Local Assortativity in Weighted and Directed Complex Networks

TL;DR

This paper introduces generalized measures of local assortativity for weighted and directed networks, demonstrating their equivalence and usefulness in analyzing network robustness and local structure.

Contribution

It proposes a unified framework for local assortativity measures applicable to weighted and directed networks, extending previous approaches and enabling detailed local analysis.

Findings

Measures are applicable to weighted and directed networks

The measures reveal local assortativity patterns related to edge weights

Application to real-world networks demonstrates practical usefulness

Abstract

Assortativity, i.e. the tendency of a vertex to bond with another based on their similarity, such as degree, is an important network characteristic that is well-known to be relevant for the network's robustness against attacks. Commonly it is analyzed on the global level, i.e. for the whole network. However, the local structure of assortativity is also of interest as it allows to assess which of the network's vertices and edges are the most endangering or the most protective ones. Hence, it is quite important to analyze the contribution of individual vertices and edges to the network's global assortativity. For unweighted networks M. Piraveenan, M. Prokopenko, and A. Y. Zomaya (2008, 2010) and Guo-Qing Zhang, Su-Qi Cheng, and Guo- Qiang Zhang (2012) suggest two allegedly different approaches to measure local assortativity. In this paper we show their equivalence and propose generalized local assortativity measures that are also applicable to weighted (un)directed networks. They allow to analyze the assortative behavior of edges and vertices as well as of entire network components. We illustrate the usefulness of our measures based on theoretical and real-world weighted networks and propose new local assortativity profiles, which provide, inter alia, information about the pattern of local assortativity with respect to edge weight.