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Logit Models and Logistic Regressions for Social Networks: I. An Introduction to Markov Graphs and p

Published online by Cambridge University Press:  01 January 2025

Stanley Wasserman  and
Philippa Pattison
Stanley Wasserman*
Affiliation:
University of Illinois
Philippa Pattison
Affiliation:
University of Melbourne
*
Requests for reprints should be sent to Stanley Wasserman, University of Illinois, 603 East Daniel Street, Champaign, IL 61820.
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Abstract

Spanning nearly sixty years of research, statistical network analysis has passed through (at least) two generations of researchers and models. Beginning in the late 1930's, the first generation of research dealt with the distribution of various network statistics, under a variety of null models. The second generation, beginning in the 1970's and continuing into the 1980's, concerned models, usually for probabilities of relational ties among very small subsets of actors, in which various simple substantive tendencies were parameterized. Much of this research, most of which utilized log linear models, first appeared in applied statistics publications.

But recent developments in social network analysis promise to bring us into a third generation. The Markov random graphs of Frank and Strauss (1986) and especially the estimation strategy for these models developed by Strauss and Ikeda (1990; described in brief in Strauss, 1992), are very recent and promising contributions to this field. Here we describe a large class of models that can be used to investigate structure in social networks. These models include several generalizations of stochastic blockmodels, as well as models parameterizing global tendencies towards clustering and centralization, and individual differences in such tendencies. Approximate model fits are obtained using Strauss and Ikeda's (1990) estimation strategy.

In this paper we describe and extend these models and demonstrate how they can be used to address a variety of substantive questions about structure in social networks.

Keywords

categorical data analysissocial network analysisrandom graphs
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 3 , September 1996 , pp. 401 - 425
Copyright
Copyright © 1996 The Psychometric Society

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Footnotes

This research was supported by grants from the Australian Research Council and the National Science Foundation (#SBR93-10184). This paper was presented at the 1994 Annual Meeting of the Psychometric Society, Champaign, Illinois, June 1994. Special thanks go to Sarah Ardu for programming assistance, Laura Koehly and Garry Robins for help with this research, and to Shizuhiko Nishisato and three reviewers for their comments. INTERNET email addresses: pattison@psych.unimelb.edu.au (PP); stanwass@uiuc.edu (SW). Affiliations: Department of Psychology, University of Melbourne (PP); Department of Psychology, Department of Statistics, and The Beckman Institute for Advanced Science and Technology, University of Illinois (SW).

References

Agresti, A. (1990). Categorical data analysis, New York: John Wiley and Sons.Google Scholar
Anderson, C. J., Wasserman, S. (1995). Logmultilinear models for valued social relations. Sociological Methods & Research, 24, 96–127.CrossRefGoogle Scholar
Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, 36, 192–236.CrossRefGoogle Scholar
Faust, K., Wasserman, S. (1992). Centrality and prestige: A review and synthesis. Journal of Quantitative Anthropology, 4, 23–78.Google Scholar
Fienberg, S. E. (1980). The analysis of cross-classified, categorical data 2nd ed.,, Cambridge, MA: The MIT Press.Google Scholar
Fienberg, S. E., Wasserman, S. (1981). Categorical data analysis of single sociometric relations. In Leinhardt, S. (Eds.), Sociological methodology 1981 (pp. 156–192). San Francisco: Jossey-Bass.Google Scholar
Frank, O., Strauss, D. (1986). Markov random graphs. Journal of the American Statistical Association, 81, 832–842.CrossRefGoogle Scholar
Holland, P. W., Leinhardt, S. (1973). The structural implications of measurement error in sociometry. Journal of Mathematical Sociology, 3, 85–111.CrossRefGoogle Scholar
Holland, P. W., Leinhardt, S. (1975). The statistical analysis of local structure in social networks. In Heise, D. R. (Eds.), Sociological methodology 1976 (pp. 1–45). San Francisco: Jossey-Bass.Google Scholar
Holland, P. W., & Leinhardt, S. (1977). Notes on the statistical analysis of social network data. Unpublished manuscript.Google Scholar
Holland, P. W., Leinhardt, S. (1978). An omnibus test for social structure using triads. Sociological Methods & Research, 7, 227–256.CrossRefGoogle Scholar
Holland, P. W., Leinhardt, S. (1979). Structural sociometry. In Holland, P. W., Leinhardt, S. (Eds.), Perspectives on social network research (pp. 63–83). New York: Academic Press.CrossRefGoogle Scholar
Holland, P. W., Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76, 33–65.CrossRefGoogle Scholar
Iacobucci, D., Wasserman, S. (1990). Social networks with two sets of actors. Psychometrika, 55, 707–720.CrossRefGoogle Scholar
Ising, E. (1925). Beitrag zur theorie des ferramagnetismus. Zeitschrift fur Physik, 31, 253–258.CrossRefGoogle Scholar
Johnsen, E. C. (1985). Network macrostructure models for the Davis-Leinhardt set of empirical sociomatrices. Social Networks, 7, 203–224.CrossRefGoogle Scholar
Johnsen, E. C. (1986). Structure and process: Agreement models for friendship formation. Social Networks, 8, 257–306.CrossRefGoogle Scholar
Kindermann, R. P., Snell, J. L. (1980). On the relation between Markov random fields and social networks. Journal of Mathematical Sociology, 8, 1–13.CrossRefGoogle Scholar
Koehly, L., & Wasserman, S. (1994). Classification of actors in a social network based on stochastic centrality and prestige. Manuscript submitted for publication.Google Scholar
Pattison, P., & Wasserman, S. (in press). Logit models and logistic regressions for social networks: II. Extensions and generalizations to valued and bivariate relations. Journal of Quantitative Anthropology.Google Scholar
Reitz, K. P. (1982). Using log linear analysis with network data: Another look at Sampson's monastery. Social Networks, 4, 243–256.CrossRefGoogle Scholar
Ripley, B. (1981). Spatial statistics, New York: Wiley.CrossRefGoogle Scholar
Sampson, S. F. (1968). A novitiate in a period of change: An experimental and case study of relationships, Ithaca, NY: Department of Sociology, Cornell University.Google Scholar
Snijders, T. A. B. (1991). Enumeration and simulation methods for 0–1 matrices with given marginals. Psychometrika, 56, 397–417.CrossRefGoogle Scholar
Speed, T. P. (1978). Relations between models for spatial data, contingency tables, and Markov fields on graphs. Supplement Advances in Applied Probability, 10, 111–122.CrossRefGoogle Scholar
Strauss, D. (1977). Clustering on colored lattices. Journal of Applied Probability, 14, 135–143.CrossRefGoogle Scholar
Strauss, D. (1986). On a general class of models for interaction. SIAM Review, 28, 513–527.CrossRefGoogle Scholar
Strauss, D. (1992). The many faces of logistic regression. The American Statistician, 46, 321–327.CrossRefGoogle Scholar
Strauss, D., Ikeda, M. (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85, 204–212.CrossRefGoogle Scholar
Vickers, M. (1981). Relational analysis: An applied evaluation, Australia: Department of Psychology, University of Melbourne.Google Scholar
Vickers, M., Chan, S. (1981). Representing classroom social structure, Melbourne: Victoria Institute of Secondary Education.Google Scholar
Walker, M. E. (1995). Statistical models for social support networks: Application of exponential models to undirected graphs with dyadic dependencies. Unpublished doctoral dissertation, University of Illinois, Department of Psychology.Google Scholar
Wang, Y. J., Wong, G. Y. (1987). Stochastic blockmodels for directed graphs. Journal of the American Statistical Association, 82, 8–19.CrossRefGoogle Scholar
Wasserman, S. (1978). Models for binary directed graphs and their applications. Advances in Applied Probability, 10, 803–818.CrossRefGoogle Scholar
Wasserman, S. (1987). Conformity of two sociometric relations. Psychometrika, 52, 3–18.CrossRefGoogle Scholar
Wasserman, S., Faust, K. (1994). Social network analysis: Methods and applications, Cambridge, England: Cambridge University Press.CrossRefGoogle Scholar
Wasserman, S., Galaskiewicz, J. (1984). Some generalizations of p1: External constraints, interactions, and non-binary relations. Social Networks, 6, 177–192.CrossRefGoogle Scholar
Wasserman, S., Iacobucci, D. (1986). Statistical analysis of discrete relational data. British Journal of Mathematical and Statistical Psychology, 39, 41–64.CrossRefGoogle Scholar