Rare events statistics of random walks on networks: localization and other dynamical phase transitions

TL;DR

This paper explores rare event statistics in random walks on complex networks using large deviations theory, revealing localization and mode-switching phase transitions influenced by the network structure and observables.

Contribution

It introduces a spectral analysis approach to rare events in network random walks, identifying localization and mode-switching phase transitions, with analytical approximations validated numerically.

Findings

Identification of localization in rare event modes.

Discovery of mode-switching phase transitions.

Analytical approximation matching numerical results.

Abstract

Rare event statistics for random walks on complex networks are investigated using the large deviations formalism. Within this formalism, rare events are realized as typical events in a suitably deformed path-ensemble, and their statistics can be studied in terms of spectral properties of a deformed Markov transition matrix. We observe two different types of phase transition in such systems: (i) rare events which are singled out for sufficiently large values of the deformation parameter may correspond to {\em localized\/} modes of the deformed transition matrix, (ii) "mode-switching transitions" may occur as the deformation parameter is varied. Details depend on the nature of the observable for which the rare event statistics is studied, as well as on the underlying graph ensemble. In the present letter we report on the statistics of the average degree of the nodes visited along a random walk trajectory in Erd\H{o}s-R\'enyi networks. Large deviations rate functions and localization properties are studied numerically. For observables of the type considered here, we also derive an analytical approximation for the Legendre transform of the large-deviations rate function, which is valid in the large connectivity limit. It is found to agree well with simulations.