Fat-tailed distribution derived from first eigenvector of symmetric random sparse matrix

TL;DR

This paper investigates the distribution of the first eigenvector of symmetric random sparse matrices, revealing fat-tailed distributions in networks with mixed degrees and Gaussian distributions in uniform degree networks, using the cavity method.

Contribution

It introduces a novel analysis of eigenvector distributions in sparse matrices with different degree structures using the cavity method.

Findings

Eigenvector distribution has fat tails in mixed-degree networks.

Eigenvector distribution is approximately Gaussian in uniform-degree networks.

The cavity method effectively analyzes eigenvector properties in sparse matrices.

Abstract

Many solutions for scientific problems rely on finding the first (largest) eigenvalue and eigenvector of a particular matrix. We explore the distribution of the first eigenvector of a symmetric random sparse matrix. To analyze the properties of the first eigenvalue/vector, we employ a methodology based on the cavity method, a well-established technique in the statistical physics. A symmetric random sparse matrix in this paper can be regarded as an adjacency matrix for a network. We show that if a network is constructed by nodes that have two different types of degrees then the distribution of its eigenvector has fat tails such as the stable distribution ($\alpha < 2 $) under a certain condition; whereas if a network is constructed with nodes that have only one type of degree, the distribution of its first eigenvector becomes the Gaussian approximately. The cavity method is used to clarify these results.