Cycles and Clustering in Multiplex Networks

TL;DR

This paper introduces a new classification and analytical framework for cycles in multiplex networks, accounting for layer-specific edges, degree distributions, and correlations, providing insights into clustering and cycle structure.

Contribution

It defines a comprehensive cycle classification in multiplex networks and derives expected cycle counts considering degree correlations, advancing understanding of multiplex network topology.

Findings

Degree correlations significantly influence cycle counts.

Assortative correlations increase the number of cycles and switches.

Disassortative correlations decrease cycle and switch counts.

Abstract

In multiplex networks, cycles cannot be characterized only by their length, as edges may occur in different layers in different combinations. We define a classification of cycles by the number of edges in each layer and the number of switches between layers. We calculate the expected number of cycles of each type in the configuration model of a large sparse multiplex network. Our method accounts for the full degree distribution including correlations between degrees in different layers. In particular, we obtain the numbers of cycles of length 3 of all possible types. Using these, we give a complete set of clustering coefficients and their expected values. We show that correlations between the degrees of a vertex in different layers strongly affect the number of cycles of a given type, and the number of switches between layers. Both increase with assortative correlations and are strongly decreased by disassortative correlations. The effect of correlations on clustering coefficients is equally pronounced.