Asymptotics of the spectral radius for directed Chung-Lu random graphs with community structure

TL;DR

This paper derives new concentration results for the spectral radius of directed Chung-Lu random graphs, including those with community structure, impacting understanding of network stability and dynamics.

Contribution

It provides the first concentration results for the spectral radius of directed Chung-Lu graphs, including models with community structure, applicable to finite and asymptotic networks.

Findings

Concentration results hold for finite and large networks.

Extended to models with community structure.

Implications for network stability and dynamics.

Abstract

The spectral radius of the adjacency matrix can impact both algorithmic efficiency as well as the stability of solutions to an underlying dynamical process. Although much research has considered the distribution of the spectral radius for undirected random graph models, as symmetric adjacency matrices are amenable to spectral analysis, very little work has focused on directed graphs. Consequently, we provide novel concentration results for the spectral radius of the directed Chung-Lu random graph model. We emphasize that our concentration results are applicable both asymptotically and to networks of finite size. Subsequently, we extend our concentration results to a generalization of the directed Chung-Lu model that allows for community structure.