Advertisement

Statistical Models for Social Networks

  • Stanley Wasserman
  • Philippa Pattison
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Abstract

Recent developments in statistical models for social networks reflect an increasing theoretical focus in the social and behavioral sciences on the interdependence of social actors in dynamic, network-based social settings (e.g., Abbott, 1997; White, 1992, 1995). As a result, a growing importance has been accorded the problem of modeling the dynamic and complex interdependencies among network ties and the actions of the individuals whom they link. Included in this problem is the identification of cohesive subgroups, or classifications of the individuals. The early focus of statistical network modeling on the mathematical and statistical properties of Bernoulli and dyad-independent random graph distributions has now been replaced by efforts to construct theoretically and empirically plausible parametric models for structural network phenomena and their changes over time.

Keywords

Social Network Random Graph Dependence Graph Random Graph Model Sociological Methodology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ABBOTT, A. (1997): Of time and space: the contemporary relevance of the Chicago School. Social Forces, 75, 1149–1182.Google Scholar
  2. ANDERSON, C., WASSERMAN, S. & FAUST, K. (1992): Building stochastic blockmodels. Social Networks, 14, 137–161.CrossRefGoogle Scholar
  3. BESAG, J.E. (1974): Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B, 36, 96–127.Google Scholar
  4. BOLLOBAS, B.(1985): Random Graphs. London: Academic Press.Google Scholar
  5. van de Bunt, G., van Duijn, M., & Snijders, T.A.B.(1995): Friendship networks and rational choice. In M.G. Everett K. & K. Rennolls (eds.), Proceedings of the International Conference on Social Networks, Volume I, London, 6th-10th July, 1995. Greenwich: University of Greenwich Press.Google Scholar
  6. CARTWRIGHT, D. & HARARY, F. (1979): Balance and cluster ability: An overview. In P. W. Holland & S. Leinhardt, (eds.), Perspectives on Social Network Research, pages 25–50. New York: Academic Press.Google Scholar
  7. CROUCH, B., & WASSERMAN, S. (1998): Fitting P*: Monte Carlo maximum likelihood estimation. Paper presented at International Conference on Social Networks, Sitges, Spain, May 28–31.Google Scholar
  8. DAVIS, J. A. (1967): Clustering and structural balance in graphs. Human Relations, 20, 181–187.CrossRefGoogle Scholar
  9. DAVIS, J.A. (1979): The Davis/Holland/Leinhardt studies: An overview. In P. W. Holland & S. Leinhardt, (eds.), Perspectives on Social Network Research, pages 51–62. New York: Academic Press.Google Scholar
  10. DOREIAN, P.(1982):Maximum likelihood methods for linear models. Sociological Methods & Research, 10, 243–269.CrossRefGoogle Scholar
  11. DOREIAN, P., & STOKMAN, F. (1997): Evolution of Social Networks. Amsterdam: Gordon & Breach.Google Scholar
  12. EMIRBAYER, M. (1997): Manifesto for a relational sociology. American Journal of Sociology. 103, 281–317.CrossRefGoogle Scholar
  13. EMIRBAYER, M., & Goodwin, J.(1994): Network analysis, culture, and the problem of agency. American Journal of Sociology, 99, 1411–1454.CrossRefGoogle Scholar
  14. ERDÖS, P.(1959): Graph theory and probability, I. Canadian Journal of Mathematics, 11, 34’38.Google Scholar
  15. ERDÖS, P.(1961): Graph theory and probability, II. Canadian Journal of Mathematics, 13, 346–352.CrossRefGoogle Scholar
  16. ERDÖS, P., & RENYI, A. (1960): On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5, 17–61.Google Scholar
  17. FAUST, K., & SKVORETZ, J. (1999): Logit models for affiliation networks. In M. Becker, & M. Sobel (eds.), Sociological Methodology 1999, pages 253–280. New York: Basil Black well.Google Scholar
  18. FIENBERG, S. E., MEYER, M. M., & WASSERMAN, S. (1985): Statistical analysis of multiple sociornetric relations. Journal of the American Statistical Association, 80, 51–67.CrossRefGoogle Scholar
  19. FIENBERG, S., & WASSERMAN, S. (1981): Categorical data analysis of single sociornetric relations. In S. Leinhardt (ed.), Sociological Methodology 1981, pages 156–192. San Francisco: Jossey-Bass.Google Scholar
  20. FRANK, O. (1977): Estimation of graph totals. Scandinavian Journal of Statistics, 4, 81–89.Google Scholar
  21. FRANK, O. (1980): Sampling and inference in a population graph. International Statistical Review, 48, 33–41.CrossRefGoogle Scholar
  22. FRANK, O. (1981): A survey of statistical methods for graph analysis. In S. Leinhardt (ed.), Sociological Methodology 1981, pages 110–155. San Francisco: Jossey-Bass.Google Scholar
  23. FRANK O. (1989): Random graph mixtures. Annals of the New York Academy of Science. 576: Graph Theory and its Applications, pages 192–199. New York: East and West.Google Scholar
  24. FRANK, O., & Nowicki, K. (1993): Exploratory statistical analysis of networks. In J. Gimbel, J.W. Kennedy, & L. V. Quintas (eds.), Quo Vadis Graph Theory? A Source Book for Challenges and Directions. Amsterdam: North-Holland, (also Annals of Discrete Mathematics, 55, 349–366.).Google Scholar
  25. FRANK, O., & Strauss, D. (1986): Markov graphs. Journal of the American Statistical Association, 81, 832–842.CrossRefGoogle Scholar
  26. FRIEDKIN, N. (1998): A Structural Theory of Social Influence. New York: Cambridge University Press.CrossRefGoogle Scholar
  27. GEYER, C., & THOMPSON, E (1992): Constrained Monte Carlo maximum likelihood for dependent data. Journal of the Royal Statistical Society, Series B, 54, 657–699.Google Scholar
  28. GILBERT, E. N. (1959): Random graphs. Annals of Mathematical Statistics, 30, 1141–1144.CrossRefGoogle Scholar
  29. GRANOVETTER, M. (1973): The strength of weak ties. American Journal of Sociology, 78, 1360–1380.CrossRefGoogle Scholar
  30. HOLLAND, P. W., LASKEY, K. B., & LEINHARDT, S. (1983): Stochastic block- models: some first steps. Social Networks, 5, 109–137.CrossRefGoogle Scholar
  31. HOLLAND, P. W., & LEINHARDT, S. (1970): A method for detecting structure in sociornetric data. American Journal of Sociology, 70, 492–513.CrossRefGoogle Scholar
  32. HOLLAND, P. W. & LEINHARDT, S. (1975): The statistical analysis of local structure in social networks. In D. R. Heise (ed.), Sociological Methodology 1976, pp. 1–45. San Francisco: Jossey-Bass.Google Scholar
  33. HOLLAND, P. W., & LEINHARDT, S. (1981): An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76, 33–50.CrossRefGoogle Scholar
  34. HUBERT, L. J., & BAKER, F. B. (1978): Evaluating the conformity of sociornetric measurements. Psychometrika, 43, 31–41.CrossRefGoogle Scholar
  35. IACOBUCCI, D., & WASSERMAN, S. (1988): A general framework for the statistical analysis of sequential dyadic interaction data. Psychological Bulletin, 103, 379–390.CrossRefGoogle Scholar
  36. KATZ, L., & POWELL, J. H. (1953): A proposed index of conformity of one sociornetric measurement to another. Psychometrika, 18, 249–256.CrossRefGoogle Scholar
  37. KATZ, L., & POWELL, J. H. (1957): Probability distributions of random variables associated with a structure of the sample space of sociometric investigations. Annals of Mathematical Statistics, 28, 442–448.CrossRefGoogle Scholar
  38. KATZ, L., & PROCTOR, C.H. (1959). The concept of configuration of interpersonal relations in a group as a time-dependent stochastic process. Psychometrika, 24, 317–327.CrossRefGoogle Scholar
  39. LAURITZEN, S. (1996): Graphical Models. Oxford: Oxford University Press.Google Scholar
  40. LAZEGA, E. and VAN DUIJN, M. (1997): Position in formal structure, personal characteristics and choices of advisors in a law firm: A logistic regression model for dyadic network data. Social Networks, 19, 375–397.CrossRefGoogle Scholar
  41. LAZEGA, E., & PATTISON, P. (1999): Multiplexity, generalized exchange and cooperation in organizations. Social Networks, 21, 67–90.CrossRefGoogle Scholar
  42. LEENDERS, R. (1995): Models for network dynamics: A Markovian framework. Journal of Mathematical Sociology, 20, 1–21.CrossRefGoogle Scholar
  43. LEENDERS, R. (1996): Evolution of friendship and best friendship choices. Journal of Mathematical Sociology, 21, 133–148.CrossRefGoogle Scholar
  44. LINDENBERG, S. (1997): Grounding groups in theory: functional, cognitive and structural interdependencies. Advances in Group Processes, 14, 281–331.Google Scholar
  45. MANTEL, N. (1967): The detection of disease clustering and a generalized regression approach. Cancer Research, 27, 209–220.Google Scholar
  46. MARSDEN, P., & FRIEDKIN, N. (1994): Network studies of social influence. In S. Wasserman J. & Galaskiewicz (eds.), Advances in Social Network Analysis (pages 3–25). Thousand Oaks, CA: Sage.Google Scholar
  47. PATTISON, P., MISCHE, A., & ROBINS, G.L. (1998): The plurality of social relations: k-partite representations of interdependent social forms. Keynote address, Conference on Ordinal and Symbolic Data Analysis, Amherst, Sept. 28–30.Google Scholar
  48. PATTISON, P. E., & WASSERMAN, S. (1999): Logit models and logistic regressions for social networks, II. Multivariate relations. British Journal of Mathematical and Statistical Psychology, 52, 169–194.CrossRefGoogle Scholar
  49. PATTISON, P., WASSERMAN, S., ROBINS, G.L., & KANFER, A.M. (in press): Statistical evaluation of algebraic constraints for social networks. Journal of Mathematical Psychology.Google Scholar
  50. RAPOPORT, A. (1949): Outline of a probabilistic approach to animal sociology, I. Bulletin of Mathematical Biophysics, 11, 183–196.CrossRefGoogle Scholar
  51. ROBINS, G.L., PATTISON, P., & ELLIOTT, P. (in press): Network models for social influence processes. Psychometrika.Google Scholar
  52. ROBINS, G.L., PATTISON, P., & WASSERMAN, S. (1999): Logit models and logistic regressions for social networks, III. Valued relations. Psychometrika, 64, 371–394.CrossRefGoogle Scholar
  53. SNIJDERS, T. A. B. (1991): Enumeration and simulation methods for 0–1 matrices with given marginals. Psychometrika, 56, 397–417.CrossRefGoogle Scholar
  54. SNIJDERS, T. A. B. (1996): Stochastic actor-oriented models for network change. Journal of Mathematical Sociology, 21, 149–172.CrossRefGoogle Scholar
  55. SNIJDERS, T. A. B., & NOWICKI, K. (1997): Estimation and prediction for stochastic blockmodels for graphs with latent block structure. Journal of Classification, 14, 75–100.CrossRefGoogle Scholar
  56. SNIJDERS, T.A.B., & VAN DUIJN, M.A.J. (1997). Simulation for statistical inference in dynamic network models. In R. Conte, R, Hegselmann, & P. Terna (eds.), Simulating Social Phenomena (pages 493–512). Berlin: Springer-Verlag.Google Scholar
  57. STRAUSS, D. (1986): On a general class of models for interaction. SIAM Review, 28, 513–527.CrossRefGoogle Scholar
  58. STRAUSS, D., & IKEDA, M. (1990): Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85, 204–212.CrossRefGoogle Scholar
  59. VAN DE BUNT, G., VAN DUIJN, M., & SNIJDERS, T. A. B. (1999): Friendship networks through time: an actor-oriented dynamic statistical network model. Computational and Mathematical Organization Theory, 5, 167–192.CrossRefGoogle Scholar
  60. VAN DUIJN, M., & SNIJDERS, T. A. B. (1997): p 2: a random effects model for directed graphs. Unpublished manuscript.Google Scholar
  61. WANG, Y. Y., & WONG, G. Y. (1987): Stochastic blockmodels for directed graphs. Journal of the American Statistical Association, 82, 8–19.CrossRefGoogle Scholar
  62. WASSERMAN, S. (1977): Random directed graph distributions and the triad census in social networks. Journal of Mathematical Sociology, 5, 61–86.CrossRefGoogle Scholar
  63. WASSERMAN, S. (1979): A stochastic model for directed graphs with transition rates determined by reciprocity. In K. Schuessler (ed.) Sociological Methodology 1980, pages 392–412. San Francisco: Jossey-Bass.Google Scholar
  64. WASSERMAN, S. (1980): Analyzing social networks as stochastic processes. Journal of the American Statistical Association, 75, 280–294.CrossRefGoogle Scholar
  65. WASSERMAN, S. (1987): Conformity of two sociornetric relations. Psychometrika, 52, 3–18.CrossRefGoogle Scholar
  66. WASSERMAN, S., & FAUST, K. (1994): Social Network Analysis: Methods and Applications. New York: Cambridge University Press.Google Scholar
  67. WASSERMAN, S., & GALASKIEWICZ, J. (1984): Some generalizations of pn External constraints, interactions, and non-binary relations. Social Networks, 6, 177–192.CrossRefGoogle Scholar
  68. WASSERMAN, S., & IACOBUCCI, D. (1986): Statistical analysis of discrete relational data. British Journal of Mathematical and Statistical Psychology, 39, 41–64.CrossRefGoogle Scholar
  69. WASSERMAN, S., & IACOBUCCI, D. (1988): Sequential social network data. Psychometrika, 53, 262–282.CrossRefGoogle Scholar
  70. WASSERMAN, S., & PATTISON, P. E. (1996): Logit models and logistic regressions for social networks, I. An introduction to Markov random graphs and p*. Psychometrika, 60, 401–425.CrossRefGoogle Scholar
  71. WASSERMAN, S., & PATTISON, P. E. (in press): Multivariate Random Graph Distributions. Springer Lecture Note Series in Statistics.Google Scholar
  72. WATTS, D. (1999): Networks, dynamics, and the small-world phenomenon. American Journal of Sociology, 105, 493–527.CrossRefGoogle Scholar
  73. WATTS, D., & STROGATZ, S. (1998): Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442.CrossRefGoogle Scholar
  74. WHITE, H. C. (1992): Identity and Control. Princeton, NJ: Princeton University Press.Google Scholar
  75. WHITE, H. C. (1995): Network switchings and Bayesian forks: reconstructing the social and behavioral sciences. Social Research, 62, 1035–1063.Google Scholar
  76. WHITE, H. C., BOORMAN, S., & BREIGER, R. L. (1976): Social structure from multiple networks, I. Blockmodels of roles and positions. American Journal of Sociology, 81, 730–780.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 2000

Authors and Affiliations

  • Stanley Wasserman
    • 1
  • Philippa Pattison
    • 2
  1. 1.Department of Psychology and Department of Statistics,Beckman Institute for Advanced Science and TechnologyUniversity of IllinoisChampaignUSA
  2. 2.School of Behavioural Science, Department of PsychologyUniversity of MelbourneParkvilleAustralia

Personalised recommendations