Perturbing General Uncorrelated Networks

TL;DR

This paper extends previous work on Erd"os-Rényi graphs to general uncorrelated networks, revealing that most results are universal except for scale-free graphs which show unique singular behaviors.

Contribution

It generalizes the analysis of perturbed random graphs to a broader class of uncorrelated networks, highlighting special properties of scale-free graphs.

Findings

Results are largely universal across uncorrelated networks.

Scale-free graphs exhibit unique singular behaviors.

Analytical and numerical studies confirm the generality and exceptions.

Abstract

This paper is a direct continuation of an earlier work, where we studied Erd\"os-R\'enyi random graphs perturbed by an interaction Hamiltonian favouring the formation of short cycles. Here, we generalize these results. We keep the same interaction Hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution. It is shown that the results obtained for Erd\"os-R\'enyi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behaviour, studied thoroughly in this paper, both analytically and numerically.