Sample-to-sample fluctuations in real-network ensembles

TL;DR

This paper investigates how real network structures, especially low-degree nodes and communities, cause significant fluctuations in eigenvalues, challenging assumptions that ensemble averages accurately represent real network dynamics.

Contribution

It reveals that low-degree nodes and communities in real networks lead to complex, multimodal eigenvalue distributions, highlighting limitations of ensemble-based modeling approaches.

Findings

Eigenvalue distributions are often broad and multimodal in real networks.

Low-degree nodes (<3) and communities significantly influence eigenvalues.

Ensemble averages may not accurately reflect real network dynamics.

Abstract

Network modeling based on ensemble averages tacitly assumes that the networks meant to be modeled are typical in the ensemble. Previous research on network eigenvalues, which govern a range of dynamical phenomena, has shown that this is indeed the case for uncorrelated networks with minimum degree $\ge 3$. Here we focus on real networks, which generally have both structural correlations and low-degree nodes. We show that: (i) the ensemble distribution of the dynamically most important eigenvalues can be not only broad and far apart from the real eigenvalue but also highly structured, often with a multimodal rather than bell-shaped form; (ii) these interesting properties are found to be due to low-degree nodes, mainly those with degree $< 3$, and network communities, which is a common form of structural correlation found in real networks. In addition to having implications for ensemble-based approaches, this shows that low-degree nodes may have a stronger influence on collective dynamics than previously anticipated from the study of computer-generated networks.