Statistical field theory of random graphs with prescribed degrees

TL;DR

This paper develops a statistical field theory framework to analyze the spectral properties of large random graphs with prescribed degree sequences, extending previous methods to more complex degree distributions.

Contribution

It introduces a novel field theory approach for graphs with prescribed degrees, deriving explicit equations for eigenvalue distributions, including regular and mixed-regular graphs.

Findings

Reproduces empirical eigenvalue distributions with high accuracy

Derives explicit equations for spectral densities of prescribed degree graphs

Extends field theory methods to complex degree distributions

Abstract

Statistical field theory methods have been very successful with a number of random graph and random matrix problems, but it is challenging to apply these methods to graphs with prescribed degree sequences due to the extensive number of constraints that enforce the desired degree at each vertex. Building on top of recent results where similar methods are applied to random regular graph counting, we develop an accurate statistical field theory description for the adjacency matrix spectra of graphs with prescribed degrees. For large graphs, the expectation values are dominated by functional saddle points satisfying explicit equations. For the case of regular graphs, this immediately leads to the known McKay distribution. We then consider mixed-regular graphs with N1 vertices of degree d1, N2 vertices of degree d2, etc, such that the ratios N_i/N are kept fixed as N goes to infinity. For such graphs, the eigenvalue densities are governed by nonlinear integral equations of the Hammerstein type. Solving these equations numerically reproduces with an excellent accuracy the empirical eigenvalue distributions.