Higher order assortativity in complex networks

TL;DR

This paper introduces a theoretical extension of assortativity in complex networks to higher orders, allowing for the analysis of node connection patterns beyond immediate neighbors, which can reveal new network features.

Contribution

The paper develops a novel higher order assortativity measure extending Newman’s index, applicable to paths, shortest paths, and random walks, enriching network analysis tools.

Findings

Higher order assortativity can distinguish networks with identical first-order assortativity.

The new measures can describe complex network features not captured by traditional assortativity.

Application to real networks demonstrates the utility of higher order measures.

Abstract

Assortativity was first introduced by Newman and has been extensively studied and applied to many real world networked systems since then. Assortativity is a graph metrics and describes the tendency of high degree nodes to be directly connected to high degree nodes and low degree nodes to low degree nodes. It can be interpreted as a first order measure of the connection between nodes, i.e. the first autocorrelation of the degree-degree vector. Even though assortativity has been used so extensively, to the author's knowledge, no attempt has been made to extend it theoretically. This is the scope of our paper. We will introduce higher order assortativity by extending the Newman index based on a suitable choice of the matrix driving the connections. Higher order assortativity will be defined for paths, shortest paths, random walks of a given time length, connecting any couple of nodes. The Newman assortativity is achieved for each of these measures when the matrix is the adjacency matrix, or, in other words, the correlation is of order 1. Our higher order assortativity indexes can be used for describing a variety of real networks, help discriminating networks having the same Newman index and may reveal new topological network features.