Quantitative modeling of degree-degree correlation in complex networks

This paper presents an approach to the modeling of degree-degree correlation in complex networks. Thus, a simple function, Δ(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 Δ(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.