Correlated Erdős–Rényi Multiplex Model

• The paper introduces a correlated multiplex model that leverages joint edge probabilities and a tunable parameter to capture dependencies across network layers.
• It employs a rigorous parametrization of edge states (p00, p10, p01, p11) to derive closed-form expressions for spectral properties, degree distributions, and recovery thresholds.
• The framework underpins practical applications in graph alignment, percolation analysis, and network detection, providing robust analytical predictions for diverse network phenomena.

A correlated Erdős–Rényi multiplex model consists of two or more random graphs (layers), all defined on a common vertex set, where presence of edges is pairwise independent across vertex pairs, but edges between the same pair in different layers are statistically correlated. In the canonical two-layer case, the joint law for edges is specified by four probabilities $p_{00}, p_{10}, p_{01}, p_{11}$ for the states (no edge, edge in layer 1 only, edge in layer 2 only, edge in both), subject to marginal constraints and a tunable correlation parameter. This model provides an analytically tractable null baseline for network alignment, detection, and percolation theory, and exhibits phase transitions with closed-form thresholds intimately connected to joint edge overlap and correlation structure (Cullina et al., 2017).

1. Model Specification and Parametrization

Let $G_1,G_2$ be two random graphs on a common vertex set $[n]$. For each unordered pair $\{i,j\}$, generate adjacency indicators $A_{ij}=1\{\text{edge in }G_1\}$ and $B_{ij}=1\{\text{edge in }G_2\}$ according to a joint law: $$P[(A_{ij},B_{ij})=(1,1)] = p_{11}, \quad P[(1,0)] = p_{10}, \quad P[(0,1)] = p_{01}, \quad P[(0,0)] = p_{00}$$ These probabilities satisfy $p_{10}+p_{11}=p$ and $p_{01}+p_{11}=p$ for marginal edge-density $p$ in each layer.

Correlation between the layers is tuned by the parameter $s\in[0,1]$: $$p_{11} = p^2 + s\,p(1-p), \qquad p_{10} = p_{01} = (1-s)\,p(1-p), \qquad p_{00} = (1-p)^2 + s\,p(1-p)$$ Here the covariance and Pearson correlation coefficient are

$$\text{Cov}(A_{ij},B_{ij}) = p_{11}-p^2 = s\,p(1-p), \qquad \rho = s$$

This framework extends to $L$ layers by specifying joint probabilities $$p_S = \mathbb{P}(A_{ij}^{(\ell)}=1\;\forall\ell\in S)$$ for each subset $S\subseteq\{1,\dots,L\}$, constrained by consistency bounds (Ganguly et al., 8 Oct 2025, Gemmetto et al., 2017, Bianconi, 2013).

2. Structural Properties: Overlap, Hamming, and Spectrum

• Edge overlap: The number of coinciding edges $O = |\mathrm{E}(G_1)\cap \mathrm{E}(G_2)|$ is binomial

$$O \sim \mathrm{Binomial}(N, p_{11}),\;\;N = \tfrac{n(n-1)}{2},\;\; \mathbb{E}[O]=N\,p_{11},\;\; \mathrm{Var}(O)=N\,p_{11}(1-p_{11})$$

• Edge Hamming distance: $H := |\mathrm{E}(G_1)\triangle \mathrm{E}(G_2)|$ has expectation $2N\,p(1-p)(1-s)$.
• Intersection and union: $G_1 \wedge G_2 \sim \mathrm{ER}(n, p_{11})$, $$G_1\vee G_2 \sim \mathrm{ER}(n, p_{10}+p_{01}+p_{11})$$.
• Degree distribution: Marginally binomial with density $p$; joint degree distributions are bivariate binomial with correlation $\rho$ (Ganguly et al., 8 Oct 2025).

All network-level statistics reduce to sums over i.i.d. pairs due to pairwise independence, allowing closed-form computation of spectral properties and degree distributions.

3. Exact and Partial Recovery Thresholds

The principal application is graph alignment: recovering the latent vertex correspondence when the labeling of one layer is permuted. The information-theoretic threshold for exact recovery is captured by the parameter

$$q=\left(\sqrt{p_{11}p_{00}}-\sqrt{p_{10}p_{01}}\right)^2$$

Exact recovery is possible if

$nq \ge 2\log n + \omega(1)$

and impossible if $nq\leq(1-\epsilon)\log n$. For symmetric marginals ($p_{10}=p_{01}$), this reduces to requiring $np s^2 \gtrsim 2\log n$ (Cullina et al., 2017, Cullina et al., 2016).

Partial recovery admits different thresholds. A fraction tending to one of correct matches is achievable when the expected degree of the intersection graph $d_{\cap}=(n-1)p_{11}$ diverges: $d_{\cap}\to\infty\Longrightarrow \text{recover %%%%24%%%% matches}$ Moreover, iterative algorithms leveraging balanced load allocations and $k$-core alignments reach the optimal fraction dictated by load-distribution tails in the intersection graph (Du, 17 Feb 2025, Cullina et al., 2018).

4. Hypothesis Testing and Detection Thresholds

Discriminating correlated from independent layer models is statistically possible exactly when the signal parameter $nps^2$ crosses a model-dependent threshold. In constant-degree ($p = \lambda/n$) regimes, detection is achievable if

$$s > \min\left\{\frac{1}{\sqrt{\lambda}}, \frac{1}{\sqrt{\alpha}}\right\}, \quad \alpha\approx 0.338\;\;\text{(Otter's constant)}$$

where the second threshold emerges from the enumeration of forest subgraphs (unlabeled trees) (Feng, 15 Jun 2025).

Efficient detection (polynomial time) generally requires tree counting above $s^2 > \alpha$; the information-computation gap may persist if $\lambda>1/\alpha$ (Ding et al., 2023, Ding et al., 2022).

5. Multiplex Substructure: Motifs, Subgraphs, and Limit Theory

A detailed asymptotic theory for submultiplex appearances generalizes the Erdős–Rényi motif theory. For any fixed submultiplex $H$:

• The threshold for the emergence of $H$ is given by

$$\Phi_H(n,p_1,p_2,p_{12}) = \min_{F\neq\emptyset}\:n^{|V(F)|}p_1^{|E(F^{(1)})\setminus E(F^{(2)})|}p_2^{|E(F^{(2)})\setminus E(F^{(1)})|}p_{12}^{|E(F^{(1)})\cap E(F^{(2)})|}$$

• The region where infinitely many copies of $H$ appear is a convex polyhedron in $(\theta_1,\theta_2,\theta_{12})$.
• In the interior ($\Phi_H \gg 1$), the count is asymptotically normal; at the threshold boundary ($\Phi_H \sim 1$) Poisson approximations apply, governed by submultiplex balance and core structure (Bhattacharya et al., 15 Nov 2025).

For large $n$, correlated multiplexes converge (in cut-metric and left-convergence) to multiplexons—constant graphon limit objects prescribing all marginal and joint edge densities (Ganguly et al., 8 Oct 2025). All higher-order statistics, e.g., cross-layer clustering, are computable in closed form.

6. Entropic Characterization and Sampling Methods

In the exponential-random-multiplex (canonical) ensemble, the joint law for layer edges is governed by Lagrange multipliers imposing average edge-number and overlap constraints: $$P(G) = \frac{1}{Z}\exp\left[\theta_1 \sum_{i<j} a_{ij}^{(1)} + \theta_2 \sum_{i<j} a_{ij}^{(2)} + \mu \sum_{i<j} a_{ij}^{(1)} a_{ij}^{(2)}\right]$$ The entropy per edge-pair is a function of the four occupation probabilities and the overlap. As correlations increase, entropy decreases due to increased redundancy between layers; the limit of maximal correlation compresses all probability mass into the joint (1,1) and (0,0) edge states (Bianconi, 2013).

Sampling proceeds by reconstructing the pairwise joint law for each dyad via exponential family parameters, yielding multiplexes with prescribed marginals and pairwise correlations (Gemmetto et al., 2017).

7. Percolation and Robustness in Correlated Multiplexes

Edge correlations (assortative or disassortative) can qualitatively alter percolation thresholds and hybrid transitions. In two-layer models with parameters $(p_1, p_2, p_{ov})$ for single and overlapped edges:

• Assortative correlation ($p_{ov}>p_1 p_2$): Lowers the critical density for emergence of a giant mutually connected component, introduces multiple hybrid transitions and possible re-entrance.
• Disassortative correlation ($p_{ov}<p_1 p_2$): Raises the threshold, splits the transition into multiple phases.

The critical surface is found via the coupled self-consistency equations for cavity probabilities, with the phase diagram computed from the joint excess-degree distribution. These phenomena underscore the non-perturbative impact of edge correlations on network connectivity (Baxter et al., 2016, Cullina et al., 2016, Baxter et al., 2016, Lee et al., 2011).

8. Applications and Extensions

Correlated ER multiplexes underpin information-theoretic analysis in network deanonymization, community detection, privacy quantification, and multiplex percolation. The model serves as a baseline against which effects of structure, heterogeneity, or further dependencies can be rigorously assessed. Extensions include multivariate Bernoulli ensembles for $L>2$ layers, hierarchical or block-based correlation, and graphon- or multiplexon-based limits for dense/sparse asymptotics (Ganguly et al., 8 Oct 2025, Chandna et al., 2022, Gemmetto et al., 2017).

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