Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

TL;DR

This paper introduces a cavity method-based approach to analyze the largest eigenvalue and eigenvector of large symmetric sparse matrices, providing a new analytical framework validated through examples.

Contribution

It develops a novel cavity method framework for eigenvalue problems in large sparse matrices, combining statistical mechanics and optimization techniques.

Findings

Method accurately predicts the first eigenvalue in example cases.

Cavity field distribution determines the eigenvalue.

Framework validated with analytical and numerical results.

Abstract

A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.