Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

TL;DR

This paper introduces a new eigenpair localization method using Frobenius inner products, enhancing community detection in networks by providing improved spectral guarantees under stochastic block models.

Contribution

It presents a novel localization result for eigenvalues and eigenvectors using Frobenius inner products, extending Perron-Frobenius theory and improving community detection guarantees.

Findings

The method generalizes Perron-Frobenius properties to matrices with negative entries.

Using the all-ones matrix X estimates the eigenvector signature.

Provides new quality guarantees for spectral community detection.

Abstract

We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$. Different choices for $X$ may be used, depending upon the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$, generalizing previous results on Perron-Frobenius properties of matrices with some negative entries. As another application we consider the problem of community detection in graphs and networks. The problem is solved by means of modularity-based spectral techniques, following the ideas pioneered by Miroslav Fiedler in mid 70s. We show that a suitable choice of $X$ can be used to provide new quality guarantees of those techniques, when the network follows a stochastic block model.