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Models for binary directed graphs and their applications

Published online by Cambridge University Press:  01 July 2016

Stanley S. Wasserman
Stanley S. Wasserman*
Affiliation:
University of Minnesota
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Abstract

The nature and historical development of both stochastic and deterministic models for binary graphs are discussed. Here the focus of applications is sociological and emphasizes representations of networks of interpersonal relations as directed graphs. Models from the natural sciences and from the social sciences are examined and suggestions for future research are given.

Keywords

GRAPH THEORYSOCIAL NETWORKSSTOCHASTIC MODELLING
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 803 - 818
Copyright
Copyright © Applied Probability Trust 1978 

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References

Bartholomew, D. G. (1973) Stochastic Models for Social Processes. Wiley, London.Google Scholar
Bernard, H. R. and Killworth, P. D. (1979) Deterministic models of social networks. In Holland, and Leinhardt, (eds) (1979).Google Scholar
Breiger, R. L., Boorman, S. A. and Arabie, P. (1975) An algorithm for clustering relational data with application to social network analysis and comparison with multidimensional scaling. J. Math. Psychol. 12, 328–383.CrossRefGoogle Scholar
Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629–641.Google Scholar
Bush, R. R. and Mosteller, F. (1955) Stochastic Models for Learning. Wiley, New York.CrossRefGoogle Scholar
Cartwright, D. and Harary, F. (1956) Structural balance: A generalization of Heider's theory. Psychol. Rev. 63, 277–293. Also in Leinhardt, (ed.) (1977).Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Translation Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
Davis, J. A. (1967) Clustering and structural balance in graphs. Human Relations 20, 181–187. Also in Leinhardt, (ed.) (1977).Google Scholar
Davis, J. A. (1979) The Davis–Holland–Leinhardt studies: An overview. In Holland, and Leinhardt, (eds.) (1979).Google Scholar
Ford, L. R. and Fulkerson, D. R. (1962) Flows in Networks. Princeton University Press, Princeton, N.J. Google Scholar
Frisch, H. L. and Hammersley, J. M. (1963) Percolation processes and related topics. SIAM J. 11, 894–918.Google Scholar
Hammersley, J. M. and Welsh, D. J. A. (1965) First-passage percolation processes, stochastic networks, and generalized renewal theory. In Bernoulli, Bayes, Laplace Anniversary Volume, Springer-Verlag, Berlin, 61–110.Google Scholar
Harary, F., Norman, R. Z. and Cartwright, D. (1965) Structural Models: An Introduction to the Theory of Directed Graphs. Wiley, New York.Google Scholar
Heider, F. (1958) The Psychology of Interpersonal Relations. Wiley, New York.CrossRefGoogle Scholar
Holland, P. W. and Leinhardt, S. (1975) Local structure in social networks. In Sociological Methodology 1976, ed. Heise, D. R., Jossey-Bass, San Francisco.Google Scholar
Holland, P. W. and Leinhardt, S. (1977a) A dynamic model for social networks. J. Math. Sociol. 5, 5–20.Google Scholar
Holland, P. W. and Leinhardt, S. (1977b) Social structure as a network process. Z. Soziol. 6, 386–402.Google Scholar
Holland, P. W. and Leinhardt, S., (eds.) (1979) Social Networks: Surveys, Advances, and Commentaries. Academic Press, New York.Google Scholar
Katz, L. and Proctor, C. H. (1959) The concept of configuration of interpersonal relations in a group as a time-dependent stochastic process. Psychometrika 24, 317–327.Google Scholar
Kingman, J. F. C. (1973) Subadditive ergodic theory (with discussion). Ann. Prob. 1, 883–909.Google Scholar
Leinhardt, S. (1977) Social networks: a developing paradigm. In Leinhardt, (ed.) (1977).Google Scholar
Leinhardt, S., (ed.) (1977) Social Networks: A Developing Paradigm. Academic Press, New York.Google Scholar
Morgan, B. J. T. (1976) Stochastic models of grouping changes. Adv. Appl. Prob. 8, 30–57.Google Scholar
Rainio, K. (1966) A study on sociometric group structure: an application of a stochastic theory of social interaction. In Sociological Theories in Progress, Vol. 1, ed. Berger, J., Zelditch, M. and Anderson, B.. Houghton Mifflin, Boston.Google Scholar
Rapoport, A. (1957) Contribution to the theory of random and biased nets. Bull. Math. Biophys. 19, 257–277. Also in Leinhardt (ed.) (1977).Google Scholar
Rapoport, A. (1963) Mathematical models of social interaction. In Handbook of Mathematical Psychology, Vol. 2, ed. Luce, R. D., Bush, R. R. and Galanter, E., Wiley, New York.Google Scholar
Smythe, R. T. (1976) Remarks on renewal theory for percolation processes. J. Appl. Prob. 13, 290–300.Google Scholar
Smythe, R. T. and Wierman, J. C. (1977) First-passage percolation on the square lattice. I. Adv. Appl. Prob. 9, 38–54.Google Scholar
Sørenson, A. B. and Hallinan, M. (1976) A stochastic model for change in group structure. Soc. Sci. Res. 5, 43–61.Google Scholar
Wasserman, S. S. (1977a) Random directed graph distributions and the triad census in social networks. J. Math. Sociol. 5, 61–86.CrossRefGoogle Scholar
Wasserman, S. S. (1977b) Stochastic Models for Directed Graphs. Ph.D. Dissertation, Harvard University.Google Scholar
White, H. C., Boorman, S. A. and Breiger, R. L. (1976) Social structure from multiple networks. I. Block models of roles and positions. Amer. J. Sociol. 81, 730–780.Google Scholar
Whittle, P. (1965a) Statistical processes of aggregation and polymerization. Proc. Camb. Phil. Soc. 61, 475–495.Google Scholar
Whittle, P. (1965b) The equilibrium statistics of a clustering process in the uncondensed phase. Proc. R. Soc. London A285, 501–519.Google Scholar