Isomorphisms in Multilayer Networks

TL;DR

This paper generalizes graph isomorphisms to multilayer networks with multiple aspects, enabling the use of existing graph isomorphism tools for multilayer network analysis and extending various network analysis methods.

Contribution

It introduces a comprehensive framework for multilayer network isomorphisms, reducing them to graph isomorphism problems, and broadens the applicability of network analysis techniques.

Findings

Multiple types of multilayer isomorphisms identified

Multilayer isomorphism problems reduced to linear-sized graph isomorphism problems

Framework enables extending analysis methods to multilayer networks

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.