Fuzzy Measures: A solution to deal with community detection problems for networks with additional information

Abstract

In this work we introduce the notion of the weighted graph associated with a fuzzy measure. Having a finite set of elements between which there exists an affinity fuzzy relation, we propose the definition of a group based on that affinity fuzzy relation between the individuals. Then, we propose an algorithm based on the Louvain’s method to deal with community detection problems with additional information independent of the graph. We also provide a particular method to solve community detection problems over extended fuzzy graphs. Finally, we test the performance of our proposal by means of some detailed computational tests calculated in several benchmark models.

Community Detection Problem Based on Polarization Measures. An application to Twitter: the COVID-19 case in Spain

“…Several authors agree on the importance of adding some additional information to the structure represented by a graph to enrich the communities detected [9,11,43,44]. In this work, it is particularly interesting the problem addressed by Gutiérrez et al [13][14][15][45][46][47][48] about the detection of communities in extended fuzzy graphs. They proposed a methodology to analyze independently the structural information of the graph and the knowledge represented by the fuzzy measure when grouping the nodes.…”

“…To face this problem, we work inspired by the idea developed in the Additional Louvain algorithm (see [14]), based on the Louvain algorithm [8]. The key point is to distinguish two different roles within the input parameters: one of them, to establish the neighbour relations, and the other, to calculate the variation of the modularity.…”

“…Definition 12 (Weighted graph associated with a fuzzy measure G µ [13,14]). Let µ : 2 V → [0, 1] denote a fuzzy measure defined over the finite set V, and let ξ : [−1, 1] 2 → [0, 1] denote a bi-variate aggregation operator.…”

“…On this assumption, a desirable partition could be P a = {{1, 4}, {2, 3}, {5, 6}, {7, 8}}. Concerning this idea of adding some additional information to the community detection problem, and based on methodology proposed by Gutierrez et al [13][14][15] about the the community detection in graphs based on fuzzy measures, in this study we define a new method with the purpose of adding extra-information related to Polarization to the community detection problem in graphs. That additional information is based on the values given by the Polarization measure developed by Guevara et al [3].…”

A New Edge betweenness Measure Using a Game Theoretical Approach : An Application to Hierarchical Community Detection

“…Let us emphasize again that this game has been built under the assumption that the only available information in the problem is the network, so only modularity function has been considered. If more information is known about the clustering problem, this information should be included in the game, to obtain more cohesive partitions [44,45].…”