Statistical and Physical Models for Social Networks and Their Evolution

Recently, the area of Social Networks has gained popularity in computer science partially due to the increased efforts in the domain of security. One of the goals of analysis in this area is to detect and evaluate relationships between individuals that may be part of terrorist networks (Krebs, 2002). It should be noted, however, that statistical analysis of social networks spans over 60 years. Since the 1970s, one of the major directions in the field was to model probabilities of relational ties between interacting units (social actors), though in the beginning only very small groups of actors were considered. Extensive introduction to earlier methods is provided by Wasserman and Faust (1994). Two of the most prominent current directions are Markov Random Fields (MRFs) introduced by Strauss and Frank (1986) and Exponential Random Graphical Models (ERGMs), also known as p∗ (Wasserman and Pattison, 1996; Anderson et al, 1999). The ERGM have been recently extended by Snijders et al (2004) in order to achieve robustness in the estimated parameters. The statistical literature on modeling Social Networks assumes that there are n entities called actors and information about binary relations between them. Binary relations are represented as an nxn matrix Y , where Yij is 1, if actor i is somehow related to j and is 0 otherwise. For example, Yij = 1 if “i considers j to be friend”. The entities are usually represented as nodes and the relations as arrows between the nodes. If matrix Y is symmetric, then the relations are represented as undirected arrows. More generally Yij can be valued and not just binary, representing the strength (or value) of the relationship between actors i and j (Robins et al, 1999). In addition, each entity can have a set of characteristics xi such as their demographic information. Then the n dimensional vector X = x1, . . . , xn is a fully observed covariate data that is taken into account in the model (e.g. Hoff et al, 2002) Predominantly the social network literature focuses on modeling P(Y—X), i.e. on probabilistically describing relations among actors as functions of their covariates and also properties of the graph, such as indegree and outdegree of individual nodes. A complete list of graph-specific properties that are being modeled can be found in (Snijders et al, 2004). Thus, the models are geared to probabilistically explain the patterns of observed links and their absence between n given entities. There are several useful properties of the stochastic models listed in a brief survey work by Smyth (2003). Some of them are: