Dynamic Erdős–Rényi Model

• Dynamic Erdős–Rényi Model is a stochastic network model where each edge independently switches between active and inactive states using Markov or renewal processes.
• It characterizes both transient and stationary network properties, with mixing and hitting times derived from exponential edge dynamics and limit theorems like the Gumbel law.
• The model underpins applications in epidemic spread, regime-switching systems, and statistical inference, providing analytic frameworks for subgraph counts and spectral analysis.

A dynamic Erdős–Rényi (ER) model extends the classical static ER random graph by allowing each edge to independently switch states—present ("on") or absent ("off")—according to continuous-time stochastic processes, most commonly two-state Markov chains or general alternating renewal processes. The motivation for dynamic ER models is to analytically capture the statistical properties of networks whose connectivity evolves at stochastic rates, with applications ranging from epidemic propagation to regime-switching systems in communication and biology. The key technical challenge is to characterize both the transient and stationary distributions, time-dependent properties such as hitting and mixing times, and limit theorems for functionals of the network, such as subgraph counts and spectral statistics.

1. Formal Definition and Edge Dynamics

For a vertex set $V = \{1,\ldots, n\}$, the dynamic ER model stipulates that each unordered pair $\{i,j\}$ evolves independently as a two-state Markov process or alternating renewal process. The basic Markov formulation (relevant for exponential on- and off-times) can be stated as:

• Transition rates: $0 \to 1$ at rate $\lambda$, $1 \to 0$ at rate $\mu$, with $A_{ij}(t)\in\{0,1\}$ the adjacency matrix entry at time $t$.
• The instantaneous distribution: $$p(t) = \mu/(\lambda+\mu) + [p(0) - \mu/(\lambda+\mu)] e^{-(\lambda+\mu)t}$$.
• Independence: All edges evolve independently and symmetrically, so adjacency matrix $A_n(t) = (A_{ij}(t))$ is symmetric.
• For general alternating renewal processes, the sojourn times ("on"/"off" durations) may follow arbitrary parametric (e.g., geometric, Weibull) or non-parametric distributions, with stationarity determined by the mean sojourn lengths.

The state space of the network at time $\{i,j\}$0 is thus a product space $\{i,j\}$1, and the model is irreducible and ergodic under mild conditions. The stationary distribution of the graph is exactly the static ER law $\{i,j\}$2 with $\{i,j\}$3 (Hazra et al., 2024, Armbruster et al., 2011, Rosengren et al., 2016, Mandjes et al., 2024).

2. Stationarity, Mixing, and Hitting Times

The time to stationarity is governed by the rates $\{i,j\}$4. For independent edges, detailed balance holds for every edge, and the stationary law factorizes as a product measure with the edge-marginal $\{i,j\}$5 (Armbruster et al., 2011, Rosengren et al., 2016). Each edge relaxes exponentially to stationarity with mixing rate $\{i,j\}$6; the time for the overall graph to approach equilibrium in total variation distance is $\{i,j\}$7 (Armbruster et al., 2011).

Strong stationary times can be precisely identified: the time to the first update for an edge (Exp$\{i,j\}$8) is minimal for reaching edge-level stationarity; for the whole graph, the maximum of $\{i,j\}$9 independent Exp$0 \to 1$0 yields a distribution $0 \to 1$1, which for large $0 \to 1$2 scales as a Gumbel law, with mean hitting time to stationarity $0 \to 1$3 (Rosengren et al., 2016).

The hitting time to a prescribed number of edges, or more general subgraph counts, follows a birth-death process approximation; exact formulae and asymptotics are known, with large deviation scaling in the supercritical regime (Rosengren et al., 2016).

3. Functional Limit Theorems for Graph Functionals

Multiple functional central limit theorems (FCLTs) have been established for dynamic ER models.

• Principal Eigenvalue: For the top eigenvalue $0 \to 1$4 of the adjacency matrix, the normalized centered process $0 \to 1$5 converges in distribution, in the Skorokhod space $0 \to 1$6, to a zero-mean Gaussian process $0 \to 1$7 with Ornstein-Uhlenbeck covariance $0 \to 1$8. This result reflects the essentially rank-one structure of the adjacency matrix plus centered noise, and is obtained via tightness and finite-dimensional convergence using uniform operator-norm tail bounds and martingale methods (Hazra et al., 2024).
• Subgraph Counts: For simultaneous counts of subgraphs $0 \to 1$9 (e.g., edges, triangles), under suitable centering and scaling, the joint process converges to a multidimensional Gaussian process whose covariance structure is determined by optimal subgraph decompositions and covariance kernel $\lambda$0 corresponding to edge process correlations (Hazra et al., 3 Feb 2025).

In both cases, the independence and identically distributed nature of the edge processes enable explicit analytic representations of means, variances, and covariances, and the limiting fluctuations reflect the aggregate edge dynamics.

4. Regime Switching and Large Deviations

Extensions of the model incorporate external regime-switching or periodically resampled transition rates (Mandjes et al., 2017, Mandjes et al., 20 Jan 2025):

• Regime-Switching: The rates $\lambda$1 depend on a background Markov or renewal process $\lambda$2. The evolution of present edges $\lambda$3 yields a coupled Markov process; exact and asymptotic expressions for means, variances, FCLTs, and sample-path large deviation principles (LDPs) are derived, characterized by joint variational problems in the space of regime-weight trajectories and edge-dynamic cumulants.
• Periodic Resampling: At discretized epochs, rates are synchronously updated across all edges, yielding time-inhomogeneous transitions. The process admits explicit moment and covariance formulas, an FCLT in appropriate scaling limits, and a sample-path LDP decoupling into a single-interval cumulant action integral.

The sample-path LDP for the edge-indicator empirical graphon (mapping the adjacency configuration to a graphon function) quantifies the probability of atypical trajectories, with action integral rate functions and associated Hamilton–Jacobi equations governing optimal path problems between graphon states, exhibiting bifurcation phenomena (Braunsteins et al., 2020).

5. Estimation, Identifiability, and Inference Problems

Parameter estimation for dynamic ER models, particularly for on/off-time distributions, is handled via method-of-moments procedures (Mandjes et al., 2024, Mandjes et al., 20 Jan 2025):

• Empirical mean and autocovariance (and, if available, higher-order subgraph counts and cross-moments) yield enough statistics to solve for the parameters of the alternating renewal or regime-switching processes.
• Conditions for asymptotic normality and identifiability are established via the Jacobian of the moment map and long-run covariance structure.
• Extensions to count wedges, triangles, stars, or cliques provide additional constraints for identifying complex parametric families and discriminating departures from the minimal model.

Simulations confirm small variance and Gaussianity of estimators in sufficiently large samples; identifiability depends critically on the chosen subgraph set and on observation lag structure.

6. Applications: Epidemic Spread and Critical Phenomena

Dynamic ER networks serve as analytically tractable substrates for modeling SI and SIR processes (Huang et al., 2024, Armbruster et al., 2011):

• Epidemic ODEs: Depending on scaling regimes for edge-flipping and contagion rates, the limiting epidemic curves collapse to classical mean-field SIR equations, except in the critical regime where the average degree is $\lambda$4 and the edge-dynamics and infection rates occur on the same time scale. Here, extra macroscopic variables (e.g., density of infectious edges) enter the limit ODEs.
• Connectivity and outbreak times: Edge turnover significantly accelerates connectivity and epidemic spread compared to static graphs, sometimes yielding $\lambda$5 scaling for the time to network-wide connectivity or full infection.

Exceptional time phenomena in the critical dynamic ER regime (at $\lambda$6) include rare intervals when the largest component's size is much larger than in any static snapshot, with precise logarithmic scaling: $\lambda$7 (Roberts et al., 2016).

7. Extensions, Limitations, and Open Problems

Current research directions include nonparametric inference, extensions to more than two regimes, observation noise, and the analysis of dynamic ER networks with inhomogeneities such as block-model structure or edge clustering. Most current limit results and estimation procedures rely fundamentally on independence across edges. The inclusion of higher-order dependencies or degree constraints constitutes a major challenge, requiring new analytic and probabilistic techniques.

The dynamic ER model thus forms a rigorous and robust platform for the study of evolving networks, with well-developed probabilistic limit theorems, parameter estimation methods, and applications across statistical physics, epidemiology, and applied probability. The foundational results link transient, stationary, and rare-event behavior to the microscopic dynamics of edge processes, supporting both theoretical and practical analyses.