Dynamic Erdős–Rényi Graphs

• Dynamic Erdős–Rényi model is a stochastic process where each edge switches independently in continuous time, modeling temporal network evolution.
• The model uses continuous-time Markov and renewal processes to rigorously analyze edge dynamics, mixing times, and phase transitions.
• Applications include studying temporal percolation, epidemic processes, and statistical inference for dynamic network parameters.

A dynamic Erdős–Rényi model is a stochastic process in which the set of edges of a graph on $n$ fixed vertices evolves dynamically, with each edge switching independently “on” (present) and “off” (absent) through a continuous-time Markov process or, more generally, an alternating renewal process. Unlike the classical static Erdős–Rényi random graph $G(n,p)$, where each edge is included independently with fixed probability $p$, the dynamic model describes a time-indexed random graph process $(G(t))_{t\ge0}$ whose microstructure, transient regime, mixing behavior, and limiting properties have been investigated rigorously. This framework allows for the study of temporal network connectivity, dynamic percolation, time to emergence of a giant component, functional limit theorems for subgraph statistics, and inference for network dynamics given partial observations (Rosengren et al., 2016).

1. Definition, Construction, and Stationary Law

The canonical dynamic Erdős–Rényi model is a Markov process $(G(t))_{t\ge0}$ on the space of simple graphs with vertex set $[n]=\{1,\dots,n\}$. For each unordered vertex pair $(u,v)$, the edge indicator $\chi_{u,v}(t)\in\{0,1\}$ follows an independent continuous-time two-state chain, switching

• $0 \to 1$ (edge appears) at rate $\lambda$,
• $1 \to 0$ (edge disappears) at rate $\mu$.

Often, one sets $\lambda = \beta/(n-1)$, $\mu = \alpha$ for constants $\alpha,\beta > 0$ (Rosengren et al., 2016). More generality is possible: each edge’s alternation can be defined through renewal or semi-Markov processes with arbitrary on- and off-time distributions (Mandjes et al., 2024), or modulated by a hidden regime process (Mandjes et al., 20 Jan 2025, Mandjes et al., 2017). A summary of key stationary properties includes:

• The stationary law for $G(t)$ is $G(n,p)$ with $p = \lambda/(\lambda+\mu)$ (Armbruster et al., 2011, Rosengren et al., 2016).
• Each edge is present at equilibrium with probability $p$; edges are independent.
• Ergodicity holds for all $\lambda,\mu > 0$.

2. Mixing, Stationarity, and Hitting Times

Stationarity and Relaxation:

• The unique stationary distribution is $G(n,p)$. For each edge process, the fastest strong stationary time is the first update time $T_{u,v}\sim \operatorname{Exp}(\alpha+\beta/(n-1))$.
• The fastest strong stationary time for the graph process is $T_s = \max_{u<v}T_{u,v}$, with sharp mixing-time bounds:

$$\mathbb{P}(T_s \le t) = \left[1 - \exp\left(-(\alpha+\beta/(n-1))\,t\right)\right]^N$$

where $N = \binom{n}{2}$ (Rosengren et al., 2016).

• As $n\to\infty$, $\mathbb{E}[T_s]=O(\log n)$ and the rescaled $T_s$ converges to a Gumbel law.

Hitting Times and Edge Dynamics:

• The process $\eta(t) = \sum_{u<v} \chi_{u,v}(t)$ (the edge-count) is a birth-death chain.
• The expected time to reach $i$ edges, $\mathbb{E}[\tau_0(i)]$, has an explicit finite-sum recursion (Rosengren et al., 2016), with three asymptotic regimes for varying edge density:
  • Subcritical: fast growth to $cn$ edges (linear time in $O(1)$).
  • Critical: $E[\tau_0([cn])] = O(\log n)$.
  • Supercritical: exponential scaling in $n$.
• Emergence of a giant component is linked to hitting a threshold number of edges; for $c$ exceeding a sharp threshold ($c>1/2$), a component of size $\varepsilon n$ appears in $G(n, [c n])$; corresponding hitting times can be bounded in the dynamic model (Rosengren et al., 2016).

3. Large Deviations and Empirical Graphon Path Theory

Sample-path large deviation principles (LDPs) have been established for the dynamic Erdős–Rényi model, both for edge-counts and for the empirical graphon trajectory as $n\to\infty$ (Braunsteins et al., 2020, Mandjes et al., 2017):

• The empirical graphon process $f_n(t)$ for the dynamic $G_n(t)$ satisfies a full LDP in the Skorokhod path space, with speed $\binom{n}{2}$ and a good rate function given by an action integral:

$$I(h) = \frac12\int_0^T \int_{[0,1]^2} \mathcal{L}(h_t(x,y),\,\partial_t h_t(x,y))\,dx\,dy\,dt$$

where $\mathcal{L}(a,v)$ encodes single-edge pathwise large deviations (Braunsteins et al., 2020).

• This LDP enables the identification of most-probable paths for producing anomalous subgraph statistics or transitioning between graphon configurations, with explicit variational formulas and the possibility of bifurcations in the optimal paths.

4. Functional Central Limit Theorems for Subgraph and Spectral Statistics

Broad generalizations of classical limit theorems have been derived for the dynamic model:

• A functional central limit theorem (fCLT) holds jointly for subgraph counts (e.g., edges, wedges, triangles), yielding convergence (with centering and scaling) of the process of normalized counts to a multidimensional Gaussian process with explicit covariance determined by the edge process’ covariance structure. Fluctuations and correlations reflect the interplay between the dynamics and the combinatorics of the subgraphs (Hazra et al., 3 Feb 2025).
• When edge dynamics are independent of $n$ (“dense” regime), all subgraph-count fluctuations are asymptotically perfectly correlated. In the sparse regime, phase-transitions in correlation structure arise.
• For the principal eigenvalue of the adjacency matrix, an fCLT asserts that, after centering, the leading eigenvalue process converges to a Gaussian process with autocovariance governed by the edge-dynamics correlation decay (Hazra et al., 2024).

5. Statistical Inference and Regime Switching

Estimating dynamic model parameters from partial or aggregated observations has received attention:

• Method-of-moments estimators can recover parameters (on/off-time means, regime sojourn times) from time series of edge counts or subgraph counts, using explicit theoretical moment formulas and their asymptotic properties (Mandjes et al., 2024, Mandjes et al., 20 Jan 2025).
• For regime-switching models (where a hidden, stochastically switching process determines which dynamic graph is observed at each time), moments of observable statistics admit closed forms in terms of underlying regime and switch parameters. Moment-equation inversion yields consistent, asymptotically normal estimators (Mandjes et al., 20 Jan 2025).
• Observation of rarer subgraph counts (triangles, $k$-cliques) can help overidentify parameters but increases estimator variance.

6. Dynamic Epidemics and Temporal Percolation

Contact and epidemic processes in the dynamic Erdős–Rényi environment are analytically tractable (Armbruster et al., 2011, Huang et al., 2024):

• For SI and SIR epidemics, functional laws of large numbers hold. Scaling regimes for infection and edge flip rates give rise to limiting ODEs, which coincide with (i) the classical mean-field equations in high-degree or fast-mixing regimes, and (ii) novel systems tracking “infectious edge” densities in the regime where edge and epidemic dynamics occur on comparable timescales.
• The dynamic model provides explicit critical thresholds and mean times for epidemic takeover and connectivity transitions, with sharp bounds in both finite/infinite rate regimes.

7. Variants, Extensions, and Regimes

Several notable variants and extensions have been studied:

• Regime-switching and periodic-resampling: Edges’ transition rates can themselves be governed by a finite-state Markov process or drawn from a random environment at periodic resampling times (Mandjes et al., 2017). This allows for nontrivial covariance structures and dynamic correlation in edge appearance.
• Special-vertex and restricted growth models: Constraints such as the $k$-process (no two special vertices can be in the same component) yield constrained dynamic random graph processes with critical phenomena at nontrivial thresholds ($k\sim n^{1/3}$) and sharp phase transitions in the size of the largest component (Logan et al., 2018).
• Critical and super-critical regimes: In the time-varying critical regime ($p=1/n$ and dynamic switching), component size maxima scale as $n^{2/3}(\log n)^{1/3}$ over $[0,1]$, a significant deviation from the static case and arising from extreme-value statistics in the dynamic ensemble (Roberts et al., 2016).

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Key references:

• (Rosengren et al., 2016) "A Dynamic Erdős-Rényi Graph Model"
• (Braunsteins et al., 2020) "A sample-path large deviation principle for dynamic Erdős-Rényi random graphs"
• (Hazra et al., 3 Feb 2025) "Functional Central Limit Theorem for the simultaneous subgraph count of dynamic Erdős-Rényi random graphs"
• (Hazra et al., 2024) "Functional Central Limit Theorem for the principal eigenvalue of dynamic Erdős-Rényi random graphs"
• (Mandjes et al., 2024) "Estimation of on- and off-time distributions in a dynamic Erdős-Rényi random graph"
• (Mandjes et al., 2017) "Dynamic Erdős-Rényi graphs"
• (Mandjes et al., 20 Jan 2025) "Inference for dynamic Erdős-Rényi random graphs under regime switching"
• (Armbruster et al., 2011) "Dynamic Network Models"
• (Huang et al., 2024) "The SIR epidemic on a dynamic Erdős-Rényi random graph"
• (Roberts et al., 2016) "Exceptional times of the critical dynamical Erdős-Rényi graph"
• (Logan et al., 2018) "A variant of the Erdos-Renyi random graph process"