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

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 $\alpha=4$. Finally, we study the transition between non-dominant and dominant layers as a function of the intra-layer activity parameter $\gamma$. 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.