An integer programming approach for finding the most and the least central cliques

Abstract

We consider the problem of finding the most and the least "influential" or "influenceable" cliques in graphs based on three classical centrality measures: degree, closeness and betweenness. In addition to standard clique betweenness, which is defined as the proportion of shortest paths between any two graph nodes that pass through the clique, we also consider its optimistic and pessimistic versions, where outside nodes may favor or try to avoid shortest paths passing through the clique, respectively. We discuss the computational complexity issues for these problems and develop linear 0-1 programming formulations for each centrality metric. Finally, we demonstrate the performance of the developed formulations on real-life and synthetic networks, and provide some interesting insights based on the obtained results. In particular, our findings indicate that there are considerable variations between the centrality values of large cliques within the same networks. Moreover, the most central cliques in graphs are not necessarily the largest ones.

Maximal Clique and Graph Coloring Heuristics for Overlapping Community Detection in Network Psychometrics

“…These cliques, along with their intensities and clique centralities are displayed in Table 2. One interesting result is that the largest cliques in the network do not have the greatest centrality, which is consistent with the findings of Vogiatzis et al (2015). For example, cliques 7 1 and 7 5 are maximum 7-cliques with clique degree centralities of D( 7 1 ) = D( 7 5 ) = 9, which means they reach 9 of the 11 symptoms that are not in their respective cliques.…”

“…As described by Vogiatzis et al (2015), the degree centrality of a k-clique (Ck), D(Ck), is the number of vertices that are not members of Ck, yet are connected by an edge to at least onemember of Ck. In this respect, the degree centrality of a clique can be perceived as a measure of the extent to which it serves as a bridge to other vertices in the network.…”

“…Our heuristics were also designed with the recognition that a clique's intensity is not the only measure of its contribution to the network. For example, clique centrality could also be considered (Vogiatzis et al, 2015). Whereas intensity measures the internal cohesion of a clique, clique centrality measures the extent to which a clique serves as a bridge to other vertices in the network, and it is well-recognized that this type of 'bridging' in networks is also important (Jones et al, 2019).…”

Maximum Closeness Centrality k-Clubs: A Study of Dock-Less Bike Sharing

“…More recently, researchers have focused on highest betweenness groups [39]. Finally, another extension of identifying highly centralized groups has to do with the added restriction that the group induces a subgraph "motif ", such as being a complete subgraph/clique [40,41], or inducing a star [42].…”

On maximum degree‐based ‐quasi‐clique problem: Complexity and exact approaches

“…The problem of finding large ‐quasi‐cliques, and dense subgraphs in general, arises in a number of application areas, typically in the contexts related to analysis of real‐life networks. Prime examples include biological , social , telecommunication , and financial networks. The choice of the appropriate clique relaxation model depends on the nature of the underlying network in the application of interest; see, for example, related discussions in and the references therein.…”