Game-Theoretic Centrality Measures for Weighted Graphs

Abstract

The betweenness centrality is one of the basic concepts in the analysis of the social networks. Initial definition for the betweenness of a node in the graph is based on the fraction of the number of geodesics (shortest paths) between any two nodes that given node lies on, to the total number of the shortest paths connecting these nodes. This method has polynomial complexity. We propose a new concept of the betweenness centrality for weighted graphs using the methods of cooperative game theory. The characteristic function is determined by special way for different coalitions (subsets of the graph). Two approaches are used to determine the characteristic function. In the first approach the characteristic function is determined via the number of direct and indirect weighted connecting paths in the coalition. In the second approach the coalition is considered as an electric network and the characteristic function is determined as a total current in this network. We use the Kirchhoff's law. After that the betweenness centrality is determined as the Myerson value. The results of computer simulations for some examples of networks, in particular, for the popular social network "VKontakte", as well as the comparing with the PageRank method are presented.

Network partitioning algorithms as cooperative games

“…The imputation specifies how the value associated with the network is distributed to the individual players. The imputation in our cooperative game will be based on the Myerson value [ 21 , 22 , 41 ] which was designed to take into account the effect of the network.…”

“…Characteristic function (payoff of coalition S ) can be defined in different ways. Here we use a general idea from [ 21 , 22 , 42 , 43 ], which is based on discounting paths. However, unlike [ 21 , 22 , 42 , 43 ], we do not consider shortest paths but rather simple paths .…”

“…The advantage of the Myerson value is in taking into account the impact of all coalitions. We extend the method developed in [ 21 , 22 ] to calculate efficiently the Myerson value in a network. A number of network centrality measures based on game-theoretic concepts have been developed, see [ 22 – 28 ] and references therein.…”

“…We extend the method developed in [ 21 , 22 ] to calculate efficiently the Myerson value in a network. A number of network centrality measures based on game-theoretic concepts have been developed, see [ 22 – 28 ] and references therein. It might be interesting to combine node ranking and clustering based on the same approach such as the Myerson value to analyze the network structure.…”

Cooperative Game Theory Approaches for Network Partitioning

“…Characteristic function (payoff of coalition S) can be determined in different ways. Here we use the approach of [10,11,14,15], which is based on discounting directed paths. The payoff to an individual player is called an imputation.…”