Dynamical systems to define centrality in social networks

In this paper, new measures of centrality that summarize the contact structure of social networks are proposed. The new measures use a cumulative nomination scheme based on the preliminary assumption that more central individuals will be nominated more often. Some of these measures are defined to characterize networks of different sizes and, by extension, networks made of many components. These new measures are applied to a network of 40 homosexuals with AIDS [Auerbach, D., Darrow, W., Jaffe, H., Curran, J., 1984. Cluster of cases of the acquired immune deficiency syndrome: patients linked by sexual contact. Am. J. Med. 76 (1984) 487–492; Klovdahl, A.S., 1985. Social networks and the spread of infectious diseases: the AIDS example. Soc. Sci. Med. 21 (1985) 1203–1216.], an illustrative multi-component network and a simulated network. They are compared to classical measures based on geodesics (closeness, eccentricity), to information-based centrality measures introduced by Stephenson and Zelen [Stephenson, K., Zelen, M., 1989. Rethinking centrality: methods and examples. Soc. Networks 11 (1989) 1–37.] and Altmann [Altmann, M., 1993. Reinterpreting network measures for models of disease transmission. Soc. Networks 15 (1993) 1–17.], and to the centrality measure of Bonacich [Bonacich, P., 1972. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 2 (1972) 113–120.]. The most basic of our measures is shown to be related to the Bonacich index of centrality for connected networks. The scaling law of the different centrality measures is examined by measuring simulated networks of various sizes. Measures based on the distribution of the components' size obey a simple proportional scaling law while those based on geodesics do not. Our new measures prove interesting because they consider all the possible paths, do not require intensive computer calculations, and can be used to compare networks of different sizes because they are independent of the size of the networks.