  Modularity for multilayer networks, also called multislice modularity, is
parametric to a resolution factor and an inter-layer coupling factor. The
former is useful to express layer-specific relevance and the latter quantifies
the strength of node linkage across the layers of a network. However, such
parameters can be set arbitrarily, thus discarding any structure information at
graph or community level. Other issues are related to the inability of properly
modeling order relations over the layers, which is required for dynamic
networks.
  In this paper we propose a new definition of modularity for multilayer
networks that aims to overcome major issues of existing multislice modularity.
We revise the role and semantics of the layer-specific resolution and
inter-layer coupling terms, and define parameter-free unsupervised approaches
for their computation, by using information from the within-layer and
inter-layer structures of the communities. Moreover, our formulation of
multilayer modularity is general enough to account for an available ordering of
the layers and relating constraints on layer coupling. Experimental evaluation
was conducted using three state-of-the-art methods for multilayer community
detection and nine real-world multilayer networks. Results have shown the
significance of our modularity, disclosing the effects of different
combinations of the resolution and inter-layer coupling functions. This work
can pave the way for the development of new optimization methods for
discovering community structures in multilayer networks.
