Dissecting localization phenomena of dynamical processes on networks

TL;DR

This paper introduces a generic method to quantify localization in dynamical processes on networks, revealing complex patterns and the coexistence of localized and delocalized states near critical transitions.

Contribution

It advances the analysis of localization phenomena by providing a new, versatile approach applicable to theory, simulations, and real data, and explores these phenomena across different network types.

Findings

Localized states can coexist with endemic phases near transitions.

Epidemic prevalence is influenced by delocalized network components.

Different critical exponents can share similar localization patterns.

Abstract

Localization phenomena permeate many branches of physics playing a fundamental role on dynamical processes evolving on heterogeneous networks. These localization analyses are frequently grounded, for example, on eigenvectors of adjacency or non-backtracking matrices which emerge in theories of dynamic processes near to an active to inactive transition. We advance in this problem gauging nodal activity to quantify the localization in dynamical processes on networks whether they are near to a transition or not. The method is generic and applicable to theory, stochastic simulations, and real data. We investigate spreading processes on a wide spectrum of networks, both analytically and numerically, showing that nodal activity can present complex patterns depending on the network structure. Using annealed networks we show that a localized state at the transition and an endemic phase just above it are not incompatible features of a spreading process. We also report that epidemic prevalence near to the transition is determined by the delocalized component of the network even when the analysis of the inverse participation ratio indicates a localized activity. Also, dynamical processes with distinct critical exponents can be described by the same localization pattern. Turning to quenched networks, a more complex picture, depending on the type of activation and on the range of degree exponent, is observed and discussed. Our work paves an important path for investigation of localized activity in spreading and other processes on networks.