Modularity in Multilayer Networks Using Redundancy-Based Resolution and Projection-Based Inter-Layer Coupling

Abstract

The generalized version of modularity for multilayer networks, a.k.a. multislice modularity, is characterized by two model parameters, namely resolution factor and inter-layer coupling factor. The former corresponds to a notion of layer-specific relevance, whereas the inter-layer coupling factor represents the strength of node connections across the network layers. Despite the potential of this approach, the setting of both parameters can be arbitrarily selected, without considering specific characteristics from the topology of the multilayer network as well as from an available community structure. Also, the multislice modularity is not designed to explicitly model order relations over the layers, which is of prior importance for dynamic networks.This paper aims to overcome the main limitations of the multislice modularity by introducing a new formulation of modularity for multilayer networks. We revise the role and semantics of both the resolution and inter-layer coupling factors based on information available from the within-layer and inter-layer structures of the multilayer communities. Also, our proposed multilayer modularity is general enough to consider orderings of network layers and their constraints on layer coupling. Experiments were carried out on synthetic and real-world multilayer networks using state-of-the-art approaches for multilayer community detection. The obtained results have shown the meaningfulness of the proposed modularity, revealing the effects of different combinations of the resolution and inter-layer coupling functions. This work also represents a starting point for the development of new optimization methods for community detection in multilayer networks.

Multilayer Network Science

“…The definition of the multilayer generalization of modularity was given by [85] a decade ago, become a standard for applications [89,90]. It assumes that the system can be represented by a three-dimensional tensor, encoding within-layer connectivity as for edge-colored multigraphs and time-varying networks[91] (see Fig.…”

Community Detection in Fully-Connected Multi-layer Networks Through Joint Nonnegative Matrix Factorization

“…The first class of methods is based on modularity maximization and generalizes the notion of modularity to multiple layers [11], [22], [23]. Principal Modularity Maximization (PMM) [18] extracts structural features for each layer by optimizing its modularity, and then applies PCA on concatenated matrix of structural feature matrices, to find the principal vectors, followed by K-means to perform community assignment.…”

Community detection in complex networks: From statistical foundations to data science applications

“…Hence, we now deal with community detection not in a single network but in a network of networks , requiring to exploit and fuse multiple aspects of disparate, yet interdependent sources of information. While some community detection methods discussed in the previous sections can be extended to the analysis of multilayer networks, extracting multilayer communities poses many new challenges due to nontrivial heterogeneous interlayer and intralayer dependencies and yet remains a substantially less developed area in complex network analysis (Amelio et al, 2020; Contisciani et al, 2020; Yuvaraj et al, 2021). Figure 9 shows a schematic representation of communities in a multilayer network.…”