Correlation-Based Diagnostics of Social Contagion Dynamics in Multiplex Networks

Multiplex contagion dynamics display localization phenomena in which spreading activity concentrates on a subset of layers, as well as delocalized regimes where layers behave collectively. We investigate how these regimes are encoded in temporal correlations of node activity. By deriving a closed-form mean-field expression for node autocorrelations in a contact-based social contagion multiplex model and validating it through simulations, we show that lag-one autocorrelations act as sensitive indicators of both activation and localization transitions. Our results establish temporal correlations as lightweight, structure-agnostic probes of multiplex spreading dynamics, particularly valuable in partially observable systems.

💡 Research Summary

The paper addresses the challenge of diagnosing the collective state of contagion processes that unfold simultaneously across multiple online platforms, modeled as a multiplex (multilayer) network. Traditional approaches rely on structural information—particularly the spectral properties of the supra‑contact matrix—to distinguish between a localized regime, where activity concentrates on a single dominant layer, and delocalized regimes, where several layers become active together. However, real‑world data collection is often limited to a subset of platforms, making structural inference infeasible.

To overcome this limitation, the authors focus on a minimal contact‑based social contagion model extended to multiplex networks. Each layer α has its own adjacency matrix Aα and a contact‑probability matrix Rα that interpolates between contact‑process (CP) and reactive‑process (RP) dynamics via a parameter γα. Inter‑layer transmission is encoded by a coupling matrix C scaled by ε = η/β, where η is the inter‑layer infection probability and β the intra‑layer infection probability. The discrete‑time mean‑field master equation (Eq. 3) governs the infection probability pi(t) of each node i, and the epidemic threshold is given by β/µ = 1/Λ̄max, where Λ̄max is the largest eigenvalue of the supra‑contact matrix (\bar R).

The core contribution is an analytical expression for the steady‑state Pearson autocorrelation of a node’s binary state at lag h, derived under the mean‑field approximation (Eq. 12). The autocorrelation depends on three quantities: the recovery probability µ, the steady‑state infection probability pαu, and the probability qαu that a node receives no infection from its neighbors. The expression predicts an exponential decay with factor qαu(1‑µ) and reduces to the known SIS result for a single layer. At the activation threshold, where infections are rare (q≈1), the lag‑1 autocorrelation simplifies to ρ(1)=1‑µ, independent of network topology.

Using this formula, the authors map autocorrelation behavior onto the multiplex phase diagram (Fig. 1). In the inactive phase (I) all pαu=0, so autocorrelations are undefined. At the activation line, ρ(1)=1‑µ for both layers. In the active‑localized regime (AL), the dominant layer shows a monotonic decline of autocorrelation as β/µ increases (because p grows and q shrinks), while the non‑dominant layer’s autocorrelation remains near the critical value 1‑µ, reflecting sporadic, coupling‑driven infections. This disparity provides a clear signature of localization. In the active‑delocalized regime induced by higher transmissibility (AD 1), both layers sustain endemic activity and their autocorrelations decay together, erasing the layer‑specific gap. In the coupling‑driven delocalized regime (AD 2), strong inter‑layer coupling (ε ≳ Λ1‑Λ2) synchronizes the dynamics, leading to nearly identical autocorrelation decay in both layers regardless of their individual structural properties.

The authors acknowledge that near the AL↔AD 2 boundary, strong inter‑layer correlations violate the independence assumption of the master equation, reducing quantitative accuracy of Eq. 12. To validate the qualitative predictions, they complement the theory with stochastic simulations of the full process and numerical integration of the mean‑field equations. The simulations confirm that lag‑1 autocorrelations reliably distinguish the three regimes: (i) a plateau at 1‑µ for one layer while the other declines (AL), (ii) simultaneous decline in both layers (AD 1 or AD 2), and (iii) absence of autocorrelation in the inactive phase.

Importantly, the study demonstrates that even when only a single layer is observable, the measured autocorrelation can reveal the hidden state of the unobserved layers. If the observed layer exhibits low activity but an autocorrelation close to 1‑µ, it likely acts as a non‑dominant layer driven by a supercritical hidden layer. Conversely, matching autocorrelation patterns across observed layers indicate a delocalized global state. Thus, temporal autocorrelations serve as lightweight, structure‑agnostic probes for multiplex contagion dynamics, offering a practical tool for monitoring and early‑warning in partially observable online ecosystems.