Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 10 September 2021 / Approved: 13 September 2021 / Online: 13 September 2021 (11:51:33 CEST)

A peer-reviewed article of this Preprint also exists.

Gómez, D.; Castro, J.; Gutiérrez, I.; Espínola, R. A New Edge Betweenness Measure Using a Game Theoretical Approach: An Application to Hierarchical Community Detection. Mathematics 2021, 9, 2666. Gómez, D.; Castro, J.; Gutiérrez, I.; Espínola, R. A New Edge Betweenness Measure Using a Game Theoretical Approach: An Application to Hierarchical Community Detection. Mathematics 2021, 9, 2666.

Journal reference: Mathematics 2021, 9, 2666
DOI: 10.3390/math9212666

Abstract

In this paper we formally define the hierarchical clustering network problem (HCNP) as the problem to find a good hierarchical partition of a network. This new problem focuses on the dynamic process of the clustering rather than on the final picture of the clustering process. To address it, we introduce a new hierarchical clustering algorithm in networks, based on a new shortest path betweenness measure. To calculate it, the communication between each pair of nodes is weighed by the importance of the nodes that establish this communication. The weights or importance associated to each pair of nodes are calculated as the Shapley value of a game, named as the linear modularity game. This new measure, (the node-game shortest path betweenness measure), is used to obtain a hierarchical partition of the network by eliminating the link with the highest value. To evaluate the performance of our algorithm, we introduce several criteria that allow us to compare different dendrograms of a network from two point of view: modularity and homogeneity. Finally, we propose a faster algorithm based on a simplification of the node-game shortest path betweenness measure, whose order is quadratic on sparse networks. This fast version is competitive from a computational point of view with other hierarchical fast algorithms, and, in general, it provides better results.

Keywords

Game Theory; Graph Theory; Hierarchical Clustering networks; Community Detection problems; Divisive Algorithms

Subject

MATHEMATICS & COMPUTER SCIENCE, Computational Mathematics

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