"Isomorphisms in Multilayer Networks"

This paper first appeared on the arXiv in 2015 and was published in advanced access about a year ago. It finally has its final volume and page coordinates, so I am finally writing a blog entry about. At this stage, the paper certainly doesn't feel "new" anymore, but it has some useful ideas in it, and it's also my first paper in an IEEE journal. This paper is also rather unusual for me, in that my coauthor Mikko Kivelä and I decided that the clearest way to present things would be to write the paper in definition–theorem–proof format. I almost never write papers that way. Anyway, here are a few details.

Title: Isomorphisms in Multilayer Networks

Authors: Mikko Kivelä and Mason A. Porter

Abstract: We extend the concept of graph isomorphisms to multilayer networks with any number of “aspects” (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks with a single aspect, permuting vertex labels, layer labels, and both vertex labels and layer labels each yield different isomorphism relations between multilayer networks. Multilayer network isomorphisms lead naturally to defining isomorphisms in any of the numerous types of networks that can be represented as a multilayer network, and we thereby obtain isomorphisms for multiplex networks, temporal networks, networks with both of these features, and more. We reduce each of the multilayer network isomorphism problems to a graph isomorphism problem, where the size of the graph isomorphism problem grows linearly with the size of the multilayer network isomorphism problem. One can thus use software that has been developed to solve graph isomorphism problems as a practical means for solving multilayer network isomorphism problems. Our theory lays a foundation for extending many network analysis methods—including motifs, graphlets, structural roles, and network alignment—to any multilayer network.