Fuzzy Sugeno λ-measures and theirs applications to community detection problems

Abstract

In this paper we propose a new framework for community detection problems. The starting point is a n-vector which defines some evidence about the elements of a finite set. This vector is used to build an interaction measure between the n elements of the set to which it refers. This interaction measure is represented by a Sugeno λ-measure to which we make it being also a fuzzy measure. Then, we obtain the weighted graph associated with this new capacity measure. To carry on with it, we make use of the Shapley value. We also introduce the notion of extended vector fuzzy graph, which relates a graph with the capacity measure introduced in this work. Finally, we use a community detection method, based on Louvain algorithm, to search a cluster structure in the weighted graph. This partition is based on the relations among the individuals obtained from the initial vector. Let us note that in the case that there exist some connections among the elements, apart from their affinity, we can combine this extra information with that given by the vector, in order to obtain groups with highly-knit elements among which there are strong relations.