Structural transitions in scale-free networks

TL;DR

This paper develops a scaling theory for generalized scale-free network models, explaining their clustering behavior and matching real network data through mean-field analysis.

Contribution

It introduces a modified preferential attachment model capturing clustering and small-world properties, supported by a mean-field theory and numerical validation.

Findings

Clustering coefficient C(k) scales as 1/k for degree k

Numerical results agree with mean-field predictions

Model reproduces clustering observed in real networks

Abstract

Real growing networks like the WWW or personal connection based networks are characterized by a high degree of clustering, in addition to the small-world property and the absence of a characteristic scale. Appropriate modifications of the (Barabasi-Albert) preferential attachment network growth capture all these aspects. We present a scaling theory to describe the behavior of the generalized models and the mean field rate equation for the problem. This is solved for a specific case with the result C(k) ~ 1/k for the clustering of a node of degree k. Numerical results agree with such a mean-field exponent which also reproduces the clustering of many real networks.