Localization transition on complex networks via spectral statistics

TL;DR

This paper investigates the spectral properties of complex networks, revealing a disorder-driven localization transition similar to Anderson's metal-insulator transition, with the transition's nature depending on network connectivity.

Contribution

It demonstrates the existence of a localization transition in various complex networks and characterizes how connectivity influences the transition and disorder requirements.

Findings

Localization transition observed in networks with small average connectivity

Critical index matches mean field predictions

High connectivity networks lack a clear transition due to compact structure

Abstract

The spectral statistics of complex networks are numerically studied. The features of the Anderson metal-insulator transition are found to be similar for a wide range of different networks. A metal-insulator transition as a function of the disorder can be observed for different classes of complex networks for which the average connectivity is small. The critical index of the transition corresponds to the mean field expectation. When the connectivity is higher, the amount of disorder needed to reach a certain degree of localization is proportional to the average connectivity, though a precise transition cannot be identified. The absence of a clear transition at high connectivity is probably due to the very compact structure of the highly connected networks, resulting in a small diameter even for a large number of sites.