Eigenvector dynamics under perturbation of modular networks

TL;DR

This paper studies how eigenvectors of modular network adjacency matrices rotate under perturbations, providing a method to estimate the number of modules and deriving detectability limits for sparse networks.

Contribution

It introduces a novel analysis of eigenvector rotation dynamics under perturbations and derives theoretical detectability limits for modular network clustering.

Findings

Eigenvectors corresponding to the largest eigenvalues form a community eigenspace that rotates together.

A method to estimate the number of modules independently of algorithms.

Identification of a 'band' where cluster detection is difficult before the detectability limit.

Abstract

Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of $q$ communities, the number of eigenvectors corresponding to the $q$ largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modules or clusters in a network can be estimated in an algorithm-independent way. A general argument and derivation for the theoretical detectability limit for sparse modular networks with $q$ communities is presented, beyond which modularity persists in the system but cannot be detected. It is shown that for detecting the clusters or modules using the adjacency matrix, there is a "band" in which it is hard to detect the clusters even before the theoretical detectability limit is reached, and for which the theoretically predicted detectability limit forms the sufficient upper bound. Analytic estimations of these bounds are presented, and empirically demonstrated.