Rate equation approach for correlations in growing network models

TL;DR

This paper introduces a rate equation method to analyze two-vertex correlations in scale-free growing networks, extending previous clustering spectrum work to include degree correlations and weighted networks.

Contribution

It develops a novel rate equation formalism for computing degree correlations in growing networks, applicable to models with high clustering and weighted interactions.

Findings

Derived explicit formulas for degree correlations in preferential attachment models

Validated the approach with extensive numerical simulations

Extended the method to weighted networks considering interaction strengths

Abstract

We propose a rate equation approach to compute two vertex correlations in scale-free growing network models based in the preferential attachment mechanism. The formalism, based on previous work of Szab\'o \textit{et al.} [Phys. Rev. E \textbf{67} 056102 (2002)] for the clustering spectrum, measuring three vertex correlations, is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree $k$, with an appropriate boundary condition. We study the properties of both two and three vertex correlations for linear preferential attachment models, and also for a model yielding a large clustering coefficient. The expressions obtained are checked by means of extensive numerical simulations. The rate equation proposed can be generalized to more sophisticated growing network models, and also extended to deal with related correlation measures. As an example, we consider the case of a recently proposed model of weighted networks, for which we are able to compute a weighted two vertex correlation function, taking into account the strength of the interactions between connected vertices.