Ensemble averageability in network spectra

TL;DR

This paper investigates whether the extreme eigenvalues of large networks can be accurately represented by ensemble averages, demonstrating that in scale-free networks these eigenvalues become sharply peaked as network size grows, which has implications for network dynamics.

Contribution

It provides the first explicit validation of ensemble averageability for extreme eigenvalues in large scale-free networks, linking spectral properties to dynamical processes.

Findings

Ensemble distributions of extreme eigenvalues converge to peaked distributions with increasing network size.

The results are significant for understanding synchronization and epidemic spreading.

Eigenvalue fluctuations diminish in large scale-free networks.

Abstract

The extreme eigenvalues of connectivity matrices govern the influence of the network structure on a number of network dynamical processes. A fundamental open question is whether the eigenvalues of large networks are well represented by ensemble averages. Here we investigate this question explicitly and validate the concept of ensemble averageability in random scale-free networks by showing that the ensemble distributions of extreme eigenvalues converge to peaked distributions as the system size increases. We discuss the significance of this result using synchronization and epidemic spreading as example processes.