So how was the mean of the responses 1? According to the data, 70% of the respondents reported knowing
zero people in prison. However, the responses show a wide range of variation, with almost 3% reporting that
they know at least 10 prisoners. Responses to some other questions of the same format, for example “How
many people do you know named Nicole?” show much less variation.
This difference in the variability of responses to these “how many X’s do you know” questions is the
manifestation of fundamental social processes at work. Through careful examination of this pattern, as
well as others in the data, we can learn about important characteristics of the social network connecting
Americans, as well as the processes that create this network.
This analysis also furthers our understanding of statistical models from two-way data, by treating overdis-
persion as a source of information not just an issue that requires correction. More specifically, we include
overdispersion as a parameter that measures the variation in the relative propensities of individuals to form
ties to a given social group, and allow it to vary among social groups. Through such modeling of the variation
of the relative propensities, we derive a new measure of social structure that only uses survey responses from
a sample of individuals not data on the complete network.
1.1
Background
Understanding the structure of social networks, and the social processes which form them, is a central concern
of sociology for both theoretical and practical reasons (Wasserman and Faust, 1994; Freeman, 2004). Social
networks have been found to have important implications for the social mobility (Lin, 1999), getting a job
(Granovetter, 1995), the dynamics of fads and fashion (Watts, 2002), attitude formation (Lee et al., 2004),
and the spread of infectious disease (Morris and Kretzchmar, 1995).
When talking about social networks, sociologists often use the word “social structure,” which in practice
has taken on many different meanings, sometimes unclear or contradictory. In this paper, as in Heckathorn
and Jeffri (2001), we generalize the conception put forth by Blau (1974) that social structure is the difference
in affiliation patterns from what would be observed if people formed friendships entirely randomly.
Sociologists are not the only scientists interested in the structure of networks. Methods presented here
can be apply to a more generally defined network, as any set of objects (nodes) connected to each other by a
set of links (edges). In addition to social networks (friendship network, collaboration networks of scientists,
sexual networks), other examples include technological networks (the Internet backbone, the World Wide
Web, the power grid) and biological networks (metabolic networks, protein interaction networks, neural
networks, food webs); for reviews see Strogatz (2001), Newman (2003b), and Watts (2004).
1.2
Overview of this paper
We show how to use “how many X’s do you know” count data to learn about the social structure of the
acquaintanceship network in the United States. More specifically, we can learn to what extent people vary
in their number of acquaintances, to what extent people vary in their propensity to form ties to people in
2
Convention