Social network modeling is generally based on graph theory, which allows for study of dynamics and emerging phenomena. However, in terms of neighborhood, the graphs are not necessarily adapted to represent complex interactions, and the neighborhood of a group of vertices can be inferred from the neighborhoods of each vertex composing that group. In our study, we consider that 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. After giving some definitions and examples of modeling, we show how some measures used in social network analysis (degree, betweenness, and closeness) can be also generalized to consider a group as a whole entity. 1. Introduction Network modeling is an area of research which covers several domains like computer sciences, physics, sociology, or biology. In social networks modeling, graphs are often used to describe the links representing relationships or flows between entities [1]. Based on graph theory, the studies consider in most cases individuals as single elements, a group being formed by several persons interacting with each other. Most of the few works on modeling groups in social networks consider a group as a combination of persons [2], not as a whole entity. As social network analysis leads to centrality notion and others sociometric features, what about group centrality? The centrality of a vertex in a graph is widely used to determine the relative “importance” of this vertex within the network [3]. Centrality measures enable us to find users who are extensively involved in relationships with other network members. There are different centralities such as degree centrality, betweenness centrality, or closeness centrality. The problem we face is the following: analyzing a vertex can be done with this kind of measure, but if we analyze a group of persons using the same measure, we will have no particular emergence of characteristics as the union property of the neighborhoods in a graph is preserved. As social networks are complex networks [4–6], emergence of phenomena can occur [7], and the behavior of a group of persons can be different from the “sum” of the person behaviors composing the group. Some work tried to capture the different scales of a network, and a group can be viewed as a community [8]; thus, in our opinion, graph theory only is inadequate to model all complex interactions occurring in a social network. Some
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