Dynamic Random Graphs

Updated 18 December 2025

Dynamic random graphs are stochastic models where network connections evolve over time through probabilistic rules such as edge rewiring and vertex turnover.
These models capture key phenomena like phase transitions, mixing times, and epidemic spreading by integrating mechanisms from dynamic Erdős–Rényi, configuration, and latent position frameworks.
Advanced inference methods and efficient computational algorithms, including Bayesian updates and geometric data structures, enable robust simulation and estimation in large-scale dynamic networks.

Dynamic random graphs are stochastic models in which the underlying network topology evolves over time according to probabilistic rules. These frameworks encompass a variety of time-varying random graph processes, including models with edge rewiring, vertex addition or removal, latent low-rank structure, geometric constraints, or dynamic community groupings. Such models are vital for the analysis of communication networks, mobile systems, social interaction graphs, functional brain networks, and other systems where connections change on timescales comparable to the phenomena of interest.

1. Foundational Models and Generative Mechanisms

Dynamic random graphs generalize the classic static models by incorporating time-dependent mechanisms for edge or vertex updates. Key classes include:

Dynamic Erdős–Rényi Model: Each of the $n(n-1)/2$ possible edges alternates between active (“ON”) and inactive (“OFF”) phases independently, governed by specified waiting time distributions for each state. The aggregate edge probability at time $t$ is a function of the “on-times” and “off-times.” The model can be further embedded in a regime-switching background, driven by an unobserved Markov or renewal process that determines which of a set of potential dynamic graphs is active at each time, see (Mandjes et al., 20 Jan 2025).
Configuration Model with Edge Rewiring: The degree sequence is fixed, but the edge assignments are dynamically rewired by randomly choosing candidate edges—locally, mesoscopically, or globally—with probabilities that modulate the speed and scope of topology change. The link between static and dynamic versions is established via the coupling of random walks and stopping times for first rewiring (Avena et al., 2020).
Dynamic Random Intersection Graphs: Model individuals joined into groups (communities) that independently switch between “active” and “inactive” states. Edges between individuals exist dynamically if any shared community is active, yielding nontrivial degree distributions, dynamic local limits, and time-dependent structure such as the size of the largest community in heavy-tailed settings (Milewska et al., 2023).
Vertex Turnover Models: The dynamic random graph with vertex removal (DRGVR) introduces birth-death dynamics for the vertex set: with probability $p > 1/2$ a new vertex is added (forming random edges), and with probability $1-p$ a uniformly chosen vertex (and its edges) is deleted. This process results in a dynamic equilibrium with phase transitions for the emergence of a giant component and precise local weak limits in terms of multitype branching structures (Díaz et al., 2022).
Random Geometric and Hyperbolic Graph Extensions: Nodes occupy positions in geometric space and move continuously or discretely; their connections depend probabilistically on distance. Efficient algorithms exist for updating these graphs’ topology under dynamic node movement, using advanced data structures such as polar quadtrees to maintain geometric querying in sublinear time (Looz et al., 2018).
Dynamic Latent Position Models: For example, the dynamic random dot product graph (RDPG) models edge formation based on time-evolving latent node embeddings, with temporal regularity imposed via Gaussian random walks or integrated processes. Bayesian frameworks enable inference and forecasting in such settings, providing both theoretical guarantees and scalable algorithms (Loyal, 24 Sep 2025).

2. Analytical Properties and Dynamic Phase Transitions

Dynamic random graphs exhibit time-dependent structural and probabilistic characteristics reflecting both their underlying update rules and macroscopic regimes:

Degree Distributions and Local Limits: Many dynamic models retain (in stationarity or in the limit) distributions resembling their static analogues. For instance, the DRGVR process converges locally to a binomial birth–death branching tree, and the dynamic intersection graph’s local limit is a time-marked multitype branching process reflecting group activity intervals (Díaz et al., 2022, Milewska et al., 2023).
Giant Component Emergence: Dynamic models often exhibit regime changes—phase transitions in the appearance of a giant component (i.e., a connected subgraph containing a positive fraction of vertices)—as key parameters cross critical thresholds. In the DRGVR, the critical edge formation parameter $\beta_c(p)$ admits explicit bounds, and above threshold the giant occupies a deterministic fraction of the vertex set in the large- $n$ limit (Díaz et al., 2022). For dynamic intersection models, the criticality is controlled by combinations of moment parameters of the group-size distribution and weight structure (Milewska et al., 2023).
Non-Self-Averaging Kinetics: In bounded-degree formation processes, the timescales for the appearance and completion of global structures (e.g., a spanning giant component or full regularity) can exhibit strong non-self-averaging: their fluctuations do not vanish as $n\to\infty$ (Ben-Naim et al., 2011).
Temporal Limit Objects and Dynamic Local Convergence: For processes such as dynamic Erdős–Rényi, intersection, or rewired configuration models, there is an established theory of dynamic local convergence. The limiting object is a time-marked rooted infinite random tree encoding the entire ON/OFF history of each incident edge during the observation window. This enables rigorous approximation of local processes such as epidemic spread by simulation on dynamic local limits (Milewska et al., 16 Jan 2025).

3. Random Walks and Dynamical Mixing

Random walks and spreading processes on dynamic random graphs exhibit behavior sensitive to the rate and scope of topological evolution:

Mixing Time and Hitting Properties: For a wide class of time-inhomogeneous Markov chains on dynamic graphs with a common stationary distribution, the mixing and hitting times are governed (up to constants) by the maximal corresponding quantities over all instantaneous graphs. For sequences of $d$ -regular dynamic graphs, mixing and hitting times scale as $O(n^2)$ , and further improvement is possible for higher isoperimetric dimension (Sauerwald et al., 2019, Shimizu et al., 2021).
Impact of Edge Rewiring: For non-backtracking walks on randomly rewired configuration models, the mixing time interpolates between the static and highly dynamic limits, with sharp phase diagrams—ranging from no cutoff to one- or two-sided cutoff—depending on rewiring speed and spatial scale (Avena et al., 2020).
Spectral and Averaging Effects: In models where the dynamic sequence of transition operators is reversible and has a fixed stationary distribution, spectral techniques extend to the dynamic setting, allowing bounds for complex quantities including coalescence times and consensus times for voter-type dynamics (Shimizu et al., 2021).

4. Inference, Estimation, and Statistical Methodology

Dynamic random graph models pose unique challenges for statistical inference due to latent structure, incomplete observation, or unobservable background processes:

Aggregate Observation and Regime Inference: When the observed process is a function of both an evolving network and an unobserved “mode” process (e.g., regime-switching of dynamic Erdős–Rényi models), moment-based estimation can be performed using only aggregate counts of subgraphs. The method-of-moments procedure solves for all time-scale and edge process parameters using combinations of mean and cross-moment constraints, resulting in consistent and asymptotically normal estimators under standard large-sample limits (Mandjes et al., 20 Jan 2025).
Bayesian and Frequentist Recovery for Latent Dynamic Structure: For dynamic RDPG models, generalized Bayesian Gibbs posteriors built on least-squares or likelihood losses can yield consistent, rate-optimal recovery of edge probabilities and (up to rotation) latent trajectories, as well as principled uncertainty quantification and forecasting (Loyal, 24 Sep 2025).
Stability and Concentration: Dynamic models with sufficient independence properties (e.g., DRGVR) admit strong concentration inequalities for general Lipschitz functionals, allowing for high-probability control of global statistics such as component sizes, maximum degree, or number of edges (Díaz et al., 2022).

5. Interacting Dynamics: Coevolution and Dynamical Systems

Dynamic random graphs often serve as substrates for interacting dynamical processes, leading to rich co-evolutionary phenomena:

Coupled Opinion–Graph Dynamics: Dense dynamic random graphs under evolving opinion models can be analyzed by coupling the opinion process with the time-varying network, producing functional laws of large numbers for both the empirical opinion densities and the empirical graphons. Depending on the feedback structure, the system admits coexistence, consensus, or polarization regimes, with rigorous characterization of limiting distributions via Beta laws and graphon-valued limits (Baldassarri et al., 18 Oct 2024).
Kuramoto Oscillators and Synchronization: The Kuramoto model of coupled oscillators on a time-varying graph (with edge dynamics governed by Markovian addition/removal or mobile random walks) exhibits rapid synchronization properties. For sufficiently fast edge dynamics, averaging principles apply and the system converges globally to synchrony, with the time-scale and robustness of synchronization enhanced by the dynamic connectivity (Groisman et al., 2022).
Epidemic Spread and Dynamic Local Limit Theory: The SIR process on dynamically evolving sparse graphs can be precisely characterized in the large- $n$ limit by reduction to the corresponding SIR process on the time-marked local weak limit. This approach rigorously captures temporal infection pathways induced by edge dynamics, which may be invisible to snapshot (static) local limits (Milewska et al., 16 Jan 2025).

6. Computational and Algorithmic Aspects

Scaling dynamic graph simulation and inference to large $n$ often requires specialized data structures and algorithms:

Efficient Updating in Geometric Models: Polar quadtrees and probabilistic neighborhood query algorithms permit processing of node movement in dynamic random hyperbolic graphs in sublinear time per update, with theoretical guarantees for query complexity and empirical speedups by several orders of magnitude, even for very large graphs ( $n\sim 10^8$ ) (Looz et al., 2018).
Scalable Bayesian Inference: Block-Gibbs sampling strategies for dynamic latent position models (e.g., dynamic RDPG) exploit sparsity structures and prior regularity for posterior sampling in time linear in the number of observed edges per iteration, facilitating scalable learning and forecasting (Loyal, 24 Sep 2025).

7. Critical Relationships and Open Problems

Dynamic random graphs unify and extend static random graph theory, while introducing new analytical and computational challenges:

Interplay Between Temporal and Structural Phase Transitions: The timing and nature of emergent macroscopic structures—giant components, clusters, synchronization—depend intricately on the rules and timescales of topology evolution, sometimes producing qualitatively different outcomes compared to static analogues (Díaz et al., 2022, Milewska et al., 2023, Ben-Naim et al., 2011).
Limits of Static Analogies: While many results for mixing times, hitting times, and epidemic dynamics transfer to dynamic settings under suitable uniformity (e.g., common stationary distribution, regular expansion), there exist finely-tuned counterexamples where even mild deviation from these conditions causes exponential slowdowns or structural instability (Shimizu et al., 2021, Avena et al., 2020, Sauerwald et al., 2019).
Theoretical and Practical Extensions: Open questions include necessary and sufficient conditions for dynamic local convergence in broad graph models, connections between aggregate statistics and microscopic dynamics, inference under partial observation or adversarial regimes, and algorithmic frameworks for coevolving multi-layer dynamic networks (Milewska et al., 16 Jan 2025, Baldassarri et al., 18 Oct 2024).