Quantitative modeling of degree-degree correlation in complex networks

TL;DR

This paper introduces a mathematical framework for modeling degree-degree correlations in complex networks, effectively capturing assortative and disassortative patterns and validated on real-world network data.

Contribution

It develops a family of models using group theory to accurately describe degree correlations in various types of networks.

Findings

Model predicts nonuniform degree correlation distributions

Identifies two assortative and two disassortative zones

Quantitatively fits real-world network data

Abstract

This paper presents an approach to the modeling of degree-degree correlation in complex networks. Thus, a simple function, \Delta(k', k), describing specific degree-to- degree correlations is considered. The function is well suited to graphically depict assortative and disassortative variations within networks. To quantify degree correlation variations, the joint probability distribution between nodes with arbitrary degrees, P(k', k), is used. Introduction of the end-degree probability function as a basic variable allows using group theory to derive mathematical models for P(k', k). In this form, an expression, representing a family of seven models, is constructed with the needed normalization conditions. Applied to \Delta(k', k), this expression predicts a nonuniform distribution of degree correlation in networks, organized in two assortative and two disassortative zones. This structure is actually observed in a set of four modeled, technological, social, and biological networks. A regression study performed on 15 actual networks shows that the model describes quantitatively degree-degree correlation variations.