Combinatorial study of degree assortativity in networks

Why are some networks degree-degree correlated (assortative), while most of the real-world ones are anticorrelated (disassortative)? Here, we prove, by combinatorial methods, that the assortativity of a network depends only on three structural factors: transitivity (clustering coefficient), intermodular connectivity, and branching. Then, a network is assortative if the contributions of the first two factors are larger than that of the third. Highly branched networks are likely to be disassortative.

Figure 1

(Left) Network of type $G^{\prime }$ obtained by linking a $k$-regular graph and a single node. This network is disassortative as indicated by its Pearson coefficient. (Right) Network of type $G^{\prime \prime }$ in which a $k$-regular graph is linked to another regular graph, e.g., a path of length 2. The network is assortative as indicated by its Pearson correlation coefficient.