Statistical Models for Networks: A Brief Review of Some Recent Research

We begin with a graph (or a directed graph), a single set of nodes N , and a set of lines or arcs L. It is common to use this mathematical concept to represent a network. We use the notation of [1], especially Chapters 13 and 15. There are extensions of these ideas to a wide range of networks, including multiple relations, affiliation relations, valued relations, and social influence and selection situations (in which information on attributes of the nodes is available), all of which can be found in the chapters of [2]. The purpose of this short exposition is to discuss the developments in statistical models for networks that have occurred over the past five years, since the publication of the statistical chapters (8, 9, 10, and 11) of Carrington, Scott, and Wasserman (which were written in 2002). The statistical modeling of social networks is advancing quite quickly. The many exciting new developments include, for instance, longitudinal models for the co-evolution of networks and behavior [3] and latent space models for social networks [4]. In this chapter, we do not intend to review all the recent advances but rather limit our scope to a few developments that we have worked on.