Isomorphisms in Multilayer Networks

We extend the concept of graph isomorphisms to multilayer networks, and we identify multiple types of isomorphisms. For example, in multilayer networks with a single “aspect” (i.e., type of layering), permuting vertex labels, layer labels, and both of types of layers each yield a different type of isomorphism. We discuss how multilayer network isomorphisms naturally lead to defining isomorphisms in any type of network that can be represented as a multilayer network. This thereby yields isomorphisms for multiplex networks, temporal networks, networks with both such features, and more. We reduce each of the multilayer network isomorphism problems to a graph isomorphism problem, and we use this reduction to prove that the multilayer network isomorphism problem is computationally equally hard as the graph 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.