Semilocalization for inhomogeneous random graphs

TL;DR

This paper studies eigenvector localization in inhomogeneous random graphs, revealing semilocalization near spectrum edges and localization at extremal eigenvalues, using a novel pruning method and local coupling techniques.

Contribution

Introduces a new pruning procedure and local coupling approach to analyze eigenvector localization in inhomogeneous random graphs.

Findings

Eigenvectors near spectrum edges are semilocalized around small vertex sets.

Extremal eigenvalues exhibit localization around single vertices.

The pruning method effectively handles highly inhomogeneous degree distributions.

Abstract

We analyse the eigenvectors of the adjacency matrix of a random inhomogeneous graph constructed from a specified degree sequence. We assume that the empirical degree sequence has bounded mean and variance. We show that near the edges of the spectrum, the eigenvectors are semilocalized in the sense that their mass concentrates around a small set of resonant vertices. For the extremal eigenvalues, we establish localization around a single vertex. In order to obtain effective estimates in the presence of highly inhomogeneous degrees, we introduce a new economical pruning procedure that carefully extracts a forest from the original graph, whose adjacency matrix is compared to that of the original graph using a suitably constructed local coupling to random trees with independent edges.