  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.
