Journal of Classification

, Volume 11, Issue 1, pp 121–149

Metric inference for social networks

  • David Banks
  • Kathleen Carley
Article

Abstract

Using a natural metric on the space of networks, we define a probability measure for network-valued random variables. This measure is indexed by two parameters, which are interpretable as a location parameter and a dispersion parameter. From this structure, one can develop maximum likelihood estimates, hypothesis tests and confidence regions, all in the context of independent and identically distributed networks. The value of this perspective is illustrated through application to portions of the friedship cognitive social structure data gathered by Krackhardt (1987).

Keywords

Random networks Random graphs Digraphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BANKS, D. L. (1989), “Bootstrapping II,” inThe Encyclopedia of Statistical Science, Ed., S. Kotz, N. Johnson, and C. Read, New York: Wiley, 17–22.Google Scholar
  2. BARTHÉLEMY, J. P., and MCMORRIS, F. R. (1986)., “The Median Procedure for n-Trees,”Journal of Classification, 3, 329–334.Google Scholar
  3. BARTHÉLEMY, J. P. and MONJARDET, B. (1981), “The Median Procedure in Cluster Analysis and Social Choice Theory,”Mathematical Social Sciences, 1, 235–268.Google Scholar
  4. BARTHÉLEMY J. P., and MONJARDET, B. (1988), “The Median Procedure in Data Analysis: New Results and Open Problems,” in:Classification and Related Methods of Data Analysis, Ed., H. H. Bock, North Holland: Elsevier, 309–316.Google Scholar
  5. BARTHÉLEMY, J. P., LECLERC, B., and MONJARDET, B. (1986), “On the Use of Ordered Sets in Problems of Comparison and Consensus of Classifications,”Journal of Classification, 3, 187–224.Google Scholar
  6. BERNARD, H. R., and KILLWORTH, P. (1977), “Informant Accuracy in Social Network Data II,”Human Communications Research, 4, 3–18.Google Scholar
  7. BERNARD, H. R., KILLWORTH, P., and SAILER, L. (1984), “The Problem of Informant Accuracy: The Validity of Retrospective Data,”Annual Review of Anthropology, 13, 495–517.Google Scholar
  8. BLOEMENA, A. R. (1964),Sampling from a Graph, Amsterdam: Mathematisch Centrum.Google Scholar
  9. BOLLOBÁS, B. (1985),Random Graphs, London: Academic Press.Google Scholar
  10. BOORMAN, S. A., and OLIVIER, D. (1973), “Metrics on Spaces of Finite Trees,”Journal of Mathematical Psychology, 10, 26–59.Google Scholar
  11. CAPOBIANCO, M. (1970), “Statistical Inference in Finite Populations Having Structure,”Transactions of the New York Academy of Sciences, 32, 401–413.Google Scholar
  12. CARLEY, K. M. (1984),Constructing Consensus, Unpublished doctoral dissertation, Harvard.Google Scholar
  13. CARLEY, K. (1986), “An Approach for Relating Social Structure to Cognitive Structure,”Journal of Mathematical Sociology, 12, 137–189.Google Scholar
  14. CARLEY, K. (1986), “Formalizing the Social Expert's Knowledge,”Sociological Methods and Research, 17, 165–232.Google Scholar
  15. CARLEY, K. (1991), “A Theory of Group Stability,”American Sociological Review, 56, pp. 331–354.Google Scholar
  16. CONDORCET, M. (1785),Essai sur l'application de l'analyse á la probabilité des décisions rendues á la pluralité des voix, Paris.Google Scholar
  17. CRAMÉR, H. (1961)Mathematical Methods of Statistics, Princeton, N.J.: Princeton University Press.Google Scholar
  18. DAY, W. H. E. (1986), “Foreword: Comparison and Consensus of Classifications,”Journal of Classification, 3, 183–186.Google Scholar
  19. EFRON, B. (1982),The Jackknife, the Bootstrap, and Other Resampling Plans, SIAM Monograph No. 38, Philadelphia: CBMS-NSF.Google Scholar
  20. EFRON, B., and TIBSHIRANI, R. (1986), “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy,”Statistical Science, 1, 54–75.Google Scholar
  21. FESTINGER, L. (1949), “The Analysis of Sociograms Using Matrix Algebra,”Human Relations, 2, 153–158.Google Scholar
  22. FIENBERG, S. E., MEYER, M. M., and WASSERMAN, S. S. (1985), “Statistical Analysis of Multiple Sociometric Relations,”Journal of the American Statistical Association, 80, 51–67.Google Scholar
  23. FISHER, N. and HALL, P. (1990), “On Bootstrap Hypothesis Testing,”Australian Journal of Statisticsl, 32, 177–190.Google Scholar
  24. FRANK, O. (1971),Statistical Inference in Graphs, Stockholm: Swedish Research Institute of National Defense.Google Scholar
  25. FRANK, O. (1988), “Random Sampling and Social Networks: A Survey of Various Approaches,”Mathématiques, Informatique and Sciences Humaines, 26, 19–33.Google Scholar
  26. FRANK, O. (1989), “Random Graph Mixtures,” inGraph Theory and Its Applications: East and West, Proceedings of the First China-USA International Graph Theory Conference Eds., M. F. Capobianco, M. Guan, D. F. Hsu and F. Tian,Annals of the New York Academy of Sciences,576, 192–199.Google Scholar
  27. FRANK, O., and STRAUSS, D. (1986), “Markov Graphs,”Journal of the American Statistical Association, 81, 832–842.Google Scholar
  28. HAMMING, R. W. (1950), “Error Detecting and Error Correcting Codes,”Bell System Technical Journal, 29, 147–160.Google Scholar
  29. HAMMING, R. W. (1980),Coding and Information Theory, Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  30. HOLLAND, P. W., and LEINHARDT, S. (1981), “An Exponential Family of Probability Distributions for Directed Graphs,”Journal of the American Statistical Association, 76, 33–65.Google Scholar
  31. HUBERT, L., and ARABIE P. (1985), “Comparing Partitions,”Journal of Classification, 2, 193–218.Google Scholar
  32. IACOBUCCI, D., and WASSERMAN, S. (1986), “Statistical Analysis of Discrete Relational Data,”British Journal of Mathematical and Statistical Psychology, 39, 41–64.Google Scholar
  33. JARDINE, N., and SIBSON, R. (1971),Mathematical Taxonomy, Chichester: Wiley.Google Scholar
  34. KATZ, L. (1947), “On the Matric Analysis of Sociometric Data,”Sociometry, 10, 233–241.Google Scholar
  35. KATZ, L. (1953), “A New Status Index Derived from Sociometric Analysis,”Psychometrika, 18, 39–43.Google Scholar
  36. KATZ, L., and POWELL, J. H. (1955), “Measurement of the Tendency Towards Reciprocation of Choice,”Sociometry and the Science of Man, 18, 659–665.Google Scholar
  37. KEMENY, J. G. (1959), “Mathematics Without Numbers,”Daedalus, 88, 577–591.Google Scholar
  38. KNOKE, D., and KUKLINSKI, J. D. (1982),Network Analysis, Beverly Hills: Sage Publications.Google Scholar
  39. KRACKHARDT, D. (1987), “Cognitive Social Structures”Social Networks, 9, 109–134.Google Scholar
  40. KRACKHARDT, D., and PORTER, L. W. (1985), “When Friends Leave: A Structural Analysis of the Relationship Between Turnover and Stayer's Attitudes,”Administrative Science Quarterly, 30, 242–261.Google Scholar
  41. LEHMANN, E. (1983),Theory of Point Estimation, New York: Wiley.Google Scholar
  42. MACEVOY, B., and FREEMAN, L. (1988),UCINET: A Microcomputer Package for Network Analysis, Mathematical Social Science Group, School of Social Sciences, University of California, Irvine.Google Scholar
  43. MALLOWS, C. (1957), “Non-Null Ranking Models I,”Biometrika, 44, 114–130.Google Scholar
  44. MARGUSH, T. (1982), “Distances Between Trees,”Discrete Applied Mathematics, 4, 281–290.Google Scholar
  45. MARGUSH, T., and MCMORRIS, F. R. (1981), “Consensus n-Trees,”Bulletin of Mathematical Biology, 43, 239–244.Google Scholar
  46. MCMORRIS, F. R. (1990), “The Median Procedure for n-Trees as a Maximum Likelihood Method,”Journal of Classification, 7, 77–80.Google Scholar
  47. MORENO, J. L. (1934),Who Shall Survive? Washington, D.C.: Nervous and Mental Disease Publishing.Google Scholar
  48. PALMER, E. (1985),Graphical Evolution: An Introduction to the Theory of Random Graphs, New York: Wiley.Google Scholar
  49. PEARSON, K. (1900), “On the Criterion That a Given System of Deviations From the Probable In the Case of a Correlated System of Random Variables Is Such That It Can Be Reasonably Supposed to Have Arisen From Random Sampling,”Philosophy Magazine, 50, 157–172.Google Scholar
  50. RANDLES R., and WOLFE, D. A. (1979),Introduction to the Theory of Nonparametric Statistics, New York: Wiley.Google Scholar
  51. ROMNEY, A. K., and FAUST, K. (1982), “Predicting the Structure of a Communications Network from Recall Data,”Social Networks, 4, 285–304.Google Scholar
  52. STRAUSS, D., and IKEDA, M. (1990), “Pseudolikelihood Estimation for Social Networks,”Journal of the American Statistical Association, 85, 204–212.Google Scholar
  53. WANG, Y., and WONG, G. Y. (1987), “Stochastic Blockmodels for Directed Graphs,”Journal of the American Statistical Association, 82, 8–19.Google Scholar
  54. WASSERMAN, S. (1987), “Conformity of Two Sociometric Relations,”Psychometrika, 52, 3–18.Google Scholar
  55. WASSERMAN, S., and ANDERSON, C. (1987), “Stochastic a posteriori Blockmodels: Construction and Assessment,”Social Networks, 9, 1–36.Google Scholar
  56. WASSERMAN, S., and GALASKIEWICZ, J. (1984), “Some Generalizations ofP i: External Constraints, Interactions, and Non-Binary Relations,”Social Networks, 6, 177–192.Google Scholar
  57. WEISBERG, S. (1980),Applied Linear Regression, New York, Wiley.Google Scholar
  58. WONG, G. Y. (1987), “Bayesian Models for Directed Graphs,”Journal of the American Statistical Association, 82, 140–148.Google Scholar
  59. YOUNG, H. P. (1988), “Condorcet Theory of Voting,”American Political Science Review, 82, 1231–1244.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • David Banks
    • 1
  • Kathleen Carley
    • 2
  1. 1.Department of StatisticsCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Social and Decision SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations