Chapter 7 - Social Networks

Abstract

Social networks are visualized using social graphs. The nodes in this graph are the participants of a social network (players) and weighted links reflect the degrees of their mutual “trust” or influence. The prospective tools in this field are the methods of game theory. First, we show how to construct the graph of social network. Then, by calculating centrality measures for nodes and edges introduced in Chapter 6 we detect active members of social networks. We demonstrate this method for a segment of the social network VKontakte and the professional mathematical network Mathnet.ru. Then we propose the game-theoretic methods for community detection in networks. The traditional methods for detecting community structure are based on selecting dense subgraphs inside the network. Here we propose to use the methods of cooperative game theory that highlight not only the link density but also the mechanisms of cluster formation. Specifically, we suggest two approaches from cooperative game theory: the first approach is based on the Myerson value, whereas the second approach is based on hedonic games. Furthermore, the modularity based approach and its generalizations can be viewed as particular cases of the hedonic games. We demonstrate the efficiency of this approach in some examples.