Classifying complex networks using unbiased local assortativity

Assortativity is a network-level measure which quantifies the tendency of nodes to mix with similar nodes in a network. Local assortativity has been introduced as a measure to analyse the contribution of individual nodes to network assortativity. In this paper we argue that there is a bias in the formulation of local assortativity which favours low-degree nodes. We show that, after the bias is removed, local assortativity of a node can be interpreted as a scaled difference between the average excess degree of the node neighbours and the expected excess degree of the network as a whole. Finally, we study the local assortativity profiles of a number of model and real world networks, demonstrating that four classes of complex networks exist: (i) assortative networks with disassortative hubs, (ii) assortative networks with assortative hubs, (iii) disassortative networks with disassortative hubs, and (iv) disassortative networks with assortative hubs.