# Spectra of random networks with arbitrary degrees

## TL;DR

This paper introduces a message passing approach to efficiently compute the spectral density of large, locally tree-like random networks with arbitrary degree distributions, including bounds on spectral edges.

## Contribution

It develops a novel message passing method and an approximation for spectral density in configuration model networks, enabling fast analysis of their spectral properties.

## Key findings

- The approximation performs well for most network densities.
- Bounds on spectral band edges help identify structural phase transitions.
- The method is applicable to a wide range of degree distributions.

## Abstract

We derive a message passing method for computing the spectra of locally tree-like networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs with arbitrary node degrees -- the so-called configuration model. We find the latter approximation to work well for all but the sparsest of networks. We also derive bounds on the position of the band edges of the spectrum, which are important for identifying structural phase transitions in networks.