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 social network. We use the notation of Wasserman and Faust (1994), especially Chapters 13 and 15. There are extensions of these ideas to a wide range of social networks, including multiple relations, affiliation relations, valued relations, and social influence and selection situations (in which information on attributes of the nodes is available). Later chapters in this volume discuss such generalizations.

The model p* was first discussed by Frank and Strauss (1986), who termed it a distribution for a Markov random graph. Further developments, especially commentary on estimation of distribution parameters, were given by Strauss and Ikeda (1990). Wasserman and Pattison (1996) further elaborated this family of models, showing how a Markov parametric assumption provides just one of many possible sets of parameters. This family, with its variety and extensions, was named p*, a label by which it has come to be known. The parameters reflect structural concerns, which are assumed to govern the probabilistic nature of the underlying social and/or behavioral process.

The development of p* presented here is different from that found in Wasserman and Pattison (1996) and Anderson, Wasserman, and Crouch (1999), but similar to the presentation in Pattison and Wasserman (1999).

Interest in social network analysis has grown massively in recent years. This growth has been matched by an increasing sophistication in the technical tools available to users. Models and Methods in Social Network Analysis (MMSNA) presents the most important of those developments in quantitative models and methods for analyzing social network data that have appeared during the 1990s. It is a collection of original chapters by leading methodologists, commissioned by the three editors to review recent advances in their particular areas of network methods.

As is well-known, social network analysis has been used since the mid-1930s to advance research in the social and behavioral sciences, but progressed slowly and linearly, until the end of the century. Sociometry (sociograms, sociomatrices), graph theory, dyads, triads, subgroups, and blockmodels – reflecting substantive concerns such as reciprocity, structural balance, transitivity, clusterability, and structural equivalence – all made their appearances and were quickly adopted by the relatively small number of “network analysts.” It was easy to trace the evolution of network theories and ideas from professors to students, from one generation to the next. The field of network analysis was even analyzed as a network (see, for example, Mullins 1973, as well as analyses by Burt in 1978, and Hummon and Carley in 1993). Many users eventually became analysts, and some even methodologists. A conference of methodologists, held at Dartmouth College in the mid-1970s, consisted of about thirty researchers (see Holland and Leinhardt 1979) and really did constitute a “who's who” of the field – an auspicious, but rather small gathering.

One of the primary uses of graph theory in social network analysis is the identification of the “most important” actors in a social network. In this chapter, we present and discuss a variety of measures designed to highlight the differences between important and non-important actors. Definitions of importance, or synonymously, prominence, have been offered by many writers. All such measures attempt to describe and measure properties of “actor location” in a social network. Actors who are the most important or the most prominent are usually located in strategic locations within the network. As far back as Moreno (1934), researchers have attempted to quantify the notions of sociometric “stars” and “isolates.”

We will discuss the most noteworthy and substantively interesting definitions of importance or prominence along with the mathematical concepts that the various definitions have spawned. Among the definitions that we will discuss in this chapter are those based on degree, closeness, betweenness, information, and simply the differential status or rank of the actors.

These definitions yield actor indices which attempt to quantify the prominence of an individual actor embedded in a network. The actor indices can also be aggregated across actors to obtain a single, group-level index which summarizes how variable or differentiated the set of actors is as a whole with respect to a given measure. We will show how to calculate both actor and group indices in this chapter.