--- title: Temporal Graph Modelling url: https://www.emergentmind.com/topics/temporal-graph-modelling-tgm type: topic --- # Temporal Graph Modelling Searching arXiv for recent and foundational papers on Temporal Graph Modelling to ground the article. Temporal Graph Modelling (TGM) denotes the study of how to represent, learn from, query, and generate graphs whose topology and/or attributes evolve over time. In the surveyed formulation, temporal graphs appear either as continuous-time event streams, \(G = \{(u_i,v_i,t_i,x_i)\}_{i=1}^M\), or as discrete snapshot sequences, \(G = \{(G_1,t_1),(G_2,t_2),\dots,(G_T,t_T)\}\), where each \(G_t=(V_t,E_t,X^v_t,X^e_t)\) aggregates events in a time window [2208.12126]. More generally, a finite, discrete Time-Varying Graph may be written as \(H=(V,T,E)\) with \(E\subseteq V\times T\times V\times T\), so that a dynamic edge \((u,t_a,v,t_b)\) links a vertex-time pair \((u,t_a)\) to \((v,t_b)\) [1402.3488]. Across these formalisms, TGM addresses non-stationarity, fine- versus coarse-grained dynamics, inductive generalization, scalability, and evaluation, while supplying the mathematical substrate for temporal reasoning, forecasting, anomaly detection, pattern matching, and generative modeling [2208.12126]. ## 1. Formal representations and problem settings A central distinction in TGM is between continuous-time interaction data and discrete-time graph sequences. In continuous-time settings, a temporal graph is modeled as timestamped interactions such as \((u,v,t,e)\), or, in event-based form, \(EB=\{(u_i,v_i,t_i,\Delta t_i)\}\), where both link start times and link durations are retained [2205.15924, 2306.07699]. In discrete-time settings, the graph is represented as a sequence of snapshots \(G_t=(V_t,E_t,X_t^V,X_t^E)\), with node features \(X_t^V\in\mathbb R^{|V_t|\times d_v}\) and edge features \(X_t^E\in\mathbb R^{|E_t|\times d_e}\) [2401.03134]. The most general representation in the provided material is the Time-Varying Graph model \(H=(V,T,E)\), with \(E\subseteq V\times T\times V\times T\) [1402.3488]. This formulation classifies dynamic edges into spatial edges, temporal edges, mixed edges, and spatial-temporal self-loops, depending on endpoint and timestamp coincidence. By defining the temporal-node set \(VT=V\times T\), each dynamic edge becomes an ordinary directed edge between temporal nodes, which yields an isomorphism to a static directed graph on \(V\times T\) [1402.3488]. The same source states that the model can represent snapshot sequences, interval-labeled edges, spatial-plus-temporal edge models, and temporal-plus-mixed edge models, and can intrinsically model cyclic behavior through regressive edges [1402.3488]. Problem formulations in TGM vary with the downstream task. Dynamic link prediction treats the input as a continuous-time temporal graph \(G=\{(s_i,d_i,t_i)\}_{i=1}^T\) and asks the model to score whether a link \((s,d)\) will occur at time \(t\) [2307.01026]. Dynamic node property prediction, described in TGB as node-affinity prediction, seeks an output vector \(y_t[u]\in\mathbb R^{|N|}\) over a target set \(N\) for each source node \(u\) and time \(t\) [2307.01026]. Temporal Knowledge Graph Forecasting uses timestamped quadruples \(G_t=\{(s_n,r_n,o_n,t)\}\) and predicts a missing object for a future-time query [2501.11911]. Spatio-temporal forecasting instead assumes a static graph \(G=(V,E)\) with time-indexed node signals \(x_{i,t}\) and masks \(m_{i,t}\) [2501.10214]. Pattern matching over temporal graphs introduces temporal basic graph patterns, where a static pattern is paired with a temporal constraint \(\Gamma\), possibly specified by a timed automaton [2205.14269]. This diversity of formulations suggests that TGM is not a single model family but a shared mathematical domain. A plausible implication is that unification efforts in TGM are driven as much by representation choice as by learning architecture. ## 2. Temporal structure: snapshots, events, continuity, and hierarchy A core methodological divide in TGM is between snapshot-based, event-stream, continuous-depth, and hierarchical formulations. The survey identifies streaming/event-stream methods such as JODIE, DyRep, HTNE, TGN, and TGAT, and snapshot/discrete methods such as DynGEM, EvolveGCN, and DySAT [2208.12126]. EvolveGCN evolves layer weights with an RNN, \(W_t^l=\mathrm{RNN}(W_{t-1}^l)\), while TGAT uses time encodings in neighborhood attention [2208.12126]. Continuous Temporal Graph Networks (CTGNs) replace discrete updates with a Neural-ODE in node-representation space, assuming that each node embedding \(h_i(t)\) evolves according to \[ \frac{d h_i(\tau)}{d\tau} = f(h_i(\tau),\tau,\mathcal N_i(t)), \] for \(\tau\in[0,\Delta t_i]\), where \(\mathcal N_i(t)\) denotes the neighborhood at the link-start time \(t\) [2205.15924]. In this formulation, link start times enter the GNN aggregator through a continuous time encoder, and link duration \(\Delta t_i\) becomes the upper limit of ODE integration [2205.15924]. The same source states that many dynamic graph networks can be viewed as specific discretizations of CTGNs [2205.15924]. TimeGraphs proposes a different structural response to temporal heterogeneity. Rather than processing all timesteps uniformly, it converts a sequence of scene graphs \(\{G_{t_1},G_{t_2},\dots,G_{t_n}\}\) into a hierarchical temporal knowledge graph \[ \mathcal G = \bigcup_{\ell=0}^L \bigl(V^\ell,\,E^\ell_{\rm intra}\,\cup\,E^\ell_{\rm inter}\bigr), \] where \(\ell=0\) is the base level and \(\ell=1,\dots,L\) are successively coarser levels of events [2401.03134]. At each level, supernodes are formed via an assignment matrix \(S^\ell\), supernode features are aggregated from constituent lower-level nodes, and the coarsened adjacency is \(A^\ell=S^\ell A^{\ell-1}(S^\ell)^\top\) [2401.03134]. The paper explicitly characterizes this as converting a non-uniformly evolving sequence of scene or interaction graphs into a single, adaptive, multi-scale graph [2401.03134]. Other works introduce alternative temporal structure assumptions. TREND uses a Hawkes-process view of link formation and models “exciting effects” through a conditional intensity \(\lambda_{i,j}(t)\) combining a base rate and decayed contributions from prior events [2203.14303]. TGNE embeds nodes as piece-wise linear trajectories of Gaussian distributions in latent space, so that continuous-time sparsity is handled through uncertainty estimates in the posterior variances \(\sigma_i^{(k)}\) [2405.17253]. The source-separation approach to computer networks assumes that each observed adjacency matrix decomposes as \[ A(t)=\sum_{\ell=1}^L w_{t,\ell}S_\ell, \] or, in factorized form, \(A(t)\approx\sum_{\ell=1}^L w_{t,\ell}U_\ell V_\ell^\top\), where only the mixing coefficients vary with time [2303.15950]. These formulations target different temporal phenomena: arbitrary event times and durations, self-excitation, non-uniform temporal salience, seasonal variation, or uncertainty under sparsity. This suggests that temporal structure in TGM is often chosen to mirror the dominant regularity in the application domain rather than to satisfy a single canonical formalism. ## 3. Learning architectures and inference mechanisms Neural architectures in TGM typically combine temporal encoding, graph aggregation, and task-specific decoding. A common continuous-time node update can be written as \[ \mathbf{h}_v(t)=\sigma\Bigl(W_1\mathbf{h}_v(t^-)+W_2\,\mathrm{AGG}\{\mathbf{h}_u(t^-)\mid u\in\mathcal N_v(t)\}+W_3\Delta t_v+W_4\mathbf{x}_{v,e,t}\Bigr), \] which is cited in the survey as a generic form for continuous-time methods [2208.12126]. TimeGraphs constructs its hierarchy through vertex infomax pooling (VIPool) within a Graph Cross Network backbone. At each scale \(\ell\), VIPool selects a subset \(\Omega^\ell\subset V^{\ell-1}\) maximizing a mutual-information-based criterion, and a greedy approximation with negative sampling makes this tractable in \(O(|V|+|E|)\) per level [2401.03134]. After hierarchy construction, a relational GCN updates node states across relation types and across hierarchy levels: \[ h_v^{(k+1),\ell}=\sigma\Bigl(\sum_{r\in\mathcal R}\sum_{u\in\mathcal N_r(v)}W_r^\ell h_u^{(k),\ell}+b_r^\ell\Bigr)+W_{\rm up}^\ell h_{\pi(v)}^{(k),\ell+1}+W_{\rm down}^\ell\sum_{c\in\mathcal C(v)}h_c^{(k),\ell-1}, \] with optional attention weights \(\alpha_{vu}^\ell\) [2401.03134]. TGN-style models instead maintain node memories. In the TGN formalism summarized in the TGNv2 paper, each event triggers message construction, message aggregation, and memory update: \[ \mathbf{s}_i(t)=\mathrm{mem}\bigl(\bar{\mathbf{m}}_i(t),\mathbf{s}_i(t^-)\bigr), \] followed by embedding computation from memories and neighbor features [2411.03596]. The TGNv2 contribution is to augment each message with source-target identification, using encoded node indices \(\phi_n(i),\phi_n(j)\), so that the model can represent persistent forecasting, moving averages, and linear autoregressive functions of past messages [2411.03596]. The paper states that no instantiation of TGN can represent the moving average of order \(k\), whereas TGNv2 can exactly represent persistent forecasting, moving averages of any order \(k\), and any linear autoregressive model of order \(k\), on temporal graphs with bounded \(|V|\) [2411.03596]. TGSL addresses incomplete and noisy graph structure by learning additional temporal edges. It first computes edge embeddings with an edge-centric time-aware GNN, then uses an RNN over a node’s recent interacted neighbors to produce a time-aware context vector \(\mathbf z_u\), samples a small candidate pool, warps embeddings to a sampled target time, scores candidate edges through inner products, and uses Gumbel-Top-K edge selection to add plausible edges [2306.07699]. The total loss combines task loss on the original graph, task loss on the augmented graph, and a contrastive regularizer [2306.07699]. TGPM is described as a “pattern-centric” framework. It constructs interaction patches using temporally-biased random walks rooted at a target node, converts each walk into a token embedding through a small Transformer, mean-pools tokens into a patch embedding, and feeds sequences of patch embeddings into a time-aware Transformer with a learned time bias matrix \(B_{\mathrm{time}}\) [2601.22454]. Pre-training uses masked token modeling and next-time prediction [2601.22454]. TG-ODE targets irregularly sampled and partially observed networked dynamical systems. Hidden states evolve between observations through a Graph Neural ODE, and at each observation time an imputed state, a reliability matrix \(U_{t_i}\), and a Graph-GRU update are combined to form the next hidden state [2412.00165]. T-GMM, by contrast, is a discrete-time architecture for spatio-temporal forecasting that combines node-level processing, patch-level subgraph encoding, and a three-dimensional MLP-Mixer that mixes across patch tokens, time tokens, and feature channels [2501.10214]. The architectural landscape therefore spans RNN-controlled GNNs, temporal attention, Neural-ODEs, Hawkes-process intensities, hierarchical pooling, Transformer encoders, contrastive graph structure learning, and MLP mixers. A plausible implication is that TGM research differentiates models less by the mere presence of temporal information than by where temporal inductive bias is inserted: in memory, in the message function, in the latent dynamics, in structure learning, or in the representation hierarchy. ## 4. Querying, prediction, and generation TGM supports predictive, inferential, and generative tasks. For predictive tasks, dynamic link prediction and node-affinity prediction are standard. TGB formalizes dynamic link prediction as ranking the true destination among negative candidates and evaluates it with filtered MRR [2307.01026]. Dynamic node property prediction is evaluated with NDCG@K [2307.01026]. The TGB paper reports that on node tasks, persistence is best on UNTrade and moving average is best on LastFM, Subreddit, and tgbn-token, while TGN is second best on LastFM [2307.01026]. The TGNv2 paper is explicitly motivated by this failure mode, stating that heuristic approaches such as persistent forecasts and moving averages over ground-truth labels significantly and consistently outperform TGNs on dynamic node affinity prediction [2411.03596]. For temporal reasoning over scene graphs, TimeGraphs reports event prediction and recognition on Football, Resistance, and MOMA [2401.03134]. For temporal knowledge graphs, TGL-LLM integrates a temporal graph learning module into an LLM-based forecasting pipeline, using RGCN plus GRU to generate historical graph embeddings, a hybrid graph tokenization to inject graph information into prompts, and a two-stage fine-tuning paradigm for graph-language alignment [2501.11911]. Pattern-querying is developed in the timed-automata framework for temporal graph patterns. A timed automaton \(A=(Q,\Sigma,C,\delta,q_0,F)\) is used to define temporal constraints over edge variables in a temporal basic graph pattern, and three evaluation schemes are provided: a baseline two-phase algorithm, an on-demand incremental algorithm, and a fully incremental partial-match algorithm [2205.14269]. The same source states that timed automata subsume standard existential temporal motifs while expressing non-existential alternation, contiguity, mandated response or timeout patterns, mutual exclusions, set-containment constraints, and arbitrary Boolean combinations of clock-gaps [2205.14269]. Generative modeling constitutes another major branch of TGM. TIGGER treats a temporal interaction graph as a corpus of temporal random walks and learns an auto-regressive factorization \[ p(S)=p(s_1)\prod_{i=2}^{\ell}p(s_i\mid s_{