Dynamic Erdős-Rényi Model

• Dynamic Erdős–Rényi models are stochastic graph processes where each potential edge independently toggles between active and inactive states according to continuous-time Markov dynamics.
• They use edge-specific appearance and disappearance rates (λ and μ) to derive stationary laws, mixing times, and transient fluctuations within evolving networks.
• These models expose key phase transitions, including the emergence of giant components, and have applications in epidemic, communication, and sensor networks.

A dynamic Erdős-Rényi model is a family of continuous- or discrete-time stochastic processes on graphs, most commonly using the complete graph on $n$ vertices as the underlying structure, where each potential edge independently switches between present and absent according to a prescribed random mechanism. In these models, the set of active edges at time $t$ is random and evolves in time, generalizing the classical static Erdős-Rényi $G(n,p)$ ensemble by incorporating temporal dynamics such as Markovian switching, regime-modulated transitions, or periodically resampled edge states. Central themes in the analysis include the characterization of stationary laws, mixing times, transient and limiting fluctuations, sample-path large deviations, and phenomena emerging from edge-count dynamics such as phase transitions and the timing of giant-component formation.

1. Core Continuous-Time Models

The canonical dynamic Erdős-Rényi process considers $N=\binom{n}{2}$ potential (undirected) edges. Each edge $e_{ij}(t)\in\{0,1\}$ independently switches from 0 (absent) to 1 (present) at rate $\lambda>0$ (appearance) and from 1 to 0 at rate $\mu>0$ (disappearance), thus forming a two-state continuous-time Markov chain ("telegraph process") per edge (Armbruster et al., 2011, Rosengren et al., 2016). The infinitesimal generator for a single edge is

$$Q = \begin{pmatrix} -\lambda & \lambda \ \mu & -\mu \end{pmatrix}.$$

For the full graph, the process $\{G(t)\}$ is Markov with $2^N$ states, but each edge evolves independently.

The stationary law is $G(n,p)$ with $p = \lambda / (\lambda+\mu)$. Each edge's process is ergodic, and at stationarity, the presence/absence of edges are independent Bernoulli($p$), recovering the classical Erdős-Rényi graph distribution.

A more refined variant uses edge-specific appearance and disappearance rates, for instance via scaling in dense graph regimes ($\lambda=\beta/(n-1)$, $\mu=\alpha$, as in (Rosengren et al., 2016)), or stochastic mechanisms for the rates themselves (Mandjes et al., 2017).

2. Generalizations: Regime-Switching and Resampling Models

Two broad classes of extensions are detailed in (Mandjes et al., 2017):

• Regime-Switching: A background finite-state irreducible Markov chain $X(t)\in\{1,\dots,d\}$ modulates the rates. When $X(t)=i$, each edge appears at rate $\lambda_i$ and disappears at rate $\mu_i$. The process is Markov in $(X(t),Y(t))$, where $Y(t)$ is the total edge count. Stationary and transient distributions for $Y(t)$ follow from coupled matrix-valued Kolmogorov equations and can be described by recursion for the factorial moments:

$$\mathbf{e}_k^\top = k! (N)_k\, \pi^\top \Lambda(\Lambda+M-Q)^{-1} \cdots \Lambda(k\Lambda+kM-Q)^{-1},$$

with explicit formulas for mean and variance.

• Periodically-Resampled: At each discrete time step, transition probabilities for edge birth ($P_m$) and death ($R_m$) are independently redrawn and then held fixed for that interval, leading to a time-inhomogeneous birth-death process for $Y(t)$. The stationary mean is

$$\mathbb{E}[Y] = N \frac{1-\bar{P}}{2-\bar{P}-\bar{R}},$$

with typically quadratic variance growth in $N$ unless $P,R$ are non-random.

These generalizations allow for modeling (i) network response to exogenous, temporally correlated environments (regime switching) and (ii) scenarios with periodic or randomly refreshed network-wide conditions (resampling).

3. Mixing Times and Strong Stationarity

Mixing time analysis quantifies how rapidly the dynamic graph approaches stationarity. For the continuous-time Markov model with uniform edge rates, the time to stationarity for $k$ fixed edges (total variation distance) decays exponentially, with mixing time $T(\varepsilon) \asymp (2/(\lambda+\mu))\log k$ (Armbruster et al., 2011).

A more precise characterization uses strong stationary times. For the "reset-on-first-update" construction (Rosengren et al., 2016):

• Each edge's time to first update (either type) is $\mathrm{Exp}(\lambda+\mu)$.
• The maximal such time over all edges, $T_s = \max_{1\le u<v\le n} T_{uv}$, is the fastest strong stationary time, with

$$\mathbb{P}(T_s \leq t) = \left(1-e^{-(\lambda+\mu)t}\right)^N.$$

For large $n$, $T_s$ is of order $\log n$, and, properly centered and scaled, converges to a Gumbel distribution.

4. Edge Count Dynamics: Birth-Death Chains and Hitting Times

Let $\eta(t)$ denote the number of edges at time $t$. For the uniform model, $\eta(t)$ is a birth-death process with: $\lambda_k = (N-k)\lambda, \quad \mu_k = k\mu$ for $0 \leq k \leq N$. The hitting time $\tau_j(i)$ to reach $i$ edges from $j$ is

$$\mathbb{E}[\tau_j(i)] = \sum_{k=j}^{i-1} \mathbb{E}[\tau_k(k+1)],$$

where the summand admits a closed formula in terms of factorials and binomial coefficients (Rosengren et al., 2016). For asymptotic regimes ($i = [cn]$, $n\to\infty$) three behaviors emerge:

• For $c<\lambda/(2\mu)$, the hitting time concentrates on a deterministic limit.
• At the critical threshold, hitting time grows logarithmically.
• For $c>\lambda/(2\mu)$, hitting times are exponentially large in $n$.

5. Limit Theorems: Functional CLTs and Large Deviations

Both regime-switching and resampling models admit functional central limit theorems under "diffusive" scaling (Mandjes et al., 2017). When the modulating background process is accelerated ($Q \mapsto N^\delta Q$, $\delta=1$) and after centering/scaling the edge count, convergence is to an Ornstein–Uhlenbeck process governed by an SDE: $$dY_\infty(t) = -\gamma^\top Y_\infty(t)\,dt + \sqrt{g'(t) + h'(t)}\,dB(t)$$ with explicit $g'(t), h'(t)$ determined by the regime/edge rates.

Large deviation principles for the rescaled edge process ($Y(t)/N$) are established in both variants. For regime-switching, a Mogulskii-type sample-path rate function is computed via variational formulas over deterministic regime paths $g(\cdot)$, while in the resampling case a single supremum defines the local rate function.

6. Dynamic Phase Transitions and Giant Component Formation

A critical event in dynamic Erdős-Rényi graphs is the time to formation of a giant component. In analogy to the static threshold, the expected time until the largest connected component exceeds size $\epsilon n$ is directly tied to the hitting time for the total edge count to surpass $[c_\epsilon n]$, with $c_\epsilon$ computed by inverting the static percolation formula: $c_\epsilon = \frac{-\ln(1 - \epsilon)}{2\epsilon}$ For the dynamic process, given $N$ edges with stationary mean $\lambda n / (2\mu)$, the mean time to giant formation is $O(\mathbb{E}[\tau_0([c'n])])$ for $c' > c_\epsilon$, with exponential or sub-exponential scaling in $n$ dependent on the regime (Rosengren et al., 2016).

7. Applications and Extensions

Dynamic Erdős-Rényi models have broad application in modeling time-evolving networks where edge connectivity is inherently stochastic, including epidemic processes (e.g., the SI contact process), communication or sensor networks, and models with time-varying node populations (Armbruster et al., 2011, Mandjes et al., 2017). The inclusion of regime-switching captures systems influenced by temporally correlated environmental states, while periodic resampling addresses systems with coordinated, scheduled resets.

Extensions addressed in recent literature include estimation of on- and off-time distributions from aggregate edge or subgraph counts using inverse-problem and method-of-moments approaches (Mandjes et al., 25 Jan 2024), the effect of demographic turnover (birth and death of nodes), and characterization of rare events via large deviation analysis.

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References

• (Armbruster et al., 2011) Armbruster, D. & Carlsson, G. Dynamic Network Models.
• (Rosengren et al., 2016) Rosengren, S. & Trapman, P. A Dynamic Erdős–Rényi Graph Model.
• (Mandjes et al., 2017) van Leeuwaarden, J. S. H., et al. Dynamic Erdős–Rényi Graphs.
• (Mandjes et al., 25 Jan 2024) Estimation of on- and off-time distributions in a dynamic Erdős-Rényi random graph.