Shapley-Based Stackelberg Leadership Formation in Networks

Abstract

In the given research we study a leadership formation of the most influential nodes in networks. Specifically, we analyze the competition between a leader and a follower based on the Stackelberg leadership model. Applying the concept of Shapley value to measure node's importance, we represent the mechanism of Shapley-based Stackelberg leadership formation in networks. The approach is tested and represented in tabular and graphical formats. The investigation of competition between network's leaders is at the core of the leadership formation analysis in social networks. The problem is closely correlated with the centrality measurement that is based on the different evaluation methods. Degree (Freeman, 1979), betweenness (Anthonisse, 1971; Freeman, 1977), and closeness (Beauchamp, 1965; Sabidussi, 1966) are the most widely known metrics that assess the structural centralities of nodes. The algorithmic measures of node's authority are well represented in Kleinberg (1999) and Page, Brin, Motwani, & Winograd (1999), where the notion of authority is given based on the analysis of link structures. An interesting approach to characterize the role of nodes within the networks is given by Scripps & Esfahanian (2007), where the community-based metric in the symbiosis with the degree-based measure is introduced in the context of the classification of nodes' roles. Another methodology for analyzing node's leadership and importance in networks is based on a game theoretic approach. Specifically, we employ the Shapley value concept developed by Aadithya, Ravindran, Michalak, & Jennings (2010) in order to analyze how the nodes' leadership positions in networks can be strengthened by establishing new links. In the given research, we analyze the leadership formation in terms of the competition between the most influential leader and its follower in a network. We interpret the interaction between the leader and the follower in social networks based on the Stackelberg competition principle (Von Stackelberg, 2010) from the game theoretic domain. More details about the Stackelberg model can be found in Simaan, & Cruz (1973), Basar, Olsder, Clsder, Basar, Baser, & Olsder (1995), and He, Prasad, Sethi, & Gutierrez (2007). Applying Shapley value to measure an agent's (i.e., nodes) leadership position in a network, we represent the mechanism of the Shapley-based Stackelberg leadership formation. It reflects the competition between the most influential agent (i.e., leader) and the second most influence agent (i.e., follower) in the network.