. Training the temporal branch across multiple domains requires careful separation between domain-general temporal aggregation and domain-specific temporal residuals. If adapters are updated too early, they may absorb patterns that should be learned by the shared backbone, weakening cross-domain transfer. Conversely, if all parameters are trained jointly without constraints, domainspecific timestamp semantics may interfere with the learning of general temporal dependencies. To avoid these issues, we adopt an alternating training and stage-wise freezing strategy. Denote Θt as the parameters of ft , Φi denote the parameters of timer for domain DiS , and Ψi denote the parameters of the adapter Γi . For mini-batch Bi ⊂ GiS , the temporal pre-training loss is computed in the same link-based form as Eq. (20): (i)
Lpre-tem (Θt , Φi , Ψi ; Bi ) X exp g e zSu , e zSv+ /τ . =− log exp g e zSu , e zSv+ /τ +exp g e zSu , e zSv− /τ + − (u,v ,v ,t)∈Bi
(28) During Phase I (backbone-timer training), we update the shared temporal backbone and the domain-specific timers, while freezing the adapters. This phase encourages ft to learn reusable temporal aggregation patterns and allows each timer to preserve its domain-specific time scale: (i)
where ωi is the frequency, and dt is the time dimension. Since (Θt , Φi ) ← (Θt , Φi )−η∇(Θt ,Φi ) Lpre-tem (Θt , Φi , Ψi ; Bi ) , (29) T i is domain-specific, each source domain can preserve its while ∇Ψi = 0, and η denotes the learning rate. Mini-batches own temporal resolution and evolution pattern without being {Bi }ni=1 are sampled in turn, with each mini-batch containing forced to share a globally aligned time scale. the same number of samples to avoid domain-size bias. Shared Backbone and Domain Adapters. After relativeAfter Phase I converges, Phase II freezes the shared backbone time encoding, the remaining challenge is how to share and timers, and only updates the adapters: temporal knowledge without erasing domain-specific dynamics. (i) A fully shared temporal model may overlook domain-specific Ψi ← Ψi − η∇Ψi Lpre-tem (Ψi ; Bi ) , (30) rhythms, whereas fully independent temporal models would prevent cross-domain knowledge transfer. To balance universal- while keeping ∇Θt = 0 and ∇Φi = 0. Each adapter is updated ity and specificity, we adopt a shared TGAT [52] backbone ft to only when its corresponding domain is sampled, and this lets model general temporal aggregation patterns, and complement adapters capture domain residual dynamics without overwriting it with lightweight residual adapters {Γi }ni=1 , implemented as the shared temporal knowledge learned by the backbone. bottleneck MLPs, to capture domain-specific temporal residuals. Denote neighbors as Ni (v, t) = {(u, t′ ) ∈ Ei : t′ < t}. B. Adaptation with Cross-Domain Negative Transfer Mitigation Messages m and time-aware attention a are: After semantic-temporal decoupled pre-training, DyGFM ob S ′ S ′ S ′ tains two types of source-domain knowledge: semantic tokens mu→v (t ) = ϕ zu (t ) , ri (t − t ) , (24) from fs and temporal tokens from ft , together with domainau→v (t) = Softmax ψ zSv (< t) , mSu→v (t′ ) , (25) specific timers {T i }ni=1 and adapters {Γi }ni=1 , which provide (u,t′ )∈Ni (v,t) reusable priors for downstream adaptation. However, not every where ψ (·) is a ranking function, and we implement it by the source domain is equally useful for a given target domain. dot product. The backbone state is: Directly mixing all source experts may introduce irrelevant X S S ′ zv (t) = AGG au→v (t) mu→v (t ) . (26) semantic or temporal knowledge, while selecting only a single source expert may overlook complementary cross-domain ′ (u,t )∈Ni (v,t) patterns. Therefore, adaptation should selectively reuse source Formally, given zSv (t), the domain adapter refines it as: knowledge according to its relevance to the target domain. To mitigate such negative transfer, we formulate crossS S S e zv (t) = zv (t) + Γi zv (t) . (27) domain adaptation as a divergence-aware Mixture-of-Experts In this way, the shared backbone learns reusable temporal mechanism. The core idea is to measure how close each aggregation knowledge across domains, while the adapters source domain is to the target domain from both semantic
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and temporal perspectives, and then route the target graph to the most relevant source-domain experts. In this way, DyGFM can balance transferable cross-domain knowledge with target-specific adaptation. Expert Pool and Prototypes. For each source domain DiS , we summarize its semantic and temporal knowledge with two prototypes. The semantic prototype sSi represents the average semantic token of the domain, while the temporal prototype tSi summarizes its dynamic evolution pattern: 1 X S sSi = S zv , |Vi | S v∈Vi
1 tSi = S |Ei |
X
e zSv (t) .
(31)
(v,t)∈EiS
Given the target dynamic graph G T , we construct its target descriptors in the same semantic-temporal manner. For semantic T T T description, we first build graph G = V , E by discarding b T and A b T denote its aligned feature matrix timestamps. Let X b T is dimension-aligned by and adjacency matrix, where X Eq. (2). The frozen semantic encoder fs⋆ produces: 1 X T bT ,A bT . sT = T zv , ZT = fs⋆ X (32) |V | T v∈V
For temporal description, we keep timestamps and aggregate dynamic embeddings with the frozen temporal encoder ft⋆ : X 1 b T , E T , rT (t) , (33) e tT = T zTv (t) , e zTv (t) = ft⋆ X |E | T
2) Temporal-wise Divergence: For temporal dynamics, we measure the discrepancy between the target and each source temporal prototype with squared Euclidean distance: 2 Divt DT , DiS = tT − tSi 2 . (38) The combined divergence is a weighted sum: Divi = λs · Divs DT , DiS + λt · Divt DT , DiS ,
(39)
where λs and λt are trade-off parameters. Routing Optimization. Based on the combined divergence, the expert router produces mixture weights α ∈ Rn by applying a softmax over negative divergences. A source domain with smaller divergence receives a larger routing weight, indicating higher relevance to the target domain: αi = Softmaxni (−γ Divi ) , Xn Xn s̄S = αi sSi , t̄S = i=1
i=1
(40) αi tSi ,
(41)
where γ is a temperature parameter. The prototypes s̄S and t̄S serve as cross-domain priors for target-domain fine-tuning. To further stabilize routing, we regularize the weights as: Xn Rrouting (α) = λ2 · αi Divi − λ3 · H (α) , (42) i=1
where H (·) denotes the Shannon entropy [117]. The first term suppresses high-divergence domains to reduce negative transfer, while the entropy term prevents the router from collapsing to a single expert too early. This regularized routing design enables (v,t)∈E DyGFM to selectively reuse relevant source-domain knowledge where rT (t) is the relative time encoding defined as in Eq. (22). while maintaining sufficient expert diversity. The pair sT , tT serves as the target-domain descriptor for subsequent routing. C. Fine-tuning with Divergence-Conditioned Prompting Divergence-aware Routing. The routing mechanism should After divergence-aware routing, DyGFM obtains sourceassign higher weights to the source domains that are semantidomain priors that are relevant to the target domain. However, cally and temporally closer to the target domain. To this end, directly updating the pre-trained encoders for each downstream we compute two types of divergences between each source task would be inefficient and may damage the transferable S S T T prototype si , ti and the target descriptor s , t . knowledge learned during pre-training. Therefore, we adapt the 1) Semantic-wise Divergence: For semantics, we paramefrozen encoders through lightweight prompts. The key idea is to terize source and target semantic representations as diagonal condition prompt generation on both target-domain descriptors Gaussian distributions: and source-target divergences, so that the prompts can adjust S S 2S T T 2T N µi = si , diag σi , N µ = s , diag σ , (34) the frozen semantic and temporal encoders according to the target domain while preserving pre-trained knowledge. Conditioned Prompt Generator. Semantic and temporal with variances computed via sample scatter around prototypes: discrepancies provide informative signals for target-domain ⊙2 1 X S σi2S = zv − sSi , (35) adaptation. A larger divergence indicates that the target domain |Vi | deviates more from the source-domain priors, and therefore v∈Vi X requires stronger prompt-based adjustment. To this end, we 1 ⊙2 e σT 2 = T zTv − sT , (36) introduce two parallel prompt generators (P , P ) to produce s t V v∈V T semantic and temporal prompts pTs , pTt : ! n where “⊙2” denotes element-wise square. The semantic diX T T 1 T S ps = P s s , Divs D , Di , (43) vergence is then defined as the closed-form KL divergence n i=1 between diagonal Gaussian distributions: ! n X 1 T T T S pt = P t t , Divt D , Di , (44) Divs DT , DiS n i=1 2 ! d 2S σk2T + µTk − µSi,k σi,k 1X = log 2T + − 1 . (37) where the prompt generators P s and P t are implemented by 2S 2 MLPs with learnable parameters Ωs and Ωt , respectively. σ σ k i,k k=1
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
Prompt Injection and Fusion. With all pre-trained encoders frozen, the generated learnable prompts are injected into the encoder inputs rather than used to update the backbone parameters. The semantic prompt adjusts the aligned target features for semantic encoding, while the temporal prompt adjusts the target features used by the temporal encoder: h i bT ,A bT , ZTp = fs⋆ pTs ; X (45) h i e T (t) = f ⋆ pT ; X b T , E T , rT (t) , Z (46) p t t
Algorithm 1: Overall pre-training pipeline of DyGFM. Input: Source dynamic graphs {GiS }ni=1 from domain {DS } with labels {Y S }; Each GiS is with node set Vi , temporal edge set Ei = {(u, v, t)}, and a feature matrix Xi ∈ R|Vi |×di ; Hidden dimension d; Learning rate η; Temperature parameter τ ; Hyper-parameter β, λ1 ; Pre-training epochs E1 , E2 , E3 . Output: Dynamic graph encoder f = fs ◦ ft with parameters θ ⋆ = θs⋆ ◦ θt⋆ ; Domain-specific timers {T i }ni=1 with parameters {Φi }ni=1 ; Domain-specific adapters {Γi }ni=1 with parameters {Ψi }ni=1 ; Semantic tokens e S (t)}n . {ZSi }ni=1 ; Temporal tokens {Z i i=1
where [· ; ·] denotes embedding concatenation. For node v at time t, the prompted representation is obtained by fusing the semantic and temporal outputs: h i e T (t) , ϕ (·) = LN (·) ◦ Linear (·) , (47) HT (t) = ϕ ZTp ; Z p where LN (·) denotes the LayerNorm operation. Task Objectives. To maintain consistency with pre-training, we use the similarity-based task objectives for downstream fine-tuning. This design aligns downstream prediction with the similarity template used in pre-training, so that prompt tuning can adapt the target task without changing the frozen encoders. 1) Temporal Node Classification: Given a labeled support T set Dsup = {(vi , yi , ti )}, define the time-aware class prototype as the mean of support embeddings of class y at time t: X 1 T hTvi (t) . (48) ht,y = T (t,y) T (t, y) (vi ,yi =y,ti =t)∈Dsup Dsup
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Initialize all learnable parameters randomly; // Semantic Branch 3 for e1 = 1, 2, · · · , E1 do 4 Remove timestamps and merge edges for each GiS ; 5 Calculate semantic tokens ZSi ← Eq. (4) with fs ; 6 Calculate IB objective LSS-IB ← Eq. (3); 7 Calculate task loss Lpre θs , ZS ← Eq. (20); 8 Calculate semantic loss Lpre-sem ← Eq. (21); 9 Update θs by minimizing Lpre-sem and back-propagation with learning rate η; 1 2
// Temporal Branch for e2 = 1, 2, · · · , E2 do The loss for query (vi , yi , ti ) is defined by a prototype-softmax: 12 For each edge (u, v, t) ∈ Ei with relative time t, Lnode (Ωs , Ωt ) calculate its relative-time encoding rSi (t) ← T Eq. (22) with {T i }ni=1 ; exp g hTvi (ti ) , hti ,yi /τ X for e3 = 1, 2, · · · , E3 do . (49) 13 =− log P T T e S (t) ← Eq. (27) T 14 Update the temporal tokens Z (vi ,yi ,ti )∈Dsup y∈Y exp g hvi (ti ) , hti ,y /τ i n with ft and {Γi }i=1 for each GiS ; + 2) Temporal Link Prediction: Given positive edge (u, v , t) // Phase I and negatives {(u, v − , t)}, we apply an InfoNCE [114] loss: 15 16 Sample mini-batch Bi ⊂ GiS for each GSi ; Llink (Ωs , Ωt ) 17 Calculate the temporal pre-training loss for X (i) exp g hTu , hTv+ /τ TGAT backbone Lpre-tem ← Eq. (28); P . =− log T T (i) exp g hTu , hv+ /τ + exp g hTu , hv− /τ 18 Update θt , Φi by minimizing Lpre-tem and (u,v + ,v − ,t) v− back-propagation with learning rate η; (50) 10 11
Overall Fine-tuning Objective. The fine-tuning objective combines downstream task loss with routing regularizer: Lftn (Ωs , Ωt , α) = Ltask + λr · Rrouting (α) ,
(51)
task ∈ {node, link} ,
(52)
where λr is a hyper-parameter. The task loss adapts the prompts to the downstream objective, while the routing regularizer keeps the adaptation aware of source-target divergence, reducing the risk of relying on mismatched source-domain knowledge. V. A LGORITHMS AND C OMPLEXITY A NALYSIS We summarize the pre-training procedure of DyGFM in Algorithm 1 and the fine-tuning procedure in Algorithm 2. The analysis below shows that both stages scale linearly with graph size, while the additional costs introduced by semantictemporal decoupling, divergence-aware routing, and prompt learning remain lightweight.
19 20
// Phase II (i) Update Ψi by minimizing Lpre-tem and back-propagation with learning rate η, while keeping the updated ∇θt = 0 and ∇Φi = 0;
A. Complexity Analysis of Algorithm 1 The pre-training stage consists of the semantic branch and the temporal branch. Their costs are mainly determined by feature alignment, message passing, and temporal aggregation. •
Semantic Branch (lines 3-9): For each source domain, feature alignment projects the input feature matrix into the shared semantic space, with complexity O (|Vi |d0 d). The GNN-based aggregation, InfoNCE-based IB objective, and link-based pre-training loss are computed over graph edges,
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leading to complexity O (|Ei |d). Therefore, the overall cost of the semantic branch for domain i is: O (|Ei |d + |Vi |d0 d) .
Algorithm 2: Overall fine-tuning pipeline of DyGFM. Input: Target dynamic graph G T with a feature matrix X ∈ R|Vi |×di ; Dynamic graph encoder f = fs ◦ ft with parameters θ ⋆ = θs⋆ ◦ θt⋆ ; Timers {T i }ni=1 with parameters {Φi }ni=1 ; Adapters {Γi }ni=1 with parameters {Ψi }ni=1 ; Semantic tokens {ZSi }ni=1 ; Temporal tokens e S (t)}n ; Hidden dimension d; Learning rate {Z i i=1 η; Temperature τ ; Hyper-parameter γ, λs , λt , λr , λ2 , λ3 ; Fine-tuning epochs E4 . Output: Fine-tuned dynamic graph learner h⋆ = g ⋆ ◦ f ⋆ , where f ⋆ = fs⋆ ◦ ft⋆ with parameters {θs⋆ , θt⋆ , Ω⋆s , Ω⋆t }.
(53)
• Temporal
Branch (lines 11-20): The temporal branch performs relative-time encoding, TGAT-based temporal propagation, and lightweight adapter refinement. Relative-time encoding and temporal message passing scale with the number of temporal edges, yielding O (|Ei |d). The timers and residual adapters introduce additional linear costs, denoted as O (|Vi |r), where r ≪ d is the adapter bottleneck dimension. Thus, the temporal branch also remains linear with respect to the node and edge counts. P Aggregating over n source domains, let N = i |Vi | and P M = i |Ei | denote the total number of nodes and edges, respectively. The overall pre-training complexity is: O (M d + N d0 d) .
(54)
The dominant computational costs arise from edge-level message passing and node-level feature alignment. Other components, including IB regularization, timers, and adapters, introduce only minor linear overhead. Therefore, DyGFM preserves scalability for large multi-domain dynamic graphs. B. Complexity Analysis of Algorithm 2 The fine-tuning stage adapts the frozen pre-trained encoders to a target dynamic graph through divergence-aware routing and lightweight prompt learning. Its computation mainly arises from prototype construction, prompt generation, and task-specific optimization. • Prototype Construction and Routing (lines 3-14): Computing source prototypes and target semantic-temporal descriptors requires aggregating semantic and temporal representations. The cost is linear in the number of source domains and the target graph size, namely O nd + M T d + N T d , where N T and M T denote the number of target nodes and temporal edges. • Prompt Generation and Injection (lines 15-19): The semantic and temporal prompts are generated by two compact MLPs, whose complexity is O (dp) with prompt dimension p ≪ d. Prompt injection only augments the encoder inputs and does not update the frozen backbone, so it introduces negligible additional cost. • Task Optimization (lines 20-24): The main cost comes from forward propagation through the frozen semantic and temporal encoders, with complexity O M T d + N T d . Computing the node classification or link prediction loss and updating prompt parameters add only linear overhead. Combining above, the overall fine-tuning complexity is: O nd + M T d + N T d + dp . (55) Since n and p are typically small, fine-tuning scales linearly with the target graph size. Moreover, because the pre-trained encoders are frozen and only lightweight prompts and routingrelated parameters are optimized, DyGFM supports efficient and parameter-efficient cross-domain adaptation.
Initialize all learnable parameters randomly; for e4 = 1, 2, · · · , E4 do 3 // Expert Pool and Prototypes 4 Calculate semantic prototype sSi ← Eq. (31); 5 Calculate temporal prototype tSi ← Eq. (31); 6 Calculate target semantic descriptor sT ← Eq. (32); 7 Calculate target temporal descriptor tT ← Eq. (33); 8 // Divergence-aware Routing 9 Calculate semantic divergence Divs DT , DiS ← Eq. (37) between target and source domain; 10 Calculate temporal divergence Divt DT , DiS ← Eq. (38) between target and source domain; 11 Calculate combined divergence Divi ← Eq. (39); 12 // Routing Optimization 13 Initialize routing weights α ← Eq. (40); 14 Calculate routing regularizer Rrouting ← Eq. (42); 15 // Conditioned Prompt Generation 16 Initialize semantic and temporal prompts (pTs , pTt ) ← Eq. (43), Eq. (44) with prompt generators (P s , P t ); 17 // Prompt Injection and Fusion Inject prompts at encoders, and obtain semantic and 18 e T (t) ← Eq. (46); temporal node embeddings ZTp , Z p 19 Obtain prompted node embeddings HT (t) ← Eq. (47) by a lightweight fusion; 20 // Task Objectives 21 Calculate node classification loss Lnode ← Eq. (49); 22 Calculate link prediction loss Llink ← Eq. (50); 23 Calculate overall fine-tuning loss Lftn ← Eq. (51); 24 Update Ωs , Ωt by minimizing Lftn and back-propagation with learning rate η; 1 2
VI. E XPERIMENT In this section, we conduct extensive experiments to evaluate the proposed DyGFM1 over the following research questions: RQ1: How effective does the proposed DyGFM transfer across different datasets and domains? ( Section VI-B) • RQ2: Which module contributes most? ( Section VI-C) • RQ3: How time-efficient in fine-tuning? ( Section VI-D) •
1 https://github.com/RingBDStack/DyGFM.
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
TABLE II S TATISTICS OF THE DYNAMIC GRAPH DATASET. Dataset
#Node
#Edge #Class #Label
Wikipedia [118] 9,227 157,474 Reddit [118] 11,000 672,447 MOOC [118] 7,144 411,749 Genre [18] 1,505 17,858,395
2 2 2 474
217 366 4,066 984
#Feat. Dim.
Time Span
172 30 d 172 30 d 172 30 d 86 1,500 d
RQ4: How interpretable are the routing weights among the source domain experts? ( Section VI-E) • RQ5: How do node embeddings evolve? ( Section VI-F) • RQ6: How sensitive to hyper-parameters? ( Section VI-G) •
A. Experimental Settings
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setting by mixing the available source-domain events during pre-training, and then adapt them to downstream target tasks with the same few-shot splits as DyGFM. • Static GFMs: GraphPrompt [84] introduces a unified pre-train-prompt paradigm for static graphs, ProG [42] provides graph prompting protocols, GCOPE [89] improves cross-domain adaptation with coordinator-based transfer, and SAMGPT [88] aligns multi-domain graphs with structure tokens and dual prompts. Since these methods are designed for static graphs, we first remove timestamps from each available source dynamic graph and merge temporal edges into a static graph for multi-domain pre-training. During downstream evaluation, we attach a TGAT [52] temporal encoder to process temporal interactions, so that static GFMs can be evaluated on the same dynamic node classification and link prediction tasks. This adaptation provides static GFMs with access to multi-domain graph knowledge while allowing them to handle temporal downstream inputs as fairly as possible. • Dynamic Graph Prompting Methods: TIGPrompt [34] introduces a temporal prompt generator to bridge temporal and semantic gaps in temporal interaction graphs, while DyGPrompt [35] designs node-time conditional prompts to capture evolving node-time patterns. These methods are closer to dynamic graph foundation modeling than conventional DGNNs, but they are mainly developed under single-domain pre-training and adaptation settings. For fair comparison, we extend them to the multi-domain scenario by mixing the available source domains during pre-training. Their prompt tuning procedures follow the original designs, with the same support, validation, and test splits as DyGFM.
1) Datasets: We conduct experiments on four widely used real-world continuous-time dynamic graph datasets, covering different domains with heterogeneous semantics and temporal granularities. The statistics are summarized in Table II. • Wikipedia [118]: a user-page editing network, where temporal edges denote editing activities and dynamic labels indicate whether a user is temporarily banned from editing. • Reddit [118]: a user-post interaction network on subreddits, where edges represent users commenting on posts and labels denote whether a user is banned from posting. • MOOC [118]: a student-course activity network from an online education platform, where temporal edges record learning behaviors and labels indicate whether a student drops out after an activity. • Genre [18]: a user-genre listening network, where edges 3) Pre-training and Fine-tuning Settings: We evaluate represent users listening to music genres over time and DyGFM and baselines under two adaptation settings: labels indicate users’ preferred music genres. • ASDA (All-Seen-Domain-Adaptation): All four domains 2) Baselines: We compare DyGFM with 12 state-of-the-art are included during pre-training, and each domain is later baselines from four categories, covering conventional DGNNs, used for downstream adaptation. dynamic graph pre-training methods, static graph foundation • LODO (Leave-One-Domain-Out): One domain is excluded models, and dynamic graph prompting methods. For fair from pre-training and used only for downstream adaptation, comparison, all baselines are adapted to the same pre-training which evaluates the transferability to unseen domains. and fine-tuning settings whenever applicable. For each domain, the events (temporal edges) are ordered • Naive DGNNs: TGN [65] uses memory modules to capture historical interactions, TGAT [52] employs time encoding chronologically to preserve causality. We use the first 80% of and temporal attention for inductive representation learning, events for pre-training and the remaining 20% for downstream ROLAND [119] adapts static GNNs to dynamic settings evaluation. The downstream portion is further split into a 1% through recurrent state updates, and TREND [14] models support set, a 1% validation set, and an 18% test set. This temporal event and node dynamics with Hawkes-process- setting simulates few-shot adaptation, where only limited targetbased mechanisms. Since these methods are not originally domain labels are available. designed for multi-domain pre-training, we train them on During pre-training, DyGFM utilizes a two-layer GAT as the available source-domain dynamic graphs under the the semantic encoder and a two-layer TGAT as the temporal same protocol as DyGFM, so that they can access the same encoder. The model is pre-trained for up to 5,000 epochs with source-domain data. an early stopping strategy. During fine-tuning, all pre-trained • Dynamic Graph Pre-training: DDGCL [120] learns tempo- encoders are frozen, and only the divergence-aware router and ral consistency through dynamic contrastive learning with divergence-conditioned prompt generators are updated. Finedebiased sampling, while CPDG [121] performs contrastive tuning is conducted for up to 300 epochs with early stopping pre-training by jointly modeling structural and temporal sig- based on validation AUC. We use Adam for optimization, nals. These methods are originally developed for dynamic with the learning rate in [10−4 ,10−1 ] and weight decay in graph pre-training, but mainly in single-domain settings. [10−5 ,10−1 ]. The fine-tuning hyper-parameters λs , λt , and λr For fair comparison, we extend them to the multi-domain are in [0, 1], [0, 1], and [0.001, 1], respectively.
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
TABLE III C OMPARISON OF NODE CLASSIFICATION AND LINK PREDICTION ( TRANSDUCTIVE AND INDUCTIVE ) PERFORMANCE UNDER ASDA AND LODO SETTINGS . R ESULTS ARE REPORTED WITH AUC-ROC (%) WITH MEAN ± STANDARD DEVIATION OVER FIVE RUNS . “bb.” DENOTES BACKBONES . Task Target
Node Classification Wikipedia Reddit
MOOC
Link Prediction (transductive) Genre Wikipedia Reddit
Methods / Setting
MOOC
Link Prediction (inductive)
Genre Wikipedia Reddit
MOOC
Genre
ASDA (All-Seen-Domain-Adaptation)
TGN arXiv’20 [65] TGAT ICLR’20 (bb.) [52] ROLAND-GCN KDD’22 [119] ROLAND-GAT KDD’22 [119] TREND WWW’22 [14]
49.5 ±5.1 65.3 ±5.3 56.2 ±5.6 60.0 ±6.6 67.0 ±6.7
47.6 ±3.9 50.0 ±4.5 45.5 ±9.0 46.8 ±8.0 62.3 ±6.7
47.2 ±7.3 47.9 ±8.2 55.3 ±6.6 48.5 ±6.6 46.4 ±8.1 43.5 ±3.9 47.2 ±4.8 44.1 ±3.9 63.3 ±4.0 47.5 ±5.3
67.3 ±4.1 54.1 ±1.1 51.4 ±1.1 54.6 ±2.1 61.9 ±2.2
70.2 ±3.3 60.9 ±2.1 57.5 ±2.4 59.0 ±2.4 77.6 ±1.1
52.6 ±3.1 64.5 ±3.5 53.6 ±1.4 62.8 ±5.1 52.6 ±1.8 53.7 ±3.7 52.5 ±1.6 54.1 ±1.3 54.8 ±3.1 52.5 ±2.5
57.2 ±2.5 52.9 ±3.9 51.5 ±1.2 54.3 ±2.3 52.5 ±4.5
67.0 ±1.8 59.5 ±2.2 55.7 ±3.7 57.4 ±2.5 66.4 ±2.8
DDGCL CIKM’21 [120] CPDG ICDE’24 [121]
63.8 ±7.4 49.2 ±4.3
52.0 ±6.4 59.3 ±9.5 46.5 ±7.7 58.7 ±4.7 51.3 ±8.6 44.8 ±5.9
52.7 ±4.2 53.9 ±2.7
60.9 ±2.5 52.7 ±5.5 63.5 ±4.8 59.6 ±1.6 53.3 ±1.9 51.1 ±2.1
50.7 ±3.6 50.7 ±3.7
54.7 ±1.2 51.3 ±1.7 50.0 ±3.3 57.1 ±3.3 52.5 ±3.9 51.0 ±4.4
GraphPrompt WWW’23 [84] ProG NeurIPS’24 [42] GCOPE KDD’24 [89] SAMGPT WWW’25 [88]
71.0 ±4.5 57.5 ±2.6 59.9 ±2.7 65.0 ±4.0
57.5 ±4.3 65.8 ±3.4 66.0 ±6.6 74.2 ±2.2
62.3 ±5.3 47.0 ±5.1 60.3 ±2.2 46.0 ±2.2 66.0 ±4.7 49.2 ±5.4 73.8 ±5.4 50.1 ±3.6
53.5 ±1.1 65.9 ±2.8 66.5 ±5.3 69.9 ±3.9
65.8 ±1.7 88.9 ±2.2 90.2 ±1.9 90.3 ±3.7
50.7 ±5.7 64.5 ±6.0 54.2 ±3.6 64.1 ±3.1 53.6 ±4.9 65.1 ±6.0 53.4 ±3.7 65.9 ±3.6
52.1 ±2.4 57.9 ±1.1 58.7 ±3.1 57.4 ±4.1
61.6 ±2.6 76.7 ±5.7 75.1 ±2.7 78.4 ±2.0
TIGPrompt arXiv’24 [34] DyGPrompt ICLR’25 [35]
69.8 ±3.9 74.0 ±3.2
65.2 ±5.9 70.4 ±4.9 49.1 ±3.9 72.8 ±3.7 76.0 ±2.7 52.2 ±5.2
60.8 ±2.8 67.5 ±4.7
86.2 ±1.8 51.6 ±2.7 66.2 ±2.9 92.7 ±2.9 55.3 ±2.1 66.5 ±2.4
51.7 ±1.2 57.6 ±2.4
64.1 ±2.4 51.8 ±4.9 52.7 ±3.9 74.5 ±2.2 53.8 ±3.3 57.5 ±2.3
DyGFM (ours)
76.5 ±3.3
77.5 ±4.3 79.8 ±6.0 55.3 ±4.2
70.5 ±1.2
94.2 ±0.6 55.4 ±3.7 67.1 ±3.4
59.6 ±0.6
80.6 ±0.3 54.0 ±4.5 58.7 ±4.6
TGN arXiv’20 [65] TGAT ICLR’20 (bb.) [52] ROLAND-GCN KDD’22 [119] ROLAND-GAT KDD’22 [119] TREND WWW’22 [14]
54.2 ±6.5 61.8 ±6.1 53.5 ±4.4 57.5 ±6.3 63.1 ±5.5
44.8 ±3.7 55.4 ±3.8 42.2 ±9.2 42.2 ±7.5 59.1 ±5.1
44.2 ±7.0 44.8 ±9.6 52.3 ±6.4 45.1 ±7.1 43.3 ±7.5 41.3 ±2.3 44.7 ±5.7 42.6 ±1.7 61.3 ±2.2 46.0 ±3.2
59.1 ±2.0 52.4 ±1.3 50.7 ±2.9 53.9 ±3.4 60.2 ±1.9
63.3 ±3.1 58.2 ±2.5 54.0 ±2.1 56.9 ±1.7 74.3 ±1.6
51.9 ±4.4 62.7 ±2.4 51.6 ±1.4 60.7 ±6.7 51.3 ±5.3 54.0 ±2.2 50.6 ±3.5 55.0 ±3.1 51.0 ±5.5 53.5 ±2.3
54.5 ±2.1 51.9 ±4.3 50.1 ±2.4 52.4 ±3.1 51.2 ±5.5
65.0 ±2.8 57.8 ±3.1 54.5 ±2.8 55.4 ±1.8 64.3 ±2.1
DDGCL CIKM’21 [120] CPDG ICDE’24 [121]
60.6 ±8.0 46.9 ±3.5
50.6 ±8.8 56.0 ±8.7 44.5 ±5.5 56.0 ±3.4 45.9 ±7.3 42.1 ±6.4
52.0 ±4.8 51.3 ±1.9
58.1 ±1.9 52.2 ±7.7 61.6 ±5.9 53.4 ±1.4 50.9 ±2.6 51.8 ±2.4
50.2 ±1.4 50.9 ±3.3
51.4 ±1.3 53.2 ±1.7 50.6 ±3.6 53.4 ±1.2 54.7 ±2.8 50.5 ±2.6
GraphPrompt WWW’23 [84] ProG NeurIPS’24 [42] GCOPE KDD’24 [89] SAMGPT WWW’25 [88]
68.2 ±3.0 53.8 ±4.3 58.0 ±2.4 70.0 ±2.4
53.7 ±3.5 62.4 ±2.0 64.7 ±6.0 72.9 ±4.1
58.1 ±3.5 43.9 ±4.8 58.4 ±4.6 43.8 ±2.9 62.2 ±4.2 49.7 ±4.9 72.3 ±7.5 51.1 ±4.1
52.3 ±1.1 63.4 ±3.3 67.6 ±5.2 68.0 ±2.7
62.4 ±1.3 86.5 ±4.7 86.5 ±2.5 86.3 ±1.2
52.2 ±5.1 62.4 ±5.6 50.2 ±4.5 61.4 ±1.9 52.2 ±2.0 64.8 ±4.6 52.6 ±1.5 66.5 ±1.3
51.9 ±1.2 55.4 ±3.9 54.2 ±1.8 55.5 ±5.8
58.6 ±2.0 73.0 ±3.3 71.3 ±3.6 73.0 ±1.4
TIGPrompt arXiv’24 [34] DyGPrompt ICLR’25 [35]
67.2 ±4.2 74.6 ±5.9
62.9 ±5.0 68.5 ±5.8 48.0 ±4.4 74.5 ±4.6 72.8 ±3.8 47.1 ±4.1
57.5 ±1.6 64.8 ±5.7
74.6 ±3.0 51.3 ±1.1 64.2 ±4.0 87.6 ±1.0 53.0 ±4.8 65.3 ±2.3
50.1 ±4.0 55.8 ±2.1
62.8 ±1.4 52.7 ±3.6 55.3 ±3.1 74.0 ±4.7 54.7 ±3.2 57.7 ±2.3
DyGFM (ours)
76.7 ±6.2
76.8 ±4.7 78.6 ±2.0 52.6 ±3.3
70.2 ±1.1
93.8 ±4.2 53.1 ±0.6 67.4 ±0.6
56.1 ±0.5
74.7 ±1.1 56.0 ±4.1 58.0 ±5.4
Methods / Setting
51.3 ±7.7 56.7 ±3.1 52.8 ±2.5 55.4 ±1.9 51.7 ±4.5 53.7 ±3.3 52.2 ±3.4 54.3 ±2.2 52.3 ±2.8 51.2 ±1.0
50.3 ±3.6 57.4 ±1.3 53.7 ±6.6 55.9 ±2.6 52.1 ±2.9 54.0 ±3.8 52.7 ±1.3 54.9 ±2.4
LODO (Leave-One-Domain-Out) 53.6 ±6.9 56.2 ±1.3 54.5 ±2.7 55.4 ±2.5 53.9 ±3.7 52.0 ±4.0 53.7 ±1.0 54.0 ±1.3 55.3 ±3.0 52.4 ±1.3
51.7 ±5.6 56.6 ±1.8 53.9 ±4.4 56.9 ±1.7 52.3 ±1.5 55.0 ±2.1 55.6 ±3.9 58.4 ±1.0
4) Downstream Tasks and Evaluation: We evaluate two the downstream portion of the same domains. This setting evaldownstream tasks: temporal node classification and temporal uates in-domain transfer efficiency under the few-shot scenario, link prediction. For temporal node classification, dynamic labels where only 1% of the target labels are available. The results are aligned with event timestamps, and the model predicts show that DyGFM consistently achieves the best performance the label of a node at the corresponding time. For temporal across all datasets and tasks. The largest improvement appears link prediction, positive and negative samples are constructed on MOOC node classification, where DyGFM outperforms at each timestamp, and performance is evaluated under both DyGPrompt by 5.0%. Compared with single-domain dynamic transductive and inductive settings. We report AUC-ROC (%) prompt baselines such as DyGPrompt and TIGPrompt, with the mean and standard deviation over five repeated runs. DyGFM shows stronger in-domain generalization by leveraging 5) Implementation Environment: All experiments are con- divergence-aware routing, which prevents overfitting to a single ducted on a server with Ubuntu 20.04 LTS, an Intel(R) Xeon(R) source. The consistent gains under both transductive and Platinum 8358 [email protected], 1TB DDR4 memory, and an inductive link prediction further indicate that semantic-temporal NVIDIA Tesla A100 GPU with 80GB memory. The software decoupling brings complementary benefits: semantic prototypes environment includes CUDA 10.1, Python 3.8.12, PyTorch provide global structural priors, while temporal tokens capture 1.9.1, and PyTorch Geometric 2.0.1. dynamic evolution. 2) LODO Setting (Table III): In this setting, one domain B. RQ1: Transfer across Domains and Tasks is excluded from pre-training and used only for fine-tuning, RQ1 evaluates the effectiveness of DyGFM in transferring directly testing cross-domain generalization to unseen domains. across different domains and downstream tasks under different The results show that DyGFM again achieves the highest scores adaptation settings. We conduct experiments on both link on most datasets and tasks, with notable relative improvements prediction and node classification over four continuous-time of 8.0% on MOOC node classification and 7.1% on Reddit dynamic graph datasets, considering three experimental setups. transductive link prediction. These improvements demonstrate 1) ASDA Setting (Table III): In this setting, all domains are that divergence-aware routing effectively mitigates negative included during pre-training, and fine-tuning is performed on transfer by dynamically adjusting domain mixture weights
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
DyGFM (-) w/o SS-IB
3
MOOC
Genre
DyGFM
77.6 75.4 78.1
76.9 77.6 78.2
74.8 78.4 78.9
51.0 54.2 53.3
Average
77.0 ±1.4
77.6 ±0.7
77.4 ±2.2
52.8 ±1.7
Source
R M G
W M G
W R G
W R M
Target
Wikipedia
MOOC
Genre
DyGFM
79.4 76.8 75.2
77.5 75.1 77.0
78.9 75.3 79.6
52.7 52.1 58.9
Average
77.1 ±2.1
76.5 ±1.3
77.9 ±2.3
54.6 ±3.8
60
55 .3 52 . 51 6 .5 54 .1
70
50 40
Wikipedia Reddit
MOOC
Genre
Link Prediction (transductive, LODO) 100 80 60
67 .4 62 . 646 .8 65 .3
Wikipedia
79 .8 73 .3 75 . 74 8 .0
Target
80
77 .5 72 . 746 .5 74 .8
W+R W+M R+M
93 .8 88 .5 90 . 91 5 .7
W+R W+G R+G
AUC-ROC (%)
W+M W+G M+G
Node Classification (ASDA)
53 .1 48 .3 51 . 522 .5
K
R+M R+G M+G
(-) w/o Routing (-) w/o Condition
70 .2 66 . 687 .9 67 .2
2
Source
AUC-ROC (%)
K
90
76 .5 73 . 73 2 .9 75 .1
TABLE IV R ESULTS UNDER THE K-LODO SETTING FOR NODE CLASSIFICATION WITH AUC-ROC (%). “W” DENOTES Wikipedia, “R” DENOTES Reddit, “M” DENOTES MOOC, AND “G” DENOTES Genre.
13
40 Wikipedia Reddit
MOOC
Genre
Fig. 3. Ablation study on node classification and transductive link prediction.
C. RQ2: Ablation Study according to semantic and temporal discrepancies. Compared We conduct ablation studies on three core modules: with static GFMs such as GCOPE and SAMGPT, DyGFM ex• DyGFM (w/o SS-IB): We disable the information bottlehibits stronger adaptability to unseen dynamic patterns and neck aligner in the semantic branch (Eq. (3)), preventing filirregular temporal shifts, since divergence-conditioned prompts tering redundant domain-specific noise during pre-training. are expected to selectively amplify transferable knowledge • DyGFM (w/o Routing): We remove the divergence-aware from semantically similar source domains. Overall, the results router and directly average source experts (Eq. (40)), ablatconfirm that DyGFM not only performs well in in-domain fewing dynamic balancing between universality and specificity. shot scenarios, but also generalizes effectively across domains • DyGFM (w/o Condition): We replace the divergencewith distinct semantics and temporal granularities. conditioned prompt generator with a randomly initialized Key Takeaway: DyGFM consistently excels under both prompt (Eq. (43), Eq. (44)), removing the conditioning on ASDA and LODO settings, demonstrating strong few-shot both specific task and divergence. adaptability and robust cross-domain generalization. Results in Figure 3 show that removing “SS-IB” noticeably 3) K-LODO Setting (Table IV): We further consider a more degrades performance across all datasets. This confirms that the challenging Leave-One-or-K-Domains-Out setting (K > 1), self-supervised information bottleneck in the semantic branch where some domains are excluded from pre-training. This setup effectively suppresses noise while preserving transferable evaluates the robustness of DyGFM under larger domain gaps. semantic priors for downstream tasks. Removing “Routing” The results show that DyGFM maintains satisfying perforleads to moderate but consistent performance drops, e.g., mance even when multiple domains are removed from pre–3.4% on average for node classification and –2.3% for training, with only minor degradation compared with the singlelink prediction. Without divergence-aware routing, the model domain LODO case. The averaged AUC-ROC across all target cannot dynamically select suitable source-domain experts under domains remains competitive for node classification, indicating domain shifts, which weakens its cross-domain adaptability. stable cross-domain generalization. We also observe that different source-domain combinations exhibit complementary Removing “Condition” also reduces performance, particularly effects. Semantically or temporally related domains, such as on domains with stronger heterogeneity such as Genre. This Wikipedia + Reddit, yield higher transferability, whereas indicates that divergence-conditioned prompts are important more dissimilar pairs, such as MOOC + Genre, show reduced for adjusting domain-specific biases and enabling fine-grained gains. This suggests that semantic proximity helps bridge adaptation under different semantic and temporal dynamics. Key Takeaway: The three modules provide complemenstructural and temporal gaps. In addition, increasing K does tary benefits: semantic compression stabilizes transferable not always improve performance. Although multi-source preknowledge, routing improves source-domain relevance, and training enriches domain coverage, it may also introduce conditioned prompting refines adaptation granularity. conflicting temporal or semantic dynamics. The slight variance across target domains, such as stronger results on MOOC and weaker results on Reddit, suggests that adaptive routing and D. RQ3: Fine-tuning Efficiency divergence-conditioned prompting can balance universality and To test fine-tuning efficiency, we compare DyGFM with two domain specificity without overfitting. competitive baselines on Reddit for inductive link prediction.
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
TIGPrompt 20
Epoch: 285
Epoch: 56
10
100 200 AUC-ROC Curvature 80.6%
80
300
DyGFM DyGPrompt 12
14
TIGPrompt 16
Fig. 4. Efficiency Analysis on Reddit.
ASDA: G
80
200 100 GPU Memory Efficiency (GB)
80
ASDA: M
1.0
60
0
40
ASDA: R
0.8
Suspended
Active
Wikipedia Wikipedia (raw) Wikipedia
Wikipedia Wikipedia (finetuned) Wikipedia
Reddit (raw) Reddit Reddit
Reddit Reddit Reddit (finetuned)
MOOC (raw) MOOC MOOC
MOOC (finetuned) MOOC MOOC
40
64.1% 0
AUC-ROC (%)
300
74.5%
60
ASDA: W
0.4 0.6 Routing Attention
20
W R M G
W R M G
W R M G
W R M G
LODO: W
LODO: R
LODO: M
LODO: G
20 Nodes
AUC-ROC (%)
0
0.2 0
Epoch: 216
0
60
0.0
Nodes
Loss
DyGFM
DyGPrompt Loss Curvature
40 60 80
18
R M G
R M G
R M G
Fig. 5. Routing Visualizations.
R M G
Fig. 6. User Node Visualizations.
Results in Figure 4 show DyGFM converges the fastest, Key Takeaway: The routing mechanism is interpretable and stabilizing within 56 epochs, compared with 216 epochs for adaptive. When the target domain is seen during pre-training, DyGPrompt and 285 epochs for TIGPrompt. This indicates it assigns higher attention to domain-specific prototypes for that divergence-conditioned prompting provides a more effec- efficient knowledge reuse. When the target domain is unseen, tive initialization for downstream adaptation. DyGFM also it flexibly redistributes weights across semantically related achieves the highest AUC-ROC of 80.6%, with its smoother source prototypes, revealing a balance between specialization convergence curves and fewer fluctuations. It suggests that and generalization. semantic-temporal priors help guide optimization more stably during fine-tuning. In terms of GPU memory usage, F. RQ5: Evolution of Node Embeddings DyGFM falls between the two baselines: it consumes slightly To examine how node representations evolve during finemore memory than DyGPrompt, but substantially less than tuning, we visualize user embeddings before and after adaptaTIGPrompt (approximately 15 GB, 14 GB, and 18 GB). Contion using UMAP [122] on three representative user interaction sidering its higher predictive performance, DyGFM achieves datasets. These datasets contain dynamic labels indicating the best accuracy-memory trade-off. whether each user is Active or Suspended, which allows us to Key Takeaway: DyGFM achieves the fastest convergence analyze the separation of behavioral states. and highest task performance with moderate GPU usage, Results in Figure 6 show that raw embeddings of users with demonstrating superior fine-tuning efficiency and scalability. different activity states are highly entangled. This suggests that pre-training mainly captures general structural and temporal regularities, rather than task-specific discriminative semantics. E. RQ4: Interpretability of Routing Weights After fine-tuning, the embeddings of the two user types become To examine whether the learned routing weights are intermore clearly separable, especially on Wikipedia and MOOC, pretable and informative, we randomly sample 100 nodes from where distinct cluster patterns emerge. each target domain and visualize the normalized routing attenKey Takeaway: Fine-tuning reorganizes the embedding tion toward different source prototypes for node classification. space into more discriminative regions while preserving The results are shown in Figure 5, where columns denote continuity with the pre-trained representations, showing that source domains (W: Wikipedia, R: Reddit, M: MOOC, G: DyGFM enhances the separability of user states without losing Genre), and each row corresponds to the routing distribution general semantic information. of one sampled node. In the ASDA setting, nodes consistently assign the highest routing weights to their own-domain prototypes, suggesting that the router effectively captures domain- G. RQ6: Hyper-parameter Sensitivity Analysis aligned semantics and reuses intra-domain knowledge. In We conduct analysis on the sensitivity of three key hyperthe LODO setting, routing patterns become softer and more parameters: λs for semantic divergence in Eq. (39), λt for distributed. Unseen target domains activate multiple source temporal divergence in Eq. (39), and λr for routing regularizaexperts, such as Genre nodes attending to both Reddit tion in Eq. (51). and MOOC, demonstrating adaptive knowledge blending across Results in Figure 7 show that λs has a moderate influence on related domains. performance, with the best accuracy achieved around λs = 0.6.
AUC-ROC (%)
AUC-ROC (%)
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
80.0
Node Classification on Wikipedia (ASDA)
75.0 70.0
0.2
80.0
λs
0.6
0.8
A PPENDIX A S UMMARY OF N OTATIONS For clarity, we summarize the main notations used throughout this paper in Table V. These notations cover the dynamic graph setting, semantic-temporal pre-training, divergence-aware routing, prompt-based fine-tuning, and complexity analysis.
75.0
TABLE V S UMMARY OF NOTATIONS . 0.1
AUC-ROC (%)
0.4
15
0.3
80.0
λt
0.5
0.7
Description
Basic notations
75.0 70.0 0.001
Notation
0.01
λr
0.1
1.0
Fig. 7. Hyper-parameter Sensitivity Analysis.
Smaller values may underemphasize semantic alignment, while larger values slightly over-regularize the semantic divergence term. For λt , the model maintains stable performance within the range of 0.1 to 0.3, but performance decreases when the value becomes overly large. This indicates that moderate temporal regularization helps preserve temporal smoothness, whereas excessive regularization may suppress useful dynamic patterns. For λr , performance shows a sharp optimum near 0.1, suggesting that routing regularization needs to balance expert flexibility and stability. A small λr may lead to unstable expert activation, while an overly large value may constrain target-domain adaptation. Key Takeaway: DyGFM is robust to a broad range of hyperparameter values, maintaining stable performance under all the three factors. This indicates that its performance does not rely heavily on finely tuned hyper-parameter choices. VII. C ONCLUSION In this paper, we proposed DyGFM, a dynamic Graph Foundation Model for multiple domains based on decoupled and divergence-conditioned prompting. DyGFM addresses the incompatibility of multi-domain dynamic graphs through semantic-temporal decoupled pre-training, where the semantic branch learns transferable feature semantics, and the temporal branch preserves domain-specific temporal dynamics. To mitigate negative transfer, DyGFM further introduces divergenceaware routing to select relevant source-domain knowledge, and employs divergence-conditioned prompts for efficient downstream fine-tuning with frozen encoders. Extensive experiments on four continuous-time dynamic graph benchmarks show that DyGFM outperforms representative DGNNs, dynamic graph pre-training methods, static GFMs, and dynamic graph prompting baselines on node classification and link prediction tasks, demonstrating strong transferability, adaptability, and fine-tuning efficiency.
G = {V, E, T }, (u, v, t) X, A, euv d0 , d, dt , r, p z, e z, h, H T (·), PE(·), ωi ϕ(·), ψ(·), AGG(·), g(·)
Dynamic graph and temporal interaction. Node features, adjacency, and edge features. Input, hidden, time, adapter, prompt dimensions. Node-level and fused representations. Time encoding terms. Message, ranking, aggregation, scoring functions.
Pre-training and adaptation setting {GiS , DiS , YiS }n i=1 T G T , DT , Dsup n, m h = g ◦ f , fs , ft θ ⋆ , fs⋆ , ft⋆
Source graphs, domains, and labels. Target graph, domain, and support set. Number of domains and shots. Learner, semantic encoder, and temporal encoder. Frozen parameters and encoders.
Semantic branch S bS bT bT XS i , Ai , Xi , X , A A i , θs ZS , ZS+ , ZT I(·; ·), qϕ (·), p(·), KL(·) N +, N −, τ LInfoNCE , LSS-IB , Lpre , Lpre-sem λ1
Source/target features and structures. Aligner and semantic parameters. Source, positive, and target representations. IB-related quantities. Positive count, negative count, and temperature. Semantic-branch losses. IB regularization weight.
Temporal branch T rS i (t), r (t), T i Γ i , Θt , Φ i , Ψi Ni (v, t), mS u→v , au→v Bi , η (i) Lpre-tem
Time embeddings and timer. Adapter and temporal parameters. Temporal neighbors, messages, and attention. Mini-batch and learning rate. Temporal pre-training loss.
Cross-domain routing S T T sS i , ti , s , t
Source and target prototypes.
G ,V ,E T 2S 2T µS i , µ , σi , σ Divs , Divt , Divi λ s , λt , γ α, αi , s̄S , t̄S Rrouting , H(α), λ2 , λ3
Timestamp-removed target graph. Gaussian statistics. Semantic, temporal, and total divergences. Routing hyper-parameters. Routing weights and priors. Routing regularization terms.
T
T
T
Prompt-based fine-tuning P s , P t , Ωs , Ωt T pT s , pt T e Zp , ZT p (t) LN(·), Linear(·) T
ht,y , Y Lnode , Llink , Ltask , Lftn λr
Prompt generators and parameters. Semantic and temporal prompts. Prompted representations. Fusion operators. Class prototype and label space. Fine-tuning losses. Routing regularization weight.
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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 16, NO. 8, AUGUST 2026
R EFERENCES [1] M. E. Newman, D. J. Watts, and S. H. Strogatz, “Random graph models of social networks,” Proceedings of the National Academy of Sciences, vol. 99, no. suppl 1, pp. 2566–2572, 2002. [2] W. Fan, Y. Ma, Q. Li, Y. He, E. Zhao, J. Tang, and D. Yin, “Graph neural networks for social recommendation,” in WWW, 2019, pp. 417–426. [3] A. Sankar, Y. Liu, J. Yu, and N. Shah, “Graph neural networks for friend ranking in large-scale social platforms,” in WWW, 2021, pp. 2535–2546. [4] R. Ying, R. He, K. Chen, P. Eksombatchai, W. L. Hamilton, and J. Leskovec, “Graph convolutional neural networks for web-scale recommender systems,” in KDD, 2018, pp. 974–983. [5] S. Wu, Y. Tang, Y. Zhu, L. Wang, X. Xie, and T. Tan, “Session-based recommendation with graph neural networks,” AAAI, vol. 33, no. 01, pp. 346–353, 2019. [6] H. Wang, F. Zhang, M. Zhang, J. Leskovec, M. Zhao, W. Li, and Z. Wang, “Knowledge-aware graph neural networks with label smoothness regularization for recommender systems,” in KDD, 2019, pp. 968–977. [7] Z. Wang, J. Zhang, J. Feng, and Z. Chen, “Knowledge graph embedding by translating on hyperplanes,” in AAAI, vol. 28, no. 1, 2014. [8] Q. Wang, Z. Mao, B. Wang, and L. Guo, “Knowledge graph embedding: A survey of approaches and applications,” IEEE TKDE, vol. 29, no. 12, pp. 2724–2743, 2017. [9] Y. Zhang and Q. Yao, “Knowledge graph reasoning with relational digraph,” in WWW, 2022, pp. 912–924. [10] X. Lin, Z. Quan, Z.-J. Wang, T. Ma, and X. Zeng, “KGNN: Knowledge graph neural network for drug-drug interaction prediction,” in IJCAI, 2021, pp. 2739–2745. [11] V. Gligorijević, P. D. Renfrew, T. Kosciolek, J. K. Leman, D. Berenberg, T. Vatanen, C. Chandler, B. C. Taylor, I. M. Fisk, H. Vlamakis et al., “Structure-based protein function prediction using graph convolutional networks,” Nature Communications, vol. 12, no. 1, p. 3168, 2021. [12] K. Jha, S. Saha, and H. Singh, “Prediction of protein-protein interaction using graph neural networks,” Scientific Reports, vol. 12, no. 1, p. 8360, 2022. [13] W. Fan, R. Jin, P. Lu, C. Tian, and R. Xu, “Towards event prediction in temporal graphs,” VLDB, vol. 15, no. 9, pp. 1861–1874, 2022. [14] Z. Wen and Y. Fang, “TREND: Temporal event and node dynamics for graph representation learning,” in WWW, 2022, pp. 1159–1169. [15] M. Liu, K. Liang, D. Hu, H. Yu, Y. Liu, L. Meng, W. Tu, S. Zhou, and X. Liu, “TMac: Temporal multi-modal graph learning for acoustic event classification,” in ACM MM, 2023, pp. 3365–3374. [16] D. Xu, W. Cheng, D. Luo, X. Liu, and X. Zhang, “Spatio-temporal attentive RNN for node classification in temporal attributed graphs,” in IJCAI, 2019, pp. 3947–3953. [17] U. Singer, I. Guy, and K. Radinsky, “Node embedding over temporal graphs,” in IJCAI, 2019, pp. 4605–4612. [18] S. Huang, F. Poursafaei, J. Danovitch, M. Fey, W. Hu, E. Rossi, J. Leskovec, M. Bronstein, G. Rabusseau, and R. Rabbany, “Temporal graph benchmark for machine learning on temporal graphs,” NeurIPS, vol. 36, pp. 2056–2073, 2023. [19] Q. Zhang, J. Chang, G. Meng, S. Xiang, and C. Pan, “Spatio-temporal graph structure learning for traffic forecasting,” in AAAI, vol. 34, no. 01, 2020, pp. 1177–1185. [20] M. Jin, Y.-F. Li, and S. Pan, “Neural temporal walks: Motif-aware representation learning on continuous-time dynamic graphs,” NeurIPS, vol. 35, pp. 19 874–19 886, 2022. [21] Y. Yang, H. Yin, J. Cao, T. Chen, Q. V. H. Nguyen, X. Zhou, and L. Chen, “Time-aware dynamic graph embedding for asynchronous structural evolution,” IEEE TKDE, vol. 35, no. 9, pp. 9656–9670, 2023. [22] L.-P. A. Xhonneux, M. Qu, and J. Tang, “Continuous graph neural networks,” in ICML, 2020, pp. 10 432–10 441. [23] J. Skarding, B. Gabrys, and K. Musial, “Foundations and modeling of dynamic networks using dynamic graph neural networks: A survey,” IEEE Access, vol. 9, pp. 79 143–79 168, 2021. [24] M. Zhang, S. Wu, X. Yu, Q. Liu, and L. Wang, “Dynamic graph neural networks for sequential recommendation,” IEEE TKDE, vol. 35, no. 5, pp. 4741–4753, 2022. [25] D. Wang, M. Jiang, M. Syed, O. Conway, V. Juneja, S. Subramanian, and N. V. Chawla, “Calendar graph neural networks for modeling time structures in spatiotemporal user behaviors,” in KDD, 2020, pp. 2581–2589. [26] J. Liu, J. Liu, K. Zhao, Y. Tang, and W. Chen, “TP-GNN: Continuous dynamic graph neural network for graph classification,” in IEEE ICDE, 2024, pp. 2848–2861.
[27] Y. Mao, Y. Hao, X. Cao, Y. Gao, C. Yao, and X. Lin, “Boosting GNNbased link prediction via pu-auc optimization,” IEEE TKDE, 2025. [28] M. Lu, Z. Han, S. X. Rao, Z. Zhang, Y. Zhao, Y. Shan, R. Raghunathan, C. Zhang, and J. Jiang, “BRIGHT-graph neural networks in real-time fraud detection,” in CIKM, 2022, pp. 3342–3351. [29] M. Duan, T. Zheng, Y. Gao, G. Wang, Z. Feng, and X. Wang, “DGAGNN: Dynamic grouping aggregation GNN for fraud detection,” in AAAI, vol. 38, no. 10, 2024, pp. 11 820–11 828. [30] Y. Dong, J. Yao, J. Wang, Y. Liang, S. Liao, and M. Xiao, “Dynamic fraud detection: Integrating reinforcement learning into graph neural networks,” in 6th International Conference on Data-driven Optimization of Complex Systems. IEEE, 2024, pp. 818–823. [31] S. Zheng, Y. Li, S. Chen, J. Xu, and Y. Yang, “Predicting drug-protein interaction using quasi-visual question answering system,” Nature Machine Intelligence, vol. 2, no. 2, pp. 134–140, 2020. [32] Z. Li, K. Meidani, P. Yadav, and A. Barati Farimani, “Graph neural networks accelerated molecular dynamics,” The Journal of Chemical Physics, vol. 156, no. 14, 2022. [33] G. Li, Y. Yuan, and R. Zhang, “A spatial-temporal graph attention network for protein-ligand binding affinity prediction based on molecular geometry,” Multimedia Systems, vol. 31, no. 2, p. 94, 2025. [34] X. Chen, S. Zhang, Y. Xiong, X. Wu, J. Zhang, X. Sun, Y. Zhang, F. Zhao, and Y. Kang, “Prompt learning on temporal interaction graphs,” arXiv preprint arXiv:2402.06326, 2024. [35] X. Yu, Z. Liu, X. Zhang, and Y. Fang, “Node-time conditional prompt learning in dynamic graphs,” in ICLR, 2025. [36] Y. Chang, X. Wang, J. Wang, Y. Wu, L. Yang, K. Zhu, H. Chen, X. Yi, C. Wang, Y. Wang et al., “A survey on evaluation of large language models,” ACM TIST, vol. 15, no. 3, pp. 1–45, 2024. [37] J. Zhang, J. Huang, S. Jin, and S. Lu, “Vision-language models for vision tasks: A survey,” IEEE TPAMI, 2024. [38] J. Achiam, S. Adler, S. Agarwal, L. Ahmad, I. Akkaya, F. L. Aleman, D. Almeida, J. Altenschmidt, S. Altman, S. Anadkat et al., “GPT-4 technical report,” arXiv preprint arXiv:2303.08774, 2023. [39] S. Khan, M. Naseer, M. Hayat, S. W. Zamir, F. S. Khan, and M. Shah, “Transformers in vision: A survey,” ACM Computing Surveys, vol. 54, no. 10s, pp. 1–41, 2022. [40] H. Mao, Z. Chen, W. Tang, J. Zhao, Y. Ma, T. Zhao, N. Shah, M. Galkin, and J. Tang, “Position: Graph foundation models are already here,” in ICML, 2024. [41] C. Shi, J. Chen, J. Liu, and C. Yang, “Graph foundation model,” Frontiers of Computer Science, vol. 18, no. 6, 2024. [42] C. Zi, H. Zhao, X. Sun, Y. Lin, H. Cheng, and J. Li, “ProG: A graph prompt learning benchmark,” NeurIPS, vol. 37, pp. 95 406–95 437, 2024. [43] C. Shi, C. Yang, Y. Fang, L. Sun, and P. S. Yu, “Lecture-style tutorial: Towards graph foundation models,” in WWW, 2024, pp. 1264–1267. [44] R. Bommasani, D. A. Hudson, E. Adeli, R. Altman, S. Arora, S. von Arx, M. S. Bernstein, J. Bohg, A. Bosselut, E. Brunskill et al., “On the opportunities and risks of foundation models,” arXiv preprint arXiv:2108.07258, 2021. [45] J. Tang, Y. Yang, W. Wei, L. Shi, L. Su, S. Cheng, D. Yin, and C. Huang, “GraphGPT: Graph instruction tuning for large language models,” in SIGIR, 2024, pp. 491–500. [46] Y. He, Y. Sui, X. He, and B. Hooi, “UniGraph: Learning a unified cross-domain foundation model for text-attributed graphs,” in KDD, 2025, pp. 448–459. [47] D. Lachi, M. Azabou, V. Arora, and E. Dyer, “GraphFM: A scalable framework for multi-graph pretraining,” arXiv preprint arXiv:2407.11907, 2024. [48] G. H. Nguyen, J. B. Lee, R. A. Rossi, N. K. Ahmed, E. Koh, and S. Kim, “Continuous-time dynamic network embeddings,” in WWW, 2018, pp. 969–976. [49] Y. Wang, Y.-Y. Chang, Y. Liu, J. Leskovec, and P. Li, “Inductive representation learning in temporal networks via causal anonymous walks,” in ICLR, 2021. [50] A. Souza, D. Mesquita, S. Kaski, and V. Garg, “Provably expressive temporal graph networks,” NeurIPS, vol. 35, pp. 32 257–32 269, 2022. [51] X. Lu, L. Sun, T. Zhu, and W. Lv, “Improving temporal link prediction via temporal walk matrix projection,” NeurIPS, vol. 37, pp. 141 153– 141 182, 2024. [52] D. Xu, C. Ruan, E. Korpeoglu, S. Kumar, and K. Achan, “Inductive representation learning on temporal graphs,” in ICLR, 2020. [53] W. Cong, S. Zhang, J. Kang, B. Yuan, H. Wu, X. Zhou, H. Tong, and M. Mahdavi, “Do we really need complicated model architectures for temporal networks?” in ICLR, 2023.
YUAN et al.: DECOUPLED AND DIVERGENCE-CONDITIONED PROMPT FOR MULTI-DOMAIN DYNAMIC GRAPH FOUNDATION MODELS
[54] L. Yu, L. Sun, B. Du, and W. Lv, “Towards better dynamic graph learning: New architecture and unified library,” in NeurIPS, 2023, pp. 67 686–67 700. [55] K. Cheng, P. Linzhi, J. Ye, L. Sun, and B. Du, “Co-neighbor encoding schema: A light-cost structure encoding method for dynamic link prediction,” in KDD, 2024, pp. 421–432. [56] S. Zhang, X. Chen, Y. Xiong, X. Wu, Y. Zhang, Y. Fu, Y. Zhao, and J. Zhang, “Towards adaptive neighborhood for advancing temporal interaction graph modeling,” in KDD, 2024, pp. 4290–4301. [57] T. Zou, Y. Mao, J. Ye, and B. Du, “Repeat-aware neighbor sampling for dynamic graph learning,” in KDD, 2024, pp. 4722–4733. [58] J. Li, Z. Han, H. Cheng, J. Su, P. Wang, J. Zhang, and L. Pan, “Predicting path failure in time-evolving graphs,” in KDD, 2019, pp. 1279–1289. [59] Y. Ma, Z. Guo, Z. Ren, J. Tang, and D. Yin, “Streaming graph neural networks,” in SIGIR, 2020, pp. 719–728. [60] Y. Chen, A. Zeng, Q. Yu, K. Zhang, C. Yuanpeng, K. Wu, G. Huzhang, H. Yu, and Z. Zhou, “Recurrent temporal revision graph networks,” NeurIPS, vol. 36, pp. 69 348–69 360, 2023. [61] Y. Zuo, G. Liu, H. Lin, J. Guo, X. Hu, and J. Wu, “Embedding temporal network via neighborhood formation,” in KDD, 2018, pp. 2857–2866. [62] R. Trivedi, M. Farajtabar, P. Biswal, and H. Zha, “DyRep: Learning representations over dynamic graphs,” in ICLR, 2019. [63] Y. Lu, X. Wang, C. Shi, P. S. Yu, and Y. Ye, “Temporal network embedding with micro-and macro-dynamics,” in CIKM, 2019, pp. 469– 478. [64] C. Chen, H. Geng, N. Yang, X. Yang, and J. Yan, “EasyDGL: Encode, train and interpret for continuous-time dynamic graph learning,” TPAMI, vol. 46, no. 12, pp. 10 845–10 862, 2024. [65] E. Rossi, B. Chamberlain, F. Frasca, D. Eynard, F. Monti, and M. Bronstein, “Temporal graph networks for deep learning on dynamic graphs,” arXiv preprint arXiv:2006.10637, 2020. [66] Y. Luo and P. Li, “Neighborhood-aware scalable temporal network representation learning,” in LOG. PMLR, 2022, pp. 1–1. [67] Y. Zhang, Y. Xiong, Y. Liao, Y. Sun, Y. Jin, X. Zheng, and Y. Zhu, “TIGER: Temporal interaction graph embedding with restarts,” in WWW, 2023, pp. 478–488. [68] J. Su, D. Zou, and C. Wu, “PRES: Toward scalable memory-based dynamic graph neural networks,” in ICLR, 2020. [69] S. Ji, M. Liu, L. Sun, C. Liu, and T. Zhu, “MemMap: An adaptive and latent memory structure for dynamic graph learning,” in KDD, 2024, pp. 1257–1268. [70] G. Sheng, J. Su, C. Huang, and C. Wu, “MSPipe: Efficient temporal GNN training via staleness-aware pipeline,” in KDD, 2024, pp. 2651– 2662. [71] Y. Tian, Y. Qi, and F. Guo, “FreeDyG: Frequency enhanced continuoustime dynamic graph model for link prediction,” in ICLR, 2024. [72] Y. Li, Y. Xu, X. Lin, W. Zhang, and Y. Zhang, “Ranking on dynamic graphs: An effective and robust band-pass disentangled approach,” in WWW, 2025, pp. 3918–3929. [73] Z. Fan, Z. Liu, J. Zhang, Y. Xiong, L. Zheng, and P. S. Yu, “Continuoustime sequential recommendation with temporal graph collaborative transformer,” in CIKM, 2021, pp. 433–442. [74] L. Huang, Y. Ma, Y. Liu, B. Danny Du, S. Wang, and D. Li, “Positionenhanced and time-aware graph convolutional network for sequential recommendations,” ACM TOIS, vol. 41, no. 1, pp. 1–32, 2023. [75] H. Tang, S. Wu, X. Sun, J. Zeng, G. Xu, and Q. Li, “TCGC: Temporal collaboration-aware graph co-evolution learning for dynamic recommendation,” ACM TOIS, vol. 43, no. 1, pp. 1–27, 2025. [76] J. Xia, D. Li, H. Gu, T. Lu, P. Zhang, L. Shang, and N. Gu, “Neural Kalman filtering for robust temporal recommendation,” in WSDM, 2024, pp. 836–845. [77] S. Zhang, L. Chen, C. Wang, S. Li, and H. Xiong, “Temporal graph contrastive learning for sequential recommendation,” in AAAI, vol. 38, no. 8, 2024, pp. 9359–9367. [78] D. Zhou, L. Zheng, J. Han, and J. He, “A data-driven graph generative model for temporal interaction networks,” in KDD, 2020, pp. 401–411. [79] S. Tian, J. Dong, J. Li, W. Zhao, X. Xu, B. Wang, B. Song, C. Meng, T. Zhang, and L. Chen, “SAD: semi-supervised anomaly detection on dynamic graphs,” in IJCAI, 2023, pp. 2306–2314. [80] X. Yang, X. Zhao, and Z. Shen, “A generalizable anomaly detection method in dynamic graphs,” in AAAI, vol. 39, no. 20, 2025, pp. 22 001– 22 009. [81] S. Ji, X. Lu, M. Liu, L. Sun, C. Liu, B. Du, and H. Xiong, “Communitybased dynamic graph learning for popularity prediction,” in KDD, 2023, pp. 930–940. [82] M. Liu, Y. Liu, K. Liang, W. Tu, S. Wang, S. Zhou, and X. Liu, “Deep temporal graph clustering,” in ICLR, 2024.
17
[83] D. Li, S. Kosugi, Y. Zhang, M. Okumura, F. Xia, and R. Jiang, “Revisiting dynamic graph clustering via matrix factorization,” in WWW, 2025, pp. 1342–1352. [84] Z. Liu, X. Yu, Y. Fang, and X. Zhang, “GraphPrompt: Unifying pretraining and downstream tasks for graph neural networks,” in WWW, 2023, pp. 417–428. [85] X. Yu, Y. Fang, Z. Liu, and X. Zhang, “HGPrompt: Bridging homogeneous and heterogeneous graphs for few-shot prompt learning,” in AAAI, vol. 38, no. 15, 2024, pp. 16 578–16 586. [86] X. Yu, C. Zhou, Y. Fang, and X. Zhang, “MultiGPrompt for multi-task pre-training and prompting on graphs,” in WWW, 2024, pp. 515–526. [87] Y. Huang, J. Zhao, D. He, X. Wang, Y. Li, Y. Huang, D. Jin, and Z. Feng, “One prompt fits all: Universal graph adaptation for pretrained models,” in NeurIPS, 2025. [88] X. Yu, Z. Gong, C. Zhou, Y. Fang, and H. Zhang, “SAMGPT: Text-free graph foundation model for multi-domain pre-training and cross-domain adaptation,” in WWW, 2025, pp. 1142–1153. [89] H. Zhao, A. Chen, X. Sun, H. Cheng, and J. Li, “All in one and one for all: A simple yet effective method towards cross-domain graph pretraining,” in KDD, 2024, pp. 4443–4454. [90] X. Yu, C. Zhou, Y. Fang, and X. Zhang, “Text-free multi-domain graph pre-training: Toward graph foundation models,” arXiv preprint arXiv:2405.13934, 2024. [91] Z. Zhao, X. Zhai, J. Yang, and C. Shi, “Towards text-free graph foundation models: Rethinking multi-domain graph contrastive learning,” arXiv preprint arXiv:2506.22510, 2025. [92] H. Yuan, Q. Sun, J. Shi, X. Fu, B. Hooi, J. Li, and P. S. Yu, “How much can transfer? BRIDGE: Bounded multi-domain graph foundation model with generalization guarantees,” in ICML, 2025. [93] S. Wang, B. Wang, Z. Shen, B. Deng et al., “Multi-domain graph foundation models: Robust knowledge transfer via topology alignment,” in ICML, 2025. [94] Z. Wang, Z. Zhang, S. Wang, X. Xie, and Z. Kang, “GMoPE: A promptexpert mixture framework for graph foundation models,” arXiv preprint arXiv:2511.03251, 2025. [95] H. Yuan, Q. Sun, J. Shi, X. Fu, B. Hooi, J. Li, and P. S. Yu, “GRAVER: Generative graph vocabularies for robust graph foundation models finetuning,” in NeurIPS, 2025. [96] Z. Guo, Q. Sun, H. Yuan, X. Fu, M. Zhou, Y. Gao, and J. Li, “GraphMoRE: Mitigating topological heterogeneity via mixture of Riemannian experts,” in AAAI, vol. 39, no. 11, 2025, pp. 11 754–11 762. [97] L. Sun, Z. Huang, S. Zhou, Q. Wan, H. Peng, and P. Yu, “RiemannGFM: Learning a graph foundation model from Riemannian geometry,” in WWW, 2025, pp. 1154–1165. [98] L. Sun, Z. Huang, S. Chen, L. Yang, J. Ye, S. Su, and P. S. Yu, “Multidomain transferable graph gluing for building graph foundation models,” in ICLR, 2026. [99] L. Sun and P. S. Yu, “A Riemannian perspective on graph foundation models: Curvature as a guiding principle,” Frontiers of Computer Science, vol. 20, no. 12, p. 2012370, 2026. [100] T. Lin, P. Yan, K. Song, Z. Jiang, Y. Kang, J. Lin, W. Yuan, J. Cao, C. Sun, and X. Liu, “LangGFM: A large language model alone can be a powerful graph foundation model,” arXiv preprint arXiv:2410.14961, 2024. [101] X. Zhu, H. Xue, Z. Zhao, W. Xu, J. Huang, M. Guo, Q. Wang, K. Zhou, I. Razzak, and Y. Zhang, “LLM as GNN: Graph vocabulary learning for text-attributed graph foundation models,” arXiv preprint arXiv:2503.03313, 2025. [102] D. Eremeev, G. Bazhenov, O. Platonov, A. Babenko, and L. Prokhorenkova, “Turning tabular foundation models into graph foundation models,” arXiv preprint arXiv:2508.20906, 2025. [103] H. Liu, J. Feng, L. Kong, N. Liang, D. Tao, Y. Chen, and M. Zhang, “One for all: Towards training one graph model for all classification tasks,” in ICLR, 2024. [104] R. Chen, T. Zhao, A. Jaiswal, N. Shah, and Z. Wang, “LLaGA: Large language and graph assistant,” arXiv preprint arXiv:2402.08170, 2024. [105] Y. Zhu, H. Shi, X. Wang, Y. Liu, Y. Wang, B. Peng, C. Hong, and S. Tang, “GraphCLIP: Enhancing transferability in graph foundation models for text-attributed graphs,” in WWW, 2025, pp. 2183–2197. [106] Y. Cheng, J. Zhu, Y. Zhao, J. Yu, J. Tan, and X. Li, “Boosting crossdomain and cross-task generalization for text-attributed graphs from structural perspective,” Frontiers of Computer Science, 2025. [107] Y. He, Y. Sui, X. He, Y. Liu, Y. Sun, and B. Hooi, “UniGraph2: Learning a unified embedding space to bind multimodal graphs,” in WWW, 2025, pp. 1759–1770.
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[108] L. Kong, J. Feng, H. Liu, C. Huang, J. Huang, Y. Chen, and M. Zhang, “GOFA: A generative one-for-all model for joint graph language modeling,” in ICLR, 2025. [109] Y. Peng, C. Yang, Z. Fan, and C. Shi, “Data-centric prompt tuning for dynamic graphs,” in CIKM, 2025, pp. 2336–2345. [110] G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Review, vol. 35, no. 4, pp. 551–566, 1993. [111] F. Rosenblatt, “The perceptron: a probabilistic model for information storage and organization in the brain.” Psychological review, vol. 65, no. 6, p. 386, 1958. [112] N. Tishby, F. C. Pereira, and W. Bialek, “The information bottleneck method,” arXiv preprint physics/0004057, 2000. [113] N. Tishby and N. Zaslavsky, “Deep learning and the information bottleneck principle,” in IEEE Information Theory Workshop (ITW). IEEE, 2015, pp. 1–5. [114] A. v. d. Oord, Y. Li, and O. Vinyals, “Representation learning with contrastive predictive coding,” arXiv preprint arXiv:1807.03748, 2018. [115] T. M. Cover, Elements of information theory. John Wiley & Sons, 1999. [116] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” NeurIPS, vol. 30, 2017. [117] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948. [118] S. Kumar, X. Zhang, and J. Leskovec, “Predicting dynamic embedding trajectory in temporal interaction networks,” in KDD, 2019, pp. 1269– 1278. [119] J. You, T. Du, and J. Leskovec, “ROLAND: Graph learning framework for dynamic graphs,” in KDD, 2022, pp. 2358–2366. [120] S. Tian, R. Wu, L. Shi, L. Zhu, and T. Xiong, “Self-supervised representation learning on dynamic graphs,” in CIKM, 2021, pp. 1814– 1823. [121] Y. Bei, H. Xu, S. Zhou, H. Chi, H. Wang, M. Zhang, Z. Li, and J. Bu, “CPDG: A contrastive pre-training method for dynamic graph neural networks,” in ICDE. IEEE, 2024, pp. 1199–1212. [122] L. McInnes, J. Healy, N. Saul, and L. Großberger, “UMAP: Uniform manifold approximation and projection,” Journal of Open Source Software, vol. 3, no. 29, 2018.
Haonan Yuan is currently a Ph.D. candidate at the Beijing Advanced Innovation Center for Big Data and Brain Computing at Beihang University. His research interests include graph foundation models, dynamic graph learning, and OOD generalization. He has published several papers on IEEE TPAMI, ICML, NeurIPS, ICLR, WWW, AAAI, etc.
Qingyun Sun is currently an Associate Professor at the School of Computer Science and Engineering at Beihang University. Her research interests include graph machine learning and data mining. She has published several papers on IEEE TPAMI, IEEE TKDE, WWW, AAAI, ICDM, CIKM, etc.
Junhua Shi is currently a Ph.D. candidate at the Beijing Advanced Innovation Center for Big Data and Brain Computing at Beihang University. His research interests include trustworthy graph foundation models.
Xingcheng Fu is currently an Associate Professor at the Key Lab of Education Blockchain and Intelligent Technology at Guangxi Normal University. His research interests include graph representation learning, complex networks, and social network analysis. He has published several papers on IEEE TKDE, WWW, AAAI, ICDM, CIKM, etc.
Jianxin Li is currently a Professor with the School of Computer Science and Engineering, and Beijing Advanced Innovation Center for Big Data and Brain Computing in Beihang University. His current research interests include social networks, machine learning, big data, and trustworthy computing. Dr. Li has published research papers in top-tier journals and conferences, including the IEEE TKDE, TDSC, JAIR, ACM TOIS, TKDD, KDD, AAAI, WWW, etc.
Philip S. Yu is a Distinguished Professor and the Wexler Chair in Information Technology at the Department of Computer Science, University of Illinois at Chicago. Before joining UIC, he was at the IBM Watson Research Center, where he built a worldrenowned data mining and database department. He is a Fellow of the ACM and IEEE. Dr. Yu was the Editor-in-Chief of ACM TKDD (2011-2017) and IEEE TKDE (2001-2004).