  To understand the structure of a network, it can be useful to break it down
into its constituent pieces. This is the approach taken in a multitude of
successful network analysis methods, such as motif analysis. These methods
require one to enumerate or sample small connected subgraphs of a network,
which can be computationally intractable if naive methods are used. Efficient
algorithms exists for both enumeration and uniform sampling of subgraphs, and
here we generalize the ESU algorithm for a very general notion of multilayer
networks. We show that multilayer network subnetwork enumeration introduces
nontrivial complications to the existing algorithm, and present two different
generalized algorithms that preserve the desired features of unbiased sampling
and trivial parallelization. We evaluate these algorithms in synthetic networks
and with real-world data, and show that neither of the algorithms is strictly
more efficient but rather the choice depends on the features of the data.
Having a general algorithm for finding subnetworks makes advanced multilayer
network analysis possible, and enables researchers to apply a variety of
methods to previously difficult-to-handle multilayer networks in a variety of
domains and across many different types of multilayer networks.
