Degree distribution of complex networks from statistical mechanics principles

TL;DR

This paper derives the emergence of scale-free degree distributions in complex networks using statistical mechanics, showing how energy considerations explain their prevalence and properties.

Contribution

It introduces an energy framework based on network indistinguishability to analyze degree distributions, providing a novel statistical mechanics perspective.

Findings

Scale-free distributions have higher energy than regular graphs.

Number of distinguishable graphs is suppressed for large networks with  2.

Energy differences explain the emergence of scale-free structures.

Abstract

In this paper we describe the emergence of scale-free degree distributions from statistical mechanics principles. We define an energy associated to a degree sequence as the logarithm of the number of indistinguishable simple networks it is possible to draw given the degree sequence. Keeping fixed the total number of nodes and links, we show that the energy of scale-free distribution is much higher than the energy associated to the degree sequence of regular random graphs. This results unable us to estimate the annealed average of the number of distinguishable simple graphs it is possible to draw given a scale-free distribution with structural cutoff. In particular we shaw that this number for large networks is strongly suppressed for power -law exponent \gamma->2.