Assortative mixing by degree makes a network more unstable

TL;DR

This paper studies how degree correlation affects network stability, finding that assortative mixing decreases stability and disassortative networks are more stable, especially in scale-free models.

Contribution

It demonstrates that assortative mixing by degree reduces network stability, providing spectral analysis insights into the stability differences between assortative and disassortative networks.

Findings

Assortative networks have higher spectral radius, indicating less stability.

Disassortative networks exhibit logarithmic scaling of eigenvalues with network size.

Scale-free networks' stability depends on degree correlation, influencing their prevalence in nature.

Abstract

We investigate the role of degree correlation among nodes on the stability of complex networks, by studying spectral properties of randomly weighted matrices constructed from directed Erd\"{o}s-R\'enyi and scale-free random graph models. We focus on the behaviour of the largest real part of the eigenvalues, $\lambda_\text{max}$, that governs the growth rate of perturbations about an equilibrium (and hence, determines stability). We find that assortative mixing by degree, where nodes with many links connect preferentially to other nodes with many links, reduces the stability of networks. In particular, for sparse scale-free networks with $N$ nodes, $\lambda_\text{max}$ scales as $N^\alpha$ for highly assortative networks, while for disassortative graphs, $\lambda_\text{max}$ scales logarithmically with $N$. This difference may be a possible reason for the prevalence of disassortative networks in nature.