Explicit objective functions in modularity-based community detection on multiplex networks

Abstract

Modularity-based community detection in multiplex networks is commonly approached through one of three strategies: early fusion (EF), which first aggregates the layers into a single network and then applies community detection; simultaneous fusion (SF), which combines information from the layers during modularity optimization; and late fusion (LF), which combines the community assignments obtained by detecting communities separately in each layer. Although recent taxonomies and surveys distinguish these strategies conceptually, their objective functions have rarely been compared analytically. We provide such a comparison for shared-node multiplex networks with normalized non-negative edge and layer weights, a common resolution parameter, and no interlayer edges. Node-attributed networks are represented by topology and attribute-similarity layers. We use a common notation for the three formulations and derive explicit relationships between their objective functions. In particular, we show that the EF objective equals the SF objective plus a non-negative heterogeneity term. We prove that the SF objective admits an optimal layer-weight vector at a simplex vertex, although nonvertex optima may also occur under ties, whereas the EF objective is concave with respect to the layer parameter for each fixed partition, and its joint optimum may occur in the interior of the simplex. For LF methods, the resulting objective and conclusions depend heavily on how the transient graph is constructed. In this study, we focus only on one of such LF methods. We complement the analytical results with brute-force experiments on synthetic multiplexes and heuristic experiments on real-world networks, including node-attributed networks represented as multiplexes. The experiments illustrate the analytical results and the influence of heuristic optimizers on the observed comparisons.