Eigenvector Localization and Universal Regime Transitions in Multiplex Networks: A Perturbative Approach

TL;DR

A unified spectral framework for understanding localization and dominance transitions in multiplex contagion dynamics is provided, supported by dynamical simulations that highlight distinct susceptibility patterns across regimes.

Abstract

We study the transition between layer-localized and delocalized regimes in a general contact-based contagion model on multiplex networks. Using the inverse participation ratio, we characterize how activity shifts from being confined to a single layer to spreading across the entire system. Through a first-order perturbative analysis of the leading eigenvector of the supra-contact probability matrix, we derive an analytical expression for the fictive coupling $p^*$ that marks the crossover between the two regimes. This result reproduces and explains previously observed numerical scalings and extends them to a broad class of contact-based processes beyond the Susceptible-Infected-Susceptible model. We also obtain an analytical expression for the IPR of the non-dominant layer in the localized regime, confirming its power-law dependence on the coupling with exponent $α=4$. Finally, we study the transition between non-dominant and dominant layers as a function of the intra-layer activity parameter $γ$. Our analytical findings are supported by dynamical simulations that highlight distinct susceptibility patterns across regimes. Altogether, this work provides a unified spectral framework for understanding localization and dominance transitions in multiplex contagion dynamics.

Figures (3)

• Figure 1: Inverse participation ratio (IPR) for the dominant (dotted line) and non-dominant (solid line) layers as a function of the coupling $p$ (left) and the rescaled coupling $p/p^*$ (right). Top panels: random regular networks with $k_1 = 10$, $k_2 = 8$, and 100 nodes. Bottom panels: Erdős--Rényi networks with $\langle k_1 \rangle = 10$, $\langle k_2 \rangle = 8$, and 1000 nodes. The blue line corresponds to the perturbative result given by Eq. (\ref{['Ipr2approx']})
• Figure 2: Representation of the inverse participation ratio $\mathrm{IPR}_2$ as a function of $\gamma$ for a two-layer multiplex network with fixed coupling $p = 0.1$. One layer has $\gamma_1 \to \infty$, while the other has variable $\gamma_2 = \gamma$. Both layers are random regular networks with 100 nodes, $k_1 = 10$, and $k_2 = 12$. The vertical dotted line indicates the transition point at $\gamma^* = 20.59$.
• Figure 3: Top panels: modified susceptibility $\chi$ as a function of $\beta/\mu$ for $\eta = 0.1$. Layer 1 has $k_1 = 10$, $\Lambda_1 \approx 11$, and $\gamma_1 \to \infty$; layer 2 has $k_2 = 30$, $\Lambda_2 \approx 31$, and variable $\gamma_2$. From left to right: (i) dominance of layer 1 ($\gamma_2 = 5$); (ii) near the dominance transition ($\gamma_2 = 13.5$); (iii) dominance of layer 2 ($\gamma_2 = 100$). The transition occurs at $\gamma^* = 13.47$. Bottom panels: modified susceptibility $\chi$ as a function of $\beta/\mu$ for different values of the coupling $\eta$. Layer 1 has $k_1 = 30$, $\Lambda_1 \approx 31$, and $\gamma_1 \to \infty$; layer 2 has $k_2 = 10$, $\Lambda_2 \approx 11$, and $\gamma_2 = 10^4$. From left to right: (iv) localized regime ($\eta = 0.01$); (v) near the localization transition ($\eta = 15$); (vi) delocalized regime ($\eta = 50$). The fictive coupling is $p^* = 20$.