Semivalues and Centrality in Social Networks

Zusammenfassung

A social network can be modelized as a graph, which shows the possible direct communications between individuals. A cooperative game in characteristic function form can be considered to reflect the interest that motivate the interactions. Then, the graph-restricted game (the Myerson-game) is used to represent the economic possibilities of coalitions taking the available communications into account. Every semivalue in a game can be considered as actor�s power index. A centrality measure is defined as the semivalue of a given symmetric communication game. Conditions like symmetry, supperaditivity or convexity are neccessary to reach desirable properties for these measures. In Gómez et al. (2001) is proved that when the chosen semivalue is the Shapley value, the centrality measure satisfies the following properties: 1. Is symmetric. 2. The centrality of a node in a disconnected subgraph coincides with the centrality of that node in the connected subgraph to which it belongs. 3. Isolated nodes have minimal centrality. 4. In a chain centrality increases from the end node to the medain node. 5. Of all connected graphs with n nodes, the minimal centrality is attained by the end nodes of a chain. 6. Of all connected graphs with n nodes the maximal centrality is attained by the hub of a star. In th is communication we e xplore the e xt ent to which previous prop erties are satisfied when we consider other semivalues.