A new approach to betweenness centrality based on the Shapley Value

In many real-life networks, such as urban structures, protein interactions and social networks, one of the key issues is to measure the centrality of nodes, i.e. to determine which nodes and edges are more central to the functioning of the entire network than others. In this paper we focus on betweenness centrality --- a metric based on which the centrality of a node is related to the number of shortest paths that pass through that node. This metric has been shown to be well suited for many, often complex, networks. In its standard form, the betweenness centrality, just like other centrality metrics, evaluates nodes based on their individual contributions to the functioning of the network. For instance, the importance of an intersection in a road network can be computed as the difference between the full capacity of this network and its capacity when the intersection is completely shut down. However, as recently argued in the literature, such an approach is inadequate for many real-life applications, as, for example, multiple nodes can fail simultaneously. Thus, what would be desirable is to refine the existing centrality metrics such that they take into account not only the functioning of nodes as individual entities but also as members of groups of nodes. One recently-proposed way of doing this is based on the Shapley Value --- a solution concept in cooperative game theory that measures in a fair way the contributions of players to all the coalitions that they could possibly participate in. Although this approach has been used to extend various centrality metrics, such an extension to betweenness centrality is yet to be developed. The main challenge when developing such a refinement is to tackle the computational complexity; the Shapley Value generally requires an exponential number of operations, making its use limited to a small number of player (or nodes in our context). Against this background, our main contribution in this paper is to refine the betweenness centrality metric based on the Shapley Value: we develop an algorithm for computing this new metric, and show that it has the same complexity as the best known algorithm due to Brandes [7] to compute the standard betweenness centrality (i.e., polynomial in the size of the network). Finally, we show that our results can be extended to another important centrality metric called stress centrality.