Correlation-Based Diagnostics of Social Contagion Dynamics in Multiplex Networks

TL;DR

The paper addresses diagnosing multiplex contagion dynamics under partial observability by deriving a mean-field autocorrelation diagnostic for node activity in a two-layer multiplex. It demonstrates that lag-1 autocorrelations capture activation and localization transitions across AL and AD regimes, validated by simulations on RR and ER networks and linked to a spectral criterion via the supra-contact matrix. The study shows that a low-activity observed layer with autocorrelation near 1 - mu indicates a non-dominant layer driven by a hidden dominant layer, enabling inference with limited data, while highlighting limitations near the AL-AD2 crossover where interlayer correlations undermine mean-field accuracy. Overall, temporal autocorrelations emerge as lightweight, structure-agnostic probes for multiplex spreading with practical relevance for data-limited, real-world systems.

Abstract

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.

Figures (11)

• Figure 1: Schematic phase diagram in the plane of transmissibility $\beta/\mu$ and interlayer coupling $\eta/\beta$ for a two-layer multiplex with $\Lambda_1>\Lambda_2$. The absorbing phase (I) lies to the left of the criticalpoint $\beta/\mu=1/\bar{\Lambda}_{max}$. In the active phase, strong coupling above $(\eta/\beta)_t\sim \Lambda_1-\Lambda_2$ yields delocalized activity (AD$_2$). For weaker coupling, increasing $\beta/\mu$ above the non-dominant activation threshold $(\beta/\mu)_c^{(2)}$ drives the system from an active--localized regime (AL) to a delocalized endemic regime (AD$_1$). The diagram is schematic and not to scale.
• Figure 2: Representative correlation functions at steady state for $\beta/\mu=0.1$, $\eta/\beta=0.01$ and $\mu=0.5$. The quantities $\rho_{u,u}^{1,1}$ and $\rho_{u,u}^{2,2}$ are autocorrelations for layers 1 and 2, while $\rho_{u,v}^{1,1}$ and $\rho_{u,v}^{2,2}$ are nearest-neighbor intralayer correlations. The interlayer correlation between replicas of the same node is $\rho_{u,u}^{1,2}$, and $\rho_{u,v}^{1,2}$ denotes interlayer correlations between different nodes. Curves labelled "T." correspond to mean-field predictions obtained from numerical integration of \ref{['eq:mstereq']}.
• Figure 3: Active--localized regime (AL) for $\eta/\beta=0.01$. Left: correlation measures at lag $h=1$ as functions of $\beta/\mu$. Right: stationary density in each layer as a function of $\beta/\mu$. Vertical dashed lines indicate the single-layer thresholds $1/\Lambda_1$ and $1/\Lambda_2$. Curves labelled "T." correspond to numerical integration of the master equation \ref{['eq:mstereq']}.
• Figure 4: Active--delocalized regime AD$_2$ ($\eta/\beta=25$). Left: correlation measures at lag $h=1$ vs. $\beta/\mu$. Right: stationary density in each layer. The vertical dashed line at $1/\Lambda_T$ indicates the spectral activation threshold based on the largest eigenvalue of the supra-contact matrix (Eq. \ref{['eq:maxeigen']}). See caption in the original version for the full list of observables.
• Figure 5: Active--localized regime close to the AL--AD$_2$ crossover ($\eta/\beta=1$). Left: correlation measures at lag $h=1$ vs. $\beta/\mu$. Right: stationary density in each layer. Vertical dashed lines indicate the single-layer thresholds $1/\Lambda_1$ and $1/\Lambda_2$. See caption in the original version for the full list of observables.