Structure of triadic relations in multiplex networks

Abstract

Recent advances in the study of networked systems have highlighted that our\ninterconnected world is composed of networks that are coupled to each other\nthrough different "layers" that each represent one of many possible subsystems\nor types of interactions. Nevertheless, it is traditional to aggregate\nmultilayer networks into a single weighted network in order to take advantage\nof existing tools. This is admittedly convenient, but it is also extremely\nproblematic, as important information can be lost as a result. It is therefore\nimportant to develop multilayer generalizations of network concepts. In this\npaper, we analyze triadic relations and generalize the idea of transitivity to\nmultiplex networks. By focusing on triadic relations, which yield the simplest\ntype of transitivity, we generalize the concept and computation of clustering\ncoefficients to multiplex networks. We show how the layered structure of such\nnetworks introduces a new degree of freedom that has a fundamental effect on\ntransitivity. We compute multiplex clustering coefficients for several real\nmultiplex networks and illustrate why one must take great care when\ngeneralizing standard network concepts to multiplex networks. We also derive\nanalytical expressions for our clustering coefficients for ensemble averages of\nnetworks in a family of random multiplex networks. Our analysis illustrates\nthat social networks have a strong tendency to promote redundancy by closing\ntriads at every layer and that they thereby have a different type of multiplex\ntransitivity from transportation networks, which do not exhibit such a\ntendency. These insights are invisible if one only studies aggregated networks.\n