First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation

TL;DR

This paper investigates the largest eigenvalue and eigenvector of large sparse symmetric matrices with bimodal degree distributions, developing approximation schemes to understand their properties and finite size effects.

Contribution

It introduces approximation methods based on solvable examples to analyze eigenvalues and eigenvectors in complex sparse matrices with degree fluctuations.

Findings

Approximation schemes align well with numerical results for large positive biases.

Large finite size effects are explained when the bias is small.

The methods provide qualitative insights into eigenvalue behavior in sparse matrices.

Abstract

The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to accurately calculate them by solving a functional equation concerning auxiliary fields which come out in an analysis based on replica/cavity methods. However, the difficulty in analytically solving this equation makes an accurate calculation infeasible in practice. To overcome this problem, we develop approximation schemes on the basis of two exceptionally solvable examples. The schemes are reasonably consistent with numerical experiments when the statistical bias of positive matrix entries is sufficiently large, and they qualitatively explain why considerably large finite size effects of the first eigenvalue can be observed when the bias is relatively small.