Dynamic Space-Time Graph Models

• Dynamic Space-Time Graph Models are a family of formalisms that jointly capture spatial graph topology and temporal evolution through representations like snapshots and continuous event streams.
• They employ methodologies such as joint space-time filtering, time augmentation, and lifted node-time graphs to integrate spatial and temporal operators effectively.
• These models are applied in tasks like ride-demand forecasting and dynamic link prediction, demonstrating scalable, robust handling of evolving relational data.

A dynamic space-time graph model is a model of graph-structured data in which spatial structure and temporal evolution are represented jointly rather than as isolated components. In this literature, “space” may denote within-snapshot graph topology, explicit geometry, or latent metric structure, while “time” may appear as ordered snapshots, continuous event timestamps, or interval-valued edge states. Early domain-specific formulations represented ride demand as a sequence of Ride Request Graphs over 5-minute intervals and modeled diffusion through “same place different time” links, whereas later work introduced joint space-time operators, lifted node-time graphs, and graph-conditioned state-space systems as general-purpose formulations (Jauhri et al., 2017, Shahzamal et al., 2018, Hadou et al., 2021, Li et al., 2024).

1. Conceptual foundations

In contemporary graph learning, a dynamic space-time graph model is not a single architecture but a family of formalisms for evolving relational data. A discrete-time temporal graph may be written as an ordered sequence of snapshots, such as $\{G(\tau_l), X(\tau_l)\}_{1\le l\le L}$, while a continuous-time dynamic graph may be written as an event stream $\mathcal{G}=\{(u,v,t)\}$. Other formulations treat a graph signal as $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$, or define a spatiotemporal graph as $(S_{1:T},G_{1:T})$ with explicit geometry $C_t\in\mathbb{R}^{N\times 3}$ separated from graph structure (Li et al., 2024, Raghuvanshi et al., 3 Jun 2026, Hadou et al., 2021, Du et al., 2022).

A central distinction concerns how “space” and “time” are coupled. Some models define a single joint operator and perform space-time filtering directly. Some convert time into graph structure by creating node-time instances $v^t$. Some treat the evolving graph as a latent dynamical system and derive continuous-time or discrete-time recurrences. Others keep spatial and temporal operators separate but coupled through shared latent variables, learned dynamic adjacencies, or tensorized graph constructions. This suggests that the topic is best understood through its modeling primitives rather than through one canonical neural architecture (Hadou et al., 2021, Sun et al., 2022, Li et al., 2024).

The same term also spans different meanings of spatiality. In graph signal processing and temporal GNNs, spatiality usually means graph topology or graph shift operators. In generative spatiotemporal models it can mean explicit coordinates or geometric embeddings. In latent-distance models it can mean trajectories in a latent metric space whose pairwise distances control edge dynamics. Accordingly, dynamic space-time graph modeling includes snapshot GNNs, event-stream models, latent-state systems, kernel methods, and probabilistic survival-process models (Du et al., 2022, Celikkanat et al., 2023).

2. Formal data objects and representations

The literature uses several recurring mathematical objects for dynamic space-time graphs. Snapshot-based models typically assume a fixed node set with time-varying edges and features; lifted models replace each node by a time-indexed family of node instances; continuous-time models operate directly on event streams; and interval-valued models represent edges as states with durations rather than instantaneous contacts.

Representation	Formal object	Representative papers
Snapshot sequence	$\{G(\tau_l),X(\tau_l)\}_{l=1}^L$ or $\{G^1,\dots,G^T\}$	(Li et al., 2024, Sun et al., 2022)
Continuous-time event stream	$\mathcal{G}=\{(u,v,t)\}$	(Raghuvanshi et al., 3 Jun 2026, Ding et al., 2024)
Interval-valued dynamic graph	$(i,j,t_k,t_{k+1})$ with dyadic state $\mathcal{G}=\{(u,v,t)\}$0	(Celikkanat et al., 2023)
Joint graph-time signal	$\mathcal{G}=\{(u,v,t)\}$1	(Hadou et al., 2021)
Lifted node-time graph	node-time instances $\mathcal{G}=\{(u,v,t)\}$2 or an extended vertex set $\mathcal{G}=\{(u,v,t)\}$3	(Sun et al., 2022, Romero et al., 2016)

For discrete-time temporal graphs, a common formulation is $\mathcal{G}=\{(u,v,t)\}$4 with node features $\mathcal{G}=\{(u,v,t)\}$5, where the observed sequence is treated as a coarse sampling of a hidden continuous graph process. For continuous-time dynamic graphs, the observed object is an ordered stream of timestamped interactions, and the graph at time $\mathcal{G}=\{(u,v,t)\}$6 is often summarized by a subgraph adjacency $\mathcal{G}=\{(u,v,t)\}$7 and normalized Laplacian $\mathcal{G}=\{(u,v,t)\}$8. For interval graphs, each dyad alternates between link-present and link-absent states over nonoverlapping intervals, which leads naturally to duration-aware likelihoods rather than snapshot reconstruction or event-point-process losses (Li et al., 2024, Raghuvanshi et al., 3 Jun 2026, Celikkanat et al., 2023).

A separate line of work lifts the dynamic problem to a static one on a larger graph. In time augmentation, each original node $\mathcal{G}=\{(u,v,t)\}$9 is expanded into $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$0, with within-time spatial edges and across-time temporal edges. In kernel reconstruction, the extended graph uses vertex-time pairs $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$1, and the dynamic signal $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$2 is reinterpreted as a static signal on that larger vertex set. This lifting is one of the most direct constructions of a space-time graph in the literal sense (Sun et al., 2022, Romero et al., 2016).

3. Joint operators, lifted graphs, and dynamic graph constructions

A foundational operator-theoretic formulation appears in “Space-Time Graph Neural Networks,” which defines a generic diffusion-inspired convolution $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$3, recovers causal time convolution from $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$4, and defines the space-time shift operator as $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$5. The resulting space-time graph filter $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$6 acts jointly in graph-frequency and time-frequency, yields a causal architecture in which $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$7-hop information is delayed by $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$8 steps, and is shown to be stable to graph perturbations and time warping under integral Lipschitz filter assumptions (Hadou et al., 2021).

A structurally different route is time augmentation. In TADGNN, the dynamic graph is converted into a time-augmented spatio-temporal graph whose vertices are node-time instances $$\mathcal{X}\in L^2(\mathbb{R}^N)\otimes L^2(\mathbb{R})$$9. The paper gives three realizations: a full upper-triangular augmentation, a self-evolution realization connecting $(S_{1:T},G_{1:T})$0 only to $(S_{1:T},G_{1:T})$1, and a disentangled realization with separate spatial and temporal graphs. Message passing over this augmented graph yields $(S_{1:T},G_{1:T})$2-hop neighborhoods in space-time without recurrence or global temporal attention, and the adaptive transition matrix is learned only over local neighborhoods of the augmented graph rather than over all time pairs (Sun et al., 2022).

A third construction is diffusion on an implicit time-unfolded graph. TiaRa defines a time-aware random walk with restart in which a surfer at node $(S_{1:T},G_{1:T})$3 in snapshot $(S_{1:T},G_{1:T})$4 may walk spatially, restart, or move forward in time to the same node in $(S_{1:T},G_{1:T})$5. Its recursion

$(S_{1:T},G_{1:T})$6

produces one augmented adjacency-like matrix per snapshot. The closed form is a geometrically decayed sum of diffusion chains across past snapshots, so the method injects temporal locality into graph augmentation without constructing an explicit $(S_{1:T},G_{1:T})$7 supra-adjacency (Lee et al., 2022).

Other constructions learn dynamic graphs directly from data. SDGL learns a static graph $(S_{1:T},G_{1:T})$8 from node embeddings for long-term associations and a dynamic graph $(S_{1:T},G_{1:T})$9 from current node features plus static structural bias for short-term dependencies; the two graph convolutions are then fused by addition. DSTGNN instead defines a Spatial Tensor Graph $C_t\in\mathbb{R}^{N\times 3}$0 and a Temporal Tensor Graph $C_t\in\mathbb{R}^{N\times 3}$1, so spatial relations vary across time and temporal relations vary across nodes; this yields a factorized tensor-graph view rather than a single unified adjacency (Li et al., 2021, Jia et al., 2020).

4. State-space and memory-based graph dynamics

A major recent development is the recasting of dynamic space-time graph modeling as graph-conditioned state-space dynamics. “State Space Models on Temporal Graphs: A First-Principles Study” derives GraphSSM from a graph-regularized extension of HiPPO. For scalar node features, the online objective is

$C_t\in\mathbb{R}^{N\times 3}$2

so graph structure enters directly through a Laplacian regularizer at the memory-compression stage. Under the scaled Legendre measure, the latent coefficients satisfy

$C_t\in\mathbb{R}^{N\times 3}$3

The decisive feature is the graph-aware input operator $C_t\in\mathbb{R}^{N\times 3}$4, which makes the system piecewise linear time-invariant between graph mutations. In discrete time, the implemented GraphSSM separates graph diffusion or message passing from the state-space recurrence, attains $C_t\in\mathbb{R}^{N\times 3}$5 FLOP complexity per layer, and empirically finds GraphSSM-S4 stronger than GraphSSM-S5 and GraphSSM-S6 on the reported benchmarks (Li et al., 2024).

The continuous-time analogue is developed in CTDG-SSM. There, a continuous-time Topology-Aware higher order polynomial projection operator yields memory coefficients

$C_t\in\mathbb{R}^{N\times 3}$6

and the resulting continuous-time memory ODE is

$C_t\in\mathbb{R}^{N\times 3}$7

The additional structural drift term $C_t\in\mathbb{R}^{N\times 3}$8 is the continuous-time signature of topology change. After zero-order hold discretization, the model updates memory with one transition acting from the graph side and one from the temporal-memory side, and the paper reports state-of-the-art results on dynamic link prediction, dynamic node classification, and long-range sequence classification (Raghuvanshi et al., 3 Jun 2026).

DyGMamba adopts selective state space models for continuous-time dynamic graphs from a different angle. A node-level SSM encodes each endpoint’s one-hop historical interaction sequence, while a time-level SSM encodes the sequence of repeated interaction intervals for the queried node pair. The time-level output is then used to construct dynamic selection weights over the node-history sequence, so long node histories are modeled linearly in sequence length but are not pooled uniformly. The paper reports that DyGMamba achieves state-of-the-art in most dynamic link prediction settings and handles much longer temporal neighborhoods than DyGFormer under the reported GPU budget (Ding et al., 2024).

DG-Mamba applies selective state-space modeling to discrete snapshot graphs with learned intra-snapshot and inter-snapshot structure. Its kernelized dynamic message passing reduces quadratic attention-like costs to linear form, and the learned inter-snapshot adjacency is injected into the discretization step of the selective SSM. The result is a space-time graph model in which topology learning, long-range temporal propagation, and self-supervised structure regularization are coupled within one framework (Yuan et al., 2024).

5. Structure learning, latent-space, generative, and reconstruction formulations

Dynamic space-time graph models also appear in explicitly probabilistic and generative forms. GraS$C_t\in\mathbb{R}^{N\times 3}$9P models interval-valued dynamic graphs by assigning each dyad a state function $v^t$0 and a sequential survival process over alternating edge-on and edge-off durations. Its state-dependent hazard is

$v^t$1

where $v^t$2 is a continuous-time latent trajectory. This makes dynamic graph modeling duration-aware: the model learns from the persistence of edges and from the persistence of their absence, rather than from instantaneous events alone (Celikkanat et al., 2023).

A generative disentangling formulation appears in STGD-VAE. The model represents a spatiotemporal graph as $v^t$3 and factorizes latent variables into time-varying factors $v^t$4 and time-invariant factors $v^t$5, corresponding to spatial, graph, and spatial-graph interaction information. The decoder factors as $v^t$6 and $v^t$7, and the variational objective is coupled to information-thresholding procedures derived from information bottleneck arguments. This makes the model primarily a generative and interpretability-oriented dynamic space-time graph model rather than a forecasting-only architecture (Du et al., 2022).

Kernel methods provide another non-neural route. “Kernel-based Reconstruction of Space-time Functions on Dynamic Graphs” converts a time-evolving graph signal reconstruction problem into kernel ridge regression on an extended graph of vertex-time pairs. For static topology, the paper defines doubly-selective kernels on the joint space-time frequency plane; for time-varying topology, it constructs a block-structured inverse kernel

$v^t$8

which leads to a batch KRR estimator and an exact online kernel Kalman filter when the inverse kernel is block tridiagonal (Romero et al., 2016).

Additional application-specific models broaden the scope of the term. Recurrent Space-time Graph Neural Networks treat multi-scale spatial regions in video as nodes with recurrent temporal states and iterative spatial message passing; the graph topology is fixed and sparse, but node states evolve over time. Ride-request modeling uses a sequence of directed spatial snapshot graphs built from $v^t$9 cells and 5-minute temporal quantization, while SPDT diffusion models represent direct and indirect co-location through active copies and delayed exposure windows (Nicolicioiu et al., 2019, Jauhri et al., 2017, Shahzamal et al., 2018).

6. Tasks, empirical evidence, and computational properties

The reported downstream tasks are heterogeneous. Snapshot-based graph models have been evaluated on temporal node classification, future link prediction, and multivariate forecasting. Continuous-time models have focused mainly on dynamic link prediction, with additional dynamic node classification and sequence classification in CTDG-SSM. Other tasks include decentralized control, graph signal reconstruction, generation, network completion, video understanding, ride-demand characterization, and epidemic-spread simulation (Li et al., 2024, Raghuvanshi et al., 3 Jun 2026, Hadou et al., 2021, Romero et al., 2016, Nicolicioiu et al., 2019).

Empirical results consistently emphasize the value of explicit space-time coupling. GraphSSM is evaluated on DBLP-3, Brain, Reddit, DBLP-10, arXiv, and Tmall; it outperforms prior methods on the reported small-scale benchmarks and scales to arXiv and Tmall, where several baselines run out of memory. TADGNN achieves the best macro-AUC on Wiki, Reddit, and ML-Rating, while GAT remains best on ML-Genre. CTDG-SSM reports large gains on benchmarks that require long-range temporal and spatial reasoning, including the sequence classification setting where second-order topology-aware filtering reaches $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$0 accuracy at sequence length $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$1. DyGMamba reports the best average rank across negative-sampling settings in dynamic link prediction, and DG-Mamba reports the best or near-best AUC under both clean and attacked settings on COLLAB, Yelp, and ACT (Li et al., 2024, Sun et al., 2022, Raghuvanshi et al., 3 Jun 2026, Ding et al., 2024, Yuan et al., 2024).

Computationally, the field divides along clear lines. Transformer-style temporal attention is repeatedly identified as quadratic in sequence length, which motivates SSM-based alternatives. GraphSSM has recurrent complexity $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$2 per layer; DyGMamba reports linear complexity in sampled history length after patching; DG-Mamba simplifies to $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$3 under the paper’s assumptions; TADGNN has $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$4 sequential operations, unlike the $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$5 sequential dependence of EvolveGCN; TiaRa produces sparse augmented adjacency matrices with average time $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$6 per time step; and the kernel Kalman filter achieves $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$7 per time slot instead of repeatedly solving growing batch KRR problems (Li et al., 2024, Ding et al., 2024, Yuan et al., 2024, Sun et al., 2022, Lee et al., 2022, Romero et al., 2016).

Several works add stability or robustness claims. ST-GNN proves first-order stability of both the filter and the full network to graph perturbations and time warping under integral Lipschitz assumptions. TiaRa shows that temporal decay ratios and sparsification thresholds affect the balance between performance and efficiency but preserve small approximation errors for moderate $\{G(\tau_l),X(\tau_l)\}_{l=1}^L$8. DG-Mamba explicitly targets robustness through structure learning and the Principle of Relevant Information regularizer, and its reported attack results are substantially stronger than those of the compared baselines (Hadou et al., 2021, Lee et al., 2022, Yuan et al., 2024).

7. Limitations, misconceptions, and open problems

A common misconception is that every dynamic space-time graph model is a single supra-graph over node-time pairs. Some models do use explicit lifted graphs, as in time augmentation and extended-graph kernel methods, but others define joint operators, separate spatial and temporal graph constructions, or graph-conditioned latent dynamical systems. TiaRa is primarily an augmentation layer; SDGL keeps graph learning and temporal convolution as separate but fused components; and DSTGNN factorizes space and time into separate tensor graphs rather than one unified adjacency (Sun et al., 2022, Romero et al., 2016, Lee et al., 2022, Li et al., 2021, Jia et al., 2020).

Another recurring limitation is the gap between theory and dynamic topology. ST-GNN’s strongest formal stability theory assumes a fixed graph operator even though dynamic graphs are handled empirically. GraphSSM is explicitly tailored to discrete-time temporal graphs, and its approximation of hidden mutation processes relies on mixed discretization rather than recovery of the exact latent graph evolution. Kernel-based joint frequency constructions lose their clean 2D spectral interpretation when topology varies over time (Hadou et al., 2021, Li et al., 2024, Romero et al., 2016).

Continuous-time models resolve snapshot discretization but introduce their own assumptions. CTDG-SSM relies on Laplacian polynomial invertibility and piecewise interpolation between graph states; DyGMamba models only one-hop temporal neighborhoods and a bounded history of pairwise time intervals; and GraS$\{G(\tau_l),X(\tau_l)\}_{l=1}^L$9P assumes alternating dyadic states with piecewise linear latent trajectories and does not provide a specialized large-scale inference scheme. This suggests that continuous-time dynamic space-time graph modeling still trades expressive temporal semantics against approximation, sampling, and scalability constraints (Raghuvanshi et al., 3 Jun 2026, Ding et al., 2024, Celikkanat et al., 2023).

Interpretability and data assumptions remain uneven across the literature. TADGNN’s experiments use one-hot node ID features because the datasets are not naturally attributed. DG-Mamba notes that its learned dynamic structures are harder to interpret. STGD-VAE provides explicit latent disentanglement but is not primarily a forecasting architecture. Domain-specific models such as ride-request graphs and SPDT diffusion are highly interpretable but encode narrower notions of space-time interaction than general temporal graph learning frameworks (Sun et al., 2022, Yuan et al., 2024, Du et al., 2022, Jauhri et al., 2017, Shahzamal et al., 2018).

A plausible implication is that future work will continue to move along three converging directions: graph-conditioned continuous-time memory, explicit modeling of topology change between observations, and structure learning that remains computationally tractable while preserving causal and temporal semantics. The recent transition from heuristic spatial-temporal coupling toward first-principles operator design, state-space derivations, and duration-aware probabilistic modeling indicates that the dynamic space-time graph model has become a unifying research object rather than a domain-specific engineering pattern (Li et al., 2024, Raghuvanshi et al., 3 Jun 2026, Celikkanat et al., 2023).