Relevance of the minimum degree to dynamic fluctuation in strongly heterogeneous networks

TL;DR

This paper investigates how the minimum degree influences dynamic fluctuations in strongly heterogeneous networks, revealing that degree-one nodes significantly affect fluctuation scaling and correlation among hubs.

Contribution

It demonstrates the crucial role of minimum degree in fluctuation behavior and clarifies conflicting results in previous studies on heterogeneous networks.

Findings

Global fluctuation diverges algebraically with system size when minimum degree is one.

Global fluctuation diverges logarithmically with system size when minimum degree is two.

Local fluctuation growth depends linearly or sub-linearly on node degree, affecting network correlations.

Abstract

The fluctuation of dynamic variables in complex networks is known to depend on the dimension and the heterogeneity of the substrate networks. Previous studies, however, have reported inconsistent results for the scaling behavior of fluctuation in strongly heterogeneous networks. To understand the origin of this conflict, we study the dynamic fluctuation on scale-free networks with a common small degree exponent but different mean degrees and minimum degrees constructed by using the configuration model and the static model. It turns out that the global fluctuation of dynamic variables diverges algebraically and logarithmically with the system size when the minimum degree is one and two, respectively. Such different global fluctuations are traced back to different, linear and sub-linear, growth of local fluctuation at individual nodes with their degrees, implying a crucial role of degree-one nodes in controlling correlation between distinct hubs.