Cavity and replica methods for the spectral density of sparse symmetric random matrices

TL;DR

This paper provides a comprehensive pedagogical overview of methods to compute the spectral density of sparse symmetric random matrices, focusing on cavity and replica techniques, and illustrating their implementation and conceptual nuances.

Contribution

It offers a detailed, accessible review of cavity and replica methods for spectral density calculation, including algorithmic implementation and conceptual insights, aimed at students and researchers.

Findings

Derivation of recursive equations for spectral density

Equivalence of cavity and replica solutions

Implementation of stochastic population dynamics algorithm

Abstract

We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.