Modeling Groups In Social Networks

Studies in social network analysis consider the behavior of a group to be equivalent to the “sum” of individual behaviors. However, a group has to be considered as a complex system where emerging phenomena can appear. In this paper, a formalism is proposed to resolve this problematic by modeling groups in social networks using pretopology as a generalization of the graph theory. We also give an example of simulation using our formalism. Introduction Network modeling is used in several research domains (physics, biology, computer sciences, ...) and is mainly based on graph theory. Especially in social networks modeling, graphs are used to describe the links representing relationships or flows between entities. The studies consider in most cases individuals as single elements, a group being formed by several persons interacting with each other. Most work on modeling social networks consider a group as a combination of persons (Everett and Borgatti, 2005), not as a whole entity. As social networks are complex networks (Newman et al., 2006), emergence of phenomena can occur, and the behavior of a group of persons can be different from the “sum” of each behavior of the persons composing the group. Thus, in our opinion, graph theory is inadequate to model all complex interactions occurring in a social network. As some group modeling approaches based on graphs have been proposed before (Everett and Borgatti, 1999, 2005), and more recently (Saha and Getoor, 2008), to our knowledge, there is no general framework for modeling a group of entities in a network. From this fact, we propose a mathematical framework for group modeling in social networks from a topological point of view. This paper is structured in two parts: first we introduce pretopology formalism, and we give some definitions and illustrations for group modeling in a social network. Then we show a simple simulation example of a model using our approach. The reader of this paper is supposed to be familiar with graph theory and some concepts from mathematical topology. Modeling groups in social networks Before giving theoretical definitions, we first illustrate the problem of a realistic group modeling by giving a simple example: we would like to represent interactions between four persons. A small social network with a simple symmetrical friendship relation shall help in illustrating our approach (Fig. 1).