Continuous-Time Dynamic Graphs

• Continuous-Time Dynamic Graphs (CTDGs) are a mathematical abstraction that represents dynamic networks via precise, timestamped events, capturing both topological and temporal nuances.
• They underpin advanced modeling techniques—such as temporal graph neural networks, event-based recurrent architectures, and ODE-based methods—for tasks like link prediction, anomaly detection, and generative modeling.
• CTDGs offer a clear advantage over discrete methods by recording individual events in real time, though challenges remain in long-range temporal credit assignment, efficiency, and interpretability.

Continuous-Time Dynamic Graphs (CTDGs) are a foundational mathematical abstraction for representing systems in which entities and their relationships evolve at arbitrary, real-valued timestamps. CTDGs underpin state-of-the-art modeling approaches in domains with temporally evolving interactions, such as social, financial, biological, and communication networks. Unlike discrete-time dynamic graphs (DTDGs), which aggregate changes into regular snapshots, CTDGs encode each individual event—node/edge addition, deletion, or attribute update—with precise timing, enabling fine-grained analysis of both topological and temporal dynamics. Across the literature, CTDGs support a diversity of learning paradigms, from probabilistic event modeling to temporal graph neural networks (TGNNs), allowing for advanced tasks like link prediction, anomaly detection, causality inference, generative modeling, and more.

1. Mathematical Foundations and Formal Definitions

CTDGs are rigorously defined in several complementary notational systems throughout recent literature:

• Edge event stream representation: A CTDG is a tuple

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}, \mathcal{F})$$

where $\mathcal{V}$ is the set of nodes, $\mathcal{E} = \{(u,v,t,F_t)\}$ is a sequence of timestamped edge events with attributes, $\mathcal{T}$ is the continuous time domain, and $\mathcal{F}$ represents node/edge feature functions (Eddin et al., 2024, Eddin et al., 2023, Xu et al., 23 Feb 2025).

• Event-based view: Each event is specified as $(u_i, v_i, t_i, \mathbf{m}_i)$ and the graph at time $t$ is updated after processing all events up to $t$ (Poštuvan et al., 2024).
• Adjacency indicator: A time-dependent adjacency matrix $A_{uv}(t)$ can be defined, but in pure CTDG models this is often implicit since edges are instantaneous and never disappear (Bravo et al., 2024).
• Underlying structure: The node set may be fixed or dynamic; events describe structural changes (edge/node add/del) as well as feature updates (Zheng et al., 2023, Zheng et al., 2024).

These definitions enable CTDGs to capture both fine-grained temporal evolution and arbitrarily complex feature or topological changes.

2. Core Principles of Temporal Modeling

CTDGs are distinguished by several key principles:

• Irregular continuous-time event streams: Interactions arrive at arbitrary timestamps. This motivates models that process each event as it occurs—without discretized time steps (Zheng et al., 2024, Bravo et al., 2024).
• Temporal dependencies: Node and edge representations must encode not only current topology but also historical context, with temporal non-stationarity and varying interaction rates (Eddin et al., 2024). Techniques such as time-encoding (e.g., sinusoidal features, time2vec) explicitly represent elapsed time since past events.
• Long-term memory: Many CTDG applications (e.g., fraud detection, recommender systems) require models to "remember" long sequences of past interactions. This has led to the development of architectures capable of efficient long-sequence encoding, e.g., through state space models (Ding et al., 2024), recurrent cells, or advanced message-passing frameworks (Ennadir et al., 2024).

CTDG models must balance the need for expressivity—capturing temporally and structurally rich phenomena—with efficiency, scalability, and robust handling of event timing.

3. Representation Learning and Model Architectures

A diverse array of model families exists for CTDGs:

• Graph Recurrent Neural Networks (GRNNs): Maintain a hidden state for each node, updated upon each event. Training typically relies on backpropagation-through-time (BPTT), with full BPTT leveraging long-range dependencies but imposing significant resource requirements. Truncated BPTT can introduce a significant "truncation gap," restricting gradient flow to immediate neighbors (Bravo et al., 2024).
• Memory-based and attention-based GNNs: Methods such as TGN, TGAT, and DyRep employ various structures for combining memory, temporal attention, and message aggregation to encode the evolving state of the graph (Zheng et al., 2024, Guo et al., 2022, Ennadir et al., 2024).
• Temporal random-walk and histogram-based models: Graph-Sprints and its deep variant DGS approximate multi-hop temporal neighborhood aggregation with streaming histogram updates, enabling low-latency inference and real-time application (Eddin et al., 2023, Eddin et al., 2024).
• Diffusion and ODE-based models: Approaches such as CTAN (Gravina et al., 2024) and CTGN (Guo et al., 2022) model node representation evolution as continuous-time ODEs, allowing efficient propagation of long-range spatio-temporal information and well-defined expressivity properties.
• Causal and interpretable GNNs: Recent architectures (e.g., SIG (Fang et al., 2024)) incorporate explicit causal reasoning on CTDGs, extracting subgraphs that explain predictions and providing guarantees on OOD robustness and interpretability.

The landscape also includes complex multi-perspective attention models (Zhu et al., 2023), latent diffusion-based augmentation frameworks (Tian et al., 2024), and variants employing spectral or Fourier domain analysis to capture global patterns (Xu et al., 23 Feb 2025).

4. Applications: Learning, Generation, Explanation, and Robustness

CTDG frameworks underpin a range of advanced tasks:

• Link prediction: Estimation of future interactions through temporal encoding and negative sampling techniques. Evaluation uses metrics such as MRR, AUC, AP, and Recall@k, with variants for transductive and inductive settings (Eddin et al., 2024, Gravina et al., 2024).
• Graph generation: Probabilistic autoregressive event models (e.g., DG-Gen (Hosseini et al., 2024)) enable scalable, assumption-free synthesis of new CTDGs for data augmentation and benchmarking.
• Anomaly detection: CTDG learning algorithms are adapted to detect temporally, structurally, or contextually anomalous events, with domain-specific synthetic generation methods to rigorously benchmark capabilities (Poštuvan et al., 2024).
• Causal interpretability: Models explicitly extract compact causal subgraphs as explanations, supporting downstream transparency and policy analysis (Fang et al., 2024).
• Adversarial robustness: CTDGs are susceptible to stealthy poisoning attacks that selectively perturb edge arrival times and endpoints; dedicated defense methodologies filter adversarial edges and regularize temporal smoothness (Lee et al., 2023).

Each application leverages CTDGs' fine temporal granularity and dynamic structure for domain-specific objectives.

5. Expressivity, Efficiency, and Theoretical Guarantees

Recent work provides rigorous analysis of CTDG model expressivity:

• Information-flow frameworks: Theoretical analysis quantifies the ability of models to propagate structural and temporal information over arbitrary distances, bounding node-level changes and characterizing message-passing limits (Ennadir et al., 2024).
• Universal approximation and temporal coherence: Frequency-domain approaches (e.g., FGAT in UniDyG (Xu et al., 23 Feb 2025)) guarantee that the model can approximate arbitrary continuous functions on dynamic graphs and that small time shifts produce bounded changes in node embeddings.
• Long-range propagation: ODE-based models with anti-symmetric vector fields (e.g., CTAN (Gravina et al., 2024)) prove stable, non-dissipative information transmission; increasing layers or integration time expands propagation radius.

Efficiency is achieved via streaming updates (Eddin et al., 2023), mixed-mode automatic differentiation (Eddin et al., 2024), push-residual schemes (Zheng et al., 2023), and patch-based state-space architectures (Ding et al., 2024), enabling scalability to billion-edge graphs.

6. Comparative Analysis: CTDGs vs. DTDGs and Related Graph Models

CTDGs exhibit distinct advantages and constraints compared to DTDGs, CTBNs, and DBNs:

• Temporal granularity: CTDGs record each event individually, capturing rapid changes and local dynamics that DTDGs (snapshots) cannot (Xu et al., 23 Feb 2025).
• Modeling asymmetry and non-exponential times: CTDGs allow holding times and transitions that are not forced to be exponential, which is restrictive in CTBNs; context-specific independence and position symmetries compress asymmetric state spaces more efficiently than DBNs (Shenvi et al., 2020).
• Inference complexity: Linear in present slice size for CTDGs, exponential in number of variables for CTBNs, O(exp(max-clique)) in DBNs.
• Model unification: Decoupled frameworks (e.g., DecoupledDGNN (Zheng et al., 2023), UniDyG (Xu et al., 23 Feb 2025)) unify continuous- and discrete-time dynamic graphs via general propagators and frequency-based message aggregation.
• Robustness and scalability: CTDG-specialized models handle temporal noise and adversarial perturbations more effectively with energy-gated units and causal filtering (Xu et al., 23 Feb 2025, Lee et al., 2023).

These properties inform the choice of modeling paradigm for specific application settings.

7. Open Challenges and Future Directions

Current limitations and ongoing research directions in CTDGs include:

• Long-range temporal credit assignment: Bridging the truncation gap in GRNNs through unbiased online gradient estimators (RTRL variants) and memory-efficient recurrent architectures (Bravo et al., 2024).
• Low-latency real-time learning: Further reduction of batch and per-event inference latency for high-frequency CTDG streams (Eddin et al., 2024).
• Handling temporal noise: Adaptive gating and robust frequency-domain techniques for filtering spurious high-frequency events (Xu et al., 23 Feb 2025).
• Self-supervised and contrastive learning: Expanding SSRL tools and pre-text objectives for CTDG pretraining, leveraging auto-regressive event order and perturbation schemes (Ennadir et al., 2024).
• Modeling heterogeneous and attributed graphs: Agile architectures for graphs with varying types, modalities, and attribute dynamics (Zheng et al., 2024).
• Interpretability, causality, and explainability: Efficient extraction of causal subgraphs and quantification of OOD robustness in temporal prediction (Fang et al., 2024).

As CTDGs continue to rise in prominence, advanced modeling, scalable learning, and principled theory remain active frontiers.