All scale-free networks are sparse

TL;DR

This paper investigates the realizability of scale-free networks with specific degree sequences, revealing phase transitions at certain exponents that explain why large such networks are inherently sparse.

Contribution

It provides analytical and numerical insights into the conditions under which scale-free networks are realizable, highlighting a fundamental sparsity constraint.

Findings

Fraction of realizable sequences drops at exponent 0 and 2

Two first-order phase transitions in realizability

Large scale-free networks are inherently sparse

Abstract

We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.