  Multi-layer networks are networks on a set of entities (nodes) with multiple
types of relations (edges) among them where each type of relation/interaction
is represented as a network layer. As with single layer networks, community
detection is an important task in multi-layer networks. A large group of
popular community detection methods in networks are based on optimizing a
quality function known as the modularity score, which is a measure of presence
of modules or communities in networks. Hence a first step in community
detection is defining a suitable modularity score that is appropriate for the
network in question. Here we introduce several multi-layer network modularity
measures under different null models of the network, motivated by empirical
observations in networks from a diverse field of applications. In particular we
define the multi-layer configuration model, the multi-layer expected degree
model and their various modifications as null models for multi-layer networks
to derive different modularities. The proposed modularities are grouped into
two categories. The first category, which is based on degree corrected
multi-layer stochastic block model, has the multi-layer expected degree model
as their null model. The second category, which is based on multi-layer
extensions of Newman-Girvan modularity, has the multi-layer configuration model
as their null model. These measures are then optimized to detect the optimal
community assignment of nodes. We compare the effectiveness of the measures in
community detection in simulated networks and then apply them to four real
networks.
