k-step betweenness centrality

Abstract

, 90 pages The notions of betweenness centrality (BC) and its extension group betweenness centrality (GBC) are widely used in social network analyses. We introduce variants of them; namely, the k-step BC and k-step GBC. The k-step GBC of a group of vertices in a network is a measure of the likelihood that at least one group member will get the information communicated between a randomly chosen pair of vertices through a randomly chosen shortest path within the first k steps of the start of the communication. The k-step GBC of a single vertex is the k-step BC of that vertex. The introduced centrality measures may find uses in applications where it is important or critical to obtain the information within a fixed time of the start of the communication. For the introduced centrality measures, we propose an algorithm that can compute successively the k-step GBC of several groups of vertices. Moreover, we propose a mixed integer programming formulation to compute the group that has the highest k-step GBC value and a heuristic approach to compute a group of vertices whose k-step GBC value is high. The performances of the proposed algorithms are evaluated through computational experiments on real and randomly generated networks.

On the Connectivity and the Diameter of Betweenness-Uniform Graphs

“…We can easily see that edge-transitive graphs are edge betweenness-uniform. The result follows from relation (1).…”

“…It is often studied from the algorithmic point of view [4,16]. Betweenness centrality and its variants are also studied from the graph-theoretical perspective [1,2,10,15,18,24]. In this paper we focus on graphs having the same betweenness on all vertices initiated in studies [8,14].…”

Statistical Analysis of Networks

“…For instance, it may be required to evaluate an influence of a particular social group on the society or to evaluate the importance of a department within an organization. Several works, starting from Everett and Borgatti, 1999, proposed various variants of group centralities: Kolaczyk et al, 2009;Veremyev et al, 2017;Akgün and Tural, 2020. Let us emphasize that in the majority of cases it is not suitable to simply sum centrality values of individual nodes. It should come without surprise that the methods of cooperative game theory are very natural to define group centrality indices, see e.g., Michalak et al, 2013;Szczepański et al, 2016. It is often important to find only Top-k central nodes of a network.…”

The Star Degree Centrality Problem: A Decomposition Approach

“…Following this, Akgün and Tural (2020) introduces an extension of group betweenness centrality called k-step centrality, where shortest path lengths are restricted by k. It is important to note that Zhao et al ( 2017) also places length restrictions on shortest paths in their study of group closeness centrality, yet they are not cited by Akgün and Tural (2020). The methodology used by Akgün and Tural (2020) incorporates algorithms developed by Brandes (2008) and Puzis et al (2007). In another study, Li et al (2020) proposes a new group centrality metric named h-index group centrality, which incorporate the concept of h-index (see Hirsch (2005)) into groups.…”

“…This metric is motivated by Katz centrality (Katz, 1953): as such, it focuses on walks of all lengths instead of solely the shortest paths, offering an alternative to closeness and betweenness centrality measures. Following this, Akgün and Tural (2020) introduces an extension of group betweenness centrality called k-step centrality, where shortest path lengths are restricted by k. It is important to note that Zhao et al ( 2017) also places length restrictions on shortest paths in their study of group closeness centrality, yet they are not cited by Akgün and Tural (2020). The methodology used by Akgün and Tural (2020) incorporates algorithms developed by Brandes (2008) and Puzis et al (2007).…”