Overlapping modularity at the critical point of k-clique percolation

One of the most remarkable social phenomena is the formation of communities in social networks corresponding to families, friendship circles, work teams, etc. Since people usually belong to several different communities at the same time, the induced overlaps result in an extremely complicated web of the communities themselves. Thus, uncovering the intricate community structure of social networks is a non-trivial task with great potential for practical applications, gaining a notable interest in the recent years. The Clique Percolation Method (CPM) is one of the earliest overlapping community finding methods, which was already used in the analysis of several different social networks. In this approach the communities correspond to k-clique percolation clusters, and the general heuristic for setting the parameters of the method is to tune the system just below the critical point of k-clique percolation. However, this rule is based on simple physical principles and its validity was never subject to quantitative analysis. Here we examine the quality of the partitioning in the vicinity of the critical point using recently introduced overlapping modularity measures. According to our results on real social- and other networks, the overlapping modularities show a maximum close to the critical point, justifying the original criteria for the optimal parameter settings.Comment: 20 pages, 6 figure

Revisiting Resolution and Inter-Layer Coupling Factors in Modularity for Multilayer Networks

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.Comment: Accepted at the IEEE/ACM Conf. on Advances in Social Network Analysis and Mining (ASONAM 2017