The rank of diluted random graphs

TL;DR

This paper derives an explicit formula for the asymptotic multiplicity of the eigenvalue zero in the adjacency matrices of large diluted random graphs, linking spectral properties to the local structure of the limiting Galton--Watson tree.

Contribution

It provides a novel explicit formula connecting the eigenvalue zero multiplicity to the degree distribution of the limiting Galton--Watson tree and analyzes the spectral measure of the adjacency operator.

Findings

Explicit formula for zero eigenvalue multiplicity in large graphs

Self-adjointness of the adjacency operator on the Galton--Watson tree

Conditions under which the atomic mass at zero matches the kernel dimension limit

Abstract

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs $(G_n)_{n\geq0}$ converging locally to a Galton--Watson tree $T$ (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function $\phi_*$ of $T$. In the first part, we show that the adjacency operator associated with $T$ is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on $\phi_*$ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of $(G_n)_{n\geq 0}$. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.