Spatio-Temporal Functional Interaction Graph

• Spatio-temporal functional interaction graphs are representations that jointly model entities, their interactions, and temporal evolution.
• They incorporate diverse constructions—such as time-unfolded graphs, graph sequences, latent adjacency, and product graphs—to capture dynamic dependencies.
• These graphs have been successfully applied in neuroimaging, traffic forecasting, and requirements engineering, enhancing prediction accuracy and interpretability.

A spatio-temporal functional interaction graph is a graph-based representation in which entities, their interactions, and their temporal evolution are modeled jointly rather than reduced to a static relation matrix or to independent per-node time series. Across the literature, the term does not denote a single canonical construction. In resting-state fMRI it may refer to a learned task-driven graph over brain regions (El-Gazzar et al., 2021), a time-unfolded graph with ROI-time nodes (Gadgil et al., 2020), or a sequence of dynamic functional-connectivity graphs (Kim et al., 2021). In graph signal processing it can be a graph over channel-time samples with explicit lagged edges (Ghoroghchian et al., 2020); in traffic forecasting it can be a persistent learned node-interaction matrix (Alisetti et al., 1 Jun 2025) or a synchronous graph over sensor-time states (Jin et al., 2022); in urban forecasting it can be a semantic-spatial neighborhood graph for intervention analysis (Zhang et al., 3 Jun 2026). This diversity indicates that the central idea is not one fixed formalism, but a family of graph constructions for representing functional dependence across space and time.

1. Conceptual scope and semantics

The defining components of a spatio-temporal functional interaction graph are the nodes, the interaction relation, and the temporal mechanism. What varies across domains is the exact meaning assigned to each. In neuroimaging, nodes are often ROIs or ICA-derived functional units, and interaction usually denotes statistical or task-driven dependence among regional signals (El-Gazzar et al., 2021). In iEEG, the graph may instead treat each channel-time pair as a separate vertex, so time is embedded directly into the vertex set rather than appended as a feature dimension (Ghoroghchian et al., 2020). In traffic forecasting, nodes are roads or sensors and the interaction relation is explicitly interpreted as predictive dependence rather than physical adjacency (Alisetti et al., 1 Jun 2025). In requirements engineering, the graph can be centered on functional features linked by cause-effect relations, with spatio-temporal information attached as feature attributes rather than encoded directly as dynamic edges (Spichkova, 28 Dec 2025).

The term functional is correspondingly heterogeneous. In the fMRI literature it can mean a relation derived from correlation or from end-to-end supervised optimization (Gadgil et al., 2020). In hierarchical graph signal processing for iEEG, it denotes empirical co-fluctuation strength based on absolute products of centered signals rather than correlation or causal influence (Ghoroghchian et al., 2020). In InterGAT it denotes predictive utility: if node $i$ assigns high weight to node $j$, then $j$'s latent signal is useful for forecasting $i$ (Alisetti et al., 1 Jun 2025). In TopFunST, functional interaction is closer to cause-effect dependency among actions or features of a system, encoded by directed edges and enriched with duration, timing, and location constraints (Spichkova, 28 Dec 2025).

A compact formal statement of the engineering-oriented variant appears in TopFunST, where a functional feature is represented as

$$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$

and the overall model is a directed graph of such features linked by Cause Effect Relations (Spichkova, 28 Dec 2025). This contrasts sharply with graph-learning approaches in which the interaction graph is not manually specified but optimized as a latent object inside a predictive model.

2. Major formal constructions

Several recurrent formalisms organize the literature. One class uses a time-unfolded graph, where each node is an entity-time pair. Another uses a graph sequence, where each time index has its own graph snapshot. A third uses a static node set with learned latent adjacency, letting time enter through node features or recurrent/temporal modules. A fourth, more formal systems-oriented class keeps a directed dependency graph and attaches time and space to node semantics.

Construction	Representative formulation	Typical use
Time-unfolded graph	$\mathcal{V}=\{v_{t,i}\}$ or $|\mathcal V|=ST$	rs-fMRI, iEEG
Graph sequence	$G_{\text{dyn}}=(G(1),\ldots,G(T))$	dynamic brain connectomics
Latent adjacency over fixed nodes	$\mathbf{I}\in\mathbb{R}^{N\times N}$ or $A'_k\in\mathbb{R}^{N\times N}$	traffic, fMRI classification
Product graph	$j$0	graph-time CNNs
State-instance graph	$j$1 sensor-time nodes in a synchronous graph	traffic forecasting
Causal feature graph	$j$2 over functional features	requirements engineering

In time-unfolded constructions, space and time are native graph dimensions. The ST-GCN formulation for rs-fMRI defines nodes $j$3 for ROI $j$4 at time $j$5, spatial edges weighted by a static functional-affinity matrix, and temporal edges linking the same ROI across adjacent time points (Gadgil et al., 2020). The hierarchical GSP approach goes further by flattening an $j$6 signal into a graph with $j$7 vertices and explicit lagged cross-time blocks $j$8 in the adjacency matrix (Ghoroghchian et al., 2020). This makes cross-time dependence part of the graph itself rather than something handled only by a temporal neural layer.

Graph-sequence constructions treat each time window as a separate graph. STAGIN forms sliding-window correlation matrices $j$9, thresholds them into binary adjacencies $j$0, and encodes the resulting sequence $j$1 with a GIN followed by a Transformer (Kim et al., 2021). This is the clearest formulation in the supplied literature where the functional graph itself is explicitly time-indexed.

Latent-adjacency constructions instead keep the node set fixed and learn the interaction operator. DAST-GCN uses layer-wise learned directed dense graphs over ROIs, with

$j$2

so each spatio-temporal block can realize a different learned functional graph (El-Gazzar et al., 2021). InterGAT replaces masked attention with a persistent symmetric interaction matrix $j$3, symmetrized, normalized, and sparsity-regularized, then propagates features by

$j$4

before temporal decoding with a GRU (Alisetti et al., 1 Jun 2025).

Product-graph constructions, exemplified by GTCNN, represent the entire spatio-temporal signal on a single product graph. The learned coupling operator

$j$5

explicitly separates self-loops, same-time spatial coupling, same-node temporal coupling, and cross-time spatial coupling (Isufi et al., 2021). This suggests a structured alternative to fully free dynamic adjacency estimation.

3. Learning paradigms and inference strategies

The literature spans at least five distinct inference paradigms. The simplest is formula-based graph construction. In the hierarchical GSP method, spatial weights are computed from absolute products of centered same-time channel magnitudes, and temporal weights from absolute products across lags; there is no separate graph-learning objective, sparse precision estimation, or diffusion fitting (Ghoroghchian et al., 2020). This makes the graph sample-specific, band-specific, and window-specific, but not learned by optimization.

A second paradigm is end-to-end supervised graph learning. DAST-GCN learns source-target dictionaries $j$6 jointly with temporal convolutions and the classifier, so the adjacency is adaptive, task-driven, and layer-wise dynamic (El-Gazzar et al., 2021). InterGAT similarly learns $j$7 directly from forecasting loss plus an $j$8 sparsity penalty, rather than deriving it from node-feature similarity at every forward pass (Alisetti et al., 1 Jun 2025). In these models the functional interaction graph is a trainable latent operator optimized for downstream prediction.

A third paradigm is structured graph search. Auto-DSTSGN does not learn arbitrary dense edges entry-by-entry; instead it searches over structured adjacency templates composed from spatial graph (SG), temporal graph (TG), and temporal self-connectivity (TC) primitives, with differentiable architecture parameters and later argmax discretization (Jin et al., 2022). This yields a block/time-position-specific graph family that remains constrained and interpretable.

A fourth paradigm is probabilistic kernel construction from dynamics. The SPDE-based GP framework derives non-separable graph kernels from the stochastic heat and wave equations, replacing continuous spatial operators with the graph Laplacian or a fractional graph Laplacian (Nikitin et al., 2021). In this setting the functional interaction graph is not a learned adjacency matrix inside a neural network but the operator governing diffusion or oscillation over node-time functions.

A fifth paradigm is causal graph pairing for factual/counterfactual modeling. CausalPOI builds a weekly local ST-FIG for a new POI, with treatment-edge weights

$j$9

and a control graph that preserves topology while removing the semantic interaction term (Zhang et al., 3 Jun 2026). This design explicitly uses the graph as a substrate for comparing treatment and control worlds in a cold-start forecasting problem.

4. Domain-specific instantiations

In neuroimaging, the concept is most often introduced as an alternative to collapsing a scan into a single static correlation matrix. ST-GCN for rs-fMRI uses a static functional-affinity matrix but processes short subsequences on a time-unfolded graph to model non-stationary BOLD dynamics (Gadgil et al., 2020). STAGIN shifts to a sequence of dynamic functional-connectivity graphs estimated by sliding-window correlation and uses spatio-temporal attention for explainability (Kim et al., 2021). DAST-GCN replaces precomputed correlation graphs with layer-wise learned directed dense graphs shared across subjects for a task, thereby making the functional graph adaptive and phenotype-driven (El-Gazzar et al., 2021). Brain-MS-G3D takes yet another route: it retains a static FC scaffold but builds short spatio-temporal windows whose stacked adjacency connects functionally related nodes across multiple nearby frames (Dahan et al., 2021).

In electrophysiology and graph signal processing, the graph can be even more explicitly space-time native. The hierarchical GSP approach for iEEG constructs a block adjacency whose $i$0 blocks capture same-time spatial relations and whose $i$1 blocks encode lagged cross-time interactions up to hop $i$2, then extracts topology and graph-spectral features such as graph Fourier band energies (Ghoroghchian et al., 2020). This is a strong example of a spatio-temporal functional interaction graph used primarily as a feature-extraction object rather than a deep latent variable.

In traffic forecasting, three distinct graph notions appear. Auto-DSTSGN uses a synchronous graph over sensor-time tuples, with SG, TG, and TC edges capturing physical adjacency, DTW-based temporal-pattern similarity, and self-links across time (Jin et al., 2022). InterGAT uses a single learned interaction matrix per attention head, interpreted as a persistent latent topology of predictive couplings among roads or sensors (Alisetti et al., 1 Jun 2025). GTCNN uses a product graph over space-time nodes with learnable coupling coefficients $i$3, emphasizing structured interaction types rather than arbitrary edge learning (Isufi et al., 2021).

In human-object interaction, STIGPN builds a dense graph over per-frame human and object instances, then parses intra-frame and inter-frame relation graphs with attention to capture salient HOI structure (Wang et al., 2021). PGCN builds graphs whose nodes are body joints and object centers, initializes skeleton edges but not human-object/object-object edges, and then learns interaction structure with spatial attention while a temporal pyramid decoder reconstructs framewise sub-action labels (Xing et al., 2024). These models are close to a functional interaction graph in the sense that high-affinity edges correspond to task-relevant relations rather than merely geometric adjacency.

In formal systems and requirements engineering, TopFunST represents a more symbolic instantiation. The graph remains a directed cause-effect graph, but each functional feature carries duration $i$4, timing constraints $i$5, and location $i$6, so spatio-temporal semantics are attached to functional units rather than learned from data (Spichkova, 28 Dec 2025). This shows that the term can denote not only neural or statistical graphs, but also engineering models of temporally and spatially constrained functional dependencies.

5. Empirical performance and interpretability

The predictive value of these constructions is domain-specific but consistently presented as evidence that purely static or purely proximity-based relations are insufficient. On UK Biobank rs-fMRI, DAST-GCN reported $i$7 accuracy for sex classification and $i$8 for age classification, outperforming a 1D-CNN, an FCN on correlations, and a static ST-GCN; in its transfer experiment, the pretrained graph variant reached $i$9 on REST-meta-MDD sex classification (El-Gazzar et al., 2021). On HCP, Brain-MS-G3D reported $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$0 for sex classification and $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$1 correlation for fluid-intelligence prediction, compared with $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$2 for a baseline that encoded space and time separately (Dahan et al., 2021). STAGIN-SERO reported $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$3 accuracy and $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$4 AUROC on HCP-Rest gender classification, and $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$5 on HCP-Task decoding (Kim et al., 2021).

In the iEEG setting, the hierarchical GSP pipeline showed a slight overall improvement over the Kaggle winner’s code, up to $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$6 AUC improvement on some subjects, and about $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$7 fewer features on average (Ghoroghchian et al., 2020). In traffic forecasting, InterGAT-GRU was reported to achieve at least a $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$8 improvement on SZ-Taxi and a $$F = \langle Id, A, R, Obj, Pre, Post, Prov, Exec, D, T, Loc\rangle$$9 improvement on Los-Loop across all forecasting horizons, with a $\mathcal{V}=\{v_{t,i}\}$0 reduction in training time relative to GAT-GRU (Alisetti et al., 1 Jun 2025). Auto-DSTSGN reported about $\mathcal{V}=\{v_{t,i}\}$1 improvements compared with the state-of-the-art methods on four real-world datasets (Jin et al., 2022). In urban cold-start forecasting, removing the ST-FIG module in CausalPOI degraded RMSE and MAE in all four US regions reported in the ablation study, which the paper interprets as evidence that explicit semantic interaction modeling matters (Zhang et al., 3 Jun 2026).

Interpretability claims vary by model. ST-GCN for rs-fMRI introduces a positive symmetric edge-importance matrix $\mathcal{V}=\{v_{t,i}\}$2 shared across layers, so diagonals and off-diagonals can be read as ROI and connection importance (Gadgil et al., 2020). STAGIN adds node-wise spatial attention and Transformer temporal attention, and its authors report attention patterns that align with neuroscientific expectations, such as DMN hyperconnectivity and SMN hypoconnectivity in female-attended resting-state clusters (Kim et al., 2021). InterGAT explicitly analyzes the learned $\mathcal{V}=\{v_{t,i}\}$3 matrix with sparsity, spectral, localization, and community-contrast measures, and reports that sparsity stabilizes between roughly $\mathcal{V}=\{v_{t,i}\}$4 and $\mathcal{V}=\{v_{t,i}\}$5 across heads (Alisetti et al., 1 Jun 2025). DAST-GCN presents a more cautious interpretation: the learned adjacency could in principle reveal biomarkers, but the graphs are dense, dynamic across layers, and sensitive to training stochasticity, so the paper stops short of a detailed biomarker analysis (El-Gazzar et al., 2021).

Residual-based evaluation provides yet another interpretive layer. The residual correlation analysis framework constructs a multiplex graph over sensor-time pairs and tests whether residuals remain spatially or temporally correlated; large local scores $\mathcal{V}=\{v_{t,i}\}$6 or smoothed $\mathcal{V}=\{v_{t,i}\}$7 indicate uncaptured interaction structure rather than just large error magnitude (Zambon et al., 2023). This suggests that a spatio-temporal functional interaction graph can also serve as a diagnostic object after prediction, not only as a predictive representation.

6. Misconceptions, limitations, and open problems

A common misconception is that dynamic always means an explicitly time-varying adjacency sequence. The literature is more fragmented. DAST-GCN is dynamic only in the sense of layer-wise graph structure learning; its adjacency is not an explicit per-time-window $\mathcal{V}=\{v_{t,i}\}$8 or a per-subject graph generated on the fly (El-Gazzar et al., 2021). ST-GCN for rs-fMRI uses dynamic node signals with a static group-level affinity matrix (Gadgil et al., 2020). InterGAT learns a persistent matrix $\mathcal{V}=\{v_{t,i}\}$9 that is static across time and samples (Alisetti et al., 1 Jun 2025). Only some models, such as STAGIN, explicitly operate on a sequence of graph snapshots (Kim et al., 2021).

Another misconception is that functional automatically implies causal or physiologically direct interaction. The iEEG GSP approach is explicit that its weights are based on absolute products of centered magnitudes, so anticorrelated and correlated relationships can both yield strong edges, and the graph should be interpreted as a co-fluctuation proxy rather than definitive physiological connectivity (Ghoroghchian et al., 2020). CausalPOI, despite its causal framing, states that its estimates should be read as localized counterfactual estimates rather than fully randomized intervention effects, because unobserved confounding remains possible (Zhang et al., 3 Jun 2026). TopFunST uses cause-effect terminology, but its short paper does not provide a full formal semantics for how $|\mathcal V|=ST$0, $|\mathcal V|=ST$1, and $|\mathcal V|=ST$2 alter edge validity (Spichkova, 28 Dec 2025).

Interpretability is similarly uneven. Attention maps, learned adjacencies, and graph spectral analyses provide salience indicators, but several papers explicitly stop short of stronger claims. STAGIN notes that attention weights are useful indicators of salience, not guaranteed causal explanations (Kim et al., 2021). DAST-GCN notes that enforcing a static sparse graph might improve robustness of visualization but likely at a cost in predictive performance (El-Gazzar et al., 2021). InterGAT notes that its learned $|\mathcal V|=ST$3 is dense and quadratic in $|\mathcal V|=ST$4, and suggests low-rank parameterizations such as LoRA-style factorizations as future work rather than implemented solutions (Alisetti et al., 1 Jun 2025).

Open problems recur across domains. The fMRI literature explicitly raises subject-specific graph learning, explicit time-varying graph sequences, sparsity or anatomical priors for interpretability, and regularization for more stable biomarkers (El-Gazzar et al., 2021). Product-graph and SPDE-based approaches suggest more structured ways to encode graph-time coupling, but they trade off flexibility against free-form learned adjacency (Isufi et al., 2021, Nikitin et al., 2021). Disentangled generative modeling adds another dimension by separating spatial, temporal, graph, and spatial-graph coupling factors, but the guarantees remain conditional on the assumed factorization and information-bottleneck thresholds (Du et al., 2022). Taken together, these works suggest that the central unresolved issue is not whether spatio-temporal functional interaction graphs are useful, but which graph semantics, temporal granularity, inductive biases, and explanatory constraints are appropriate for a given scientific or engineering domain.