Link-space formalism for network analysis

TL;DR

The paper introduces the link-space formalism for analyzing degree-degree correlations in networks, providing analytical solutions for various models and exploring the effects of correlations on network properties.

Contribution

It presents a novel formalism for detailed network analysis, enabling analytical solutions for degree correlations in multiple network models.

Findings

Analytical solutions for link-space matrices in key network models.

Demonstration of how correlations affect degree distributions.

Derivation of non-assortative network structures for arbitrary degree distributions.

Abstract

We introduce the link-space formalism for analyzing network models with degree-degree correlations. The formalism is based on a statistical description of the fraction of links l_{i,j} connecting nodes of degrees i and j. To demonstrate its use, we apply the framework to some pedagogical network models, namely, random-attachment, Barabasi-Albert preferential attachment and the classical Erdos and Renyi random graph. For these three models the link-space matrix can be solved analytically. We apply the formalism to a simple one-parameter growing network model whose numerical solution exemplifies the effect of degree-degree correlations for the resulting degree distribution. We also employ the formalism to derive the degree distributions of two very simple network decay models, more specifically, that of random link deletion and random node deletion. The formalism allows detailed analysis of the correlations within networks and we also employ it to derive the form of a perfectly non-assortative network for arbitrary degree distribution.