Skip to main content
SpringerLink
Log in

Algebraic Statistics for p 1 Random Graph Models: Markov Bases and Their Uses

  • Conference paper
  • First Online:
Looking Back

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 202))

Abstract

The term social network connotes a social structure composed of individuals (or organizations), typically labeled as nodes, linked by one or more relations, such as friendship, information sharing, financial transaction, and so on. Social network analysis uses a graphical representation where the individuals correspond to nodes in the graph and the presence of a relationship to edges. The term social network now also describes specific social structures on the World Wide Web and many authors have examined the Web itself in various forms as a social network.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • ti2 Team. (2008). 4ti2 – A software package for algebraic, geometric and combinatorial problems on linear spaces [Computer software]. Available from http://www.4ti2.de.

  • Airoldi, E. M., Blei, D. M., Fienberg, S. E., & Xing, E. P. (2008). Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9, 1823–1856.

    Google Scholar 

  • Barndorff-Nielsen, O. E. (1978). Information and exponential families in statistical theory. New York, NY: Wiley.

    MATH  Google Scholar 

  • Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (2007). Discrete multivariate analysis: Theory and practice. New York, NY: Springer-Verlag (Original work published 1975).

    MATH  Google Scholar 

  • Brown, L. D. (1986). Fundamentals of statistical exponential families. Hayward, CA: Institute of Mathematical Statistics.

    MATH  Google Scholar 

  • Cox, D., Little, J., & O’Shea, D. (2005). Using algebraic geometry. Berlin, Germany: Springer-Verlag.

    MATH  Google Scholar 

  • Cox, D., Little, J., & O’Shea, D. (2007). Ideals, varieties, and algorithms. Berlin, Germany: Springer-Verlag.

    Book  MATH  Google Scholar 

  • Diaconis, P., & Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distribution. Annals of Statistics, 26, 363–397.

    Article  MathSciNet  MATH  Google Scholar 

  • Dobra, A., Fienberg, S. E., Rinaldo, A., Slavkovic, A. B., & Zhou, Y. (2008). Algebraic statistics and contingency table problems: Log-linear models, likelihood estimation, and disclosure limitation. In M. Putinar & S. Sullivant (Eds.), Emerging applications of algebraic geometry, pp. 63–88. New York, NY: Springer-Verlag.

    Google Scholar 

  • Drton, M., Sturmfels, B., & Sullivant, S. (2009). Lectures on algebraic statistics. Basel, Switzerland: Springer Basel AG.

    Book  MATH  Google Scholar 

  • Erdös, P., & Rényi, A. (1960). The evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 5, 17–61.

    MATH  Google Scholar 

  • Eriksson, N., Fienberg, S. E., Rinaldo, A., & Sullivant, S. (2006). Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models. Journal of Symbolic Computation, 41, 222–233.

    Article  MathSciNet  MATH  Google Scholar 

  • Fienberg, S. E. (2007). Editorial, expanding the statistical toolkit with algebraic statistics. Statistica Sinica, 17, 1261–1272.

    MathSciNet  MATH  Google Scholar 

  • Fienberg, S. E., Meyer, M. M., & Wasserman, S. S. (1985). Statistical analysis of multiple sociometric relations. Journal of the American Statistical Association, 80, 51–67.

    Article  Google Scholar 

  • Fienberg, S. E., & Wasserman, S. S. (1981a). Categorical data analysis of single sociometric relations. Sociological Methodology, 1981, 156–192.

    Article  Google Scholar 

  • Fienberg, S. E., & Wasserman, S. S. (1981b). Discussion of Holland, P. W. and Leinhardt, S., An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76, 54–57.

    MathSciNet  Google Scholar 

  • Fischer, G. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59–77.

    Article  MathSciNet  MATH  Google Scholar 

  • Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81, 832–842.

    Article  MathSciNet  MATH  Google Scholar 

  • Gawrilow, E., & Joswig, M. (2008). Polymake – A software package for analyzing convex polytopes [Computer software]. Available from http://www.math.tu-berlin.de/polymake/.

  • Geiger, D., Meek, C., & Sturmfels, B. (2006). On the toric algebra of graphical models. Annals of Statistics, 34(3), 1462–1492.

    Article  MathSciNet  Google Scholar 

  • Gibilisco, P., Riccomagno, E., Rogantin, M. P., & Wynn, H. P. (Eds.). (2009). Algebraic and geometric methods in statistics. Cambridge, England: Cambridge University Press.

    Google Scholar 

  • Haberman, S. J. (1974). The analysis of frequency data. Chicago, IL: University of Chicago Press.

    MATH  Google Scholar 

  • Haberman, S. J. (1977). Maximum likelihood estimates in exponential response models. Annals of Statistics, 5(5), 815–841.

    Article  MathSciNet  MATH  Google Scholar 

  • Haberman, S. J. (1981). An exponential family of probability distributions for directed graphs: Comment. Journal of the American Statistical Association, 76, 60–61.

    MathSciNet  Google Scholar 

  • Holland, P. W., & Leinhardt, S. (1970). A method for detecting structure in sociometric data. American Journal of Sociology, 70, 492–513.

    Article  Google Scholar 

  • Holland, P. W., & Leinhardt, S. (1971). Transitivity in structural models of small groups. Small Group Research, 2, 107–124.

    Article  Google Scholar 

  • Holland, P. W., & Leinhardt, S. (1976). Local structure in social networks. Sociological Methodology, 7, 1–45.

    Article  Google Scholar 

  • Holland, P. W., & Leinhardt, S. (1978). An omnibus test for social structure using triads. Sociological Methods Research, 7, 227–256.

    Article  Google Scholar 

  • Holland, P. W., & Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs (with discussion). Journal of the American Statistical Association, 76, 33–65.

    Article  MathSciNet  MATH  Google Scholar 

  • Hunter, D. R., Goodreau, S., & Handcock, M. S. (2008). Goodness of fit of social network models. Journal of the American Statistical Association, 103, 248–258.

    Article  MathSciNet  MATH  Google Scholar 

  • Katz, L. (1951). The distribution of the number of isolates in a social group. Annals of Mathematical Statistics, 23, 271–276.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Milgram, S. (1967). The small world problem. Psychology Today, 1(1), 60–67.

    Google Scholar 

  • Moreno, J. L. (1934). Who shall survive? Washington, DC: Nervous and Mental Disease Publishing Company.

    Google Scholar 

  • Moreno, J. L. (1951). Sociometry, Experimental method and the science of society. An approach to a new political orientation. New York, NY: Beacon House.

    Google Scholar 

  • Ohsugi, H., & Hibi, T. (1999). Toric ideals generated by quadratic binomials. Journal of Algebra, 218(2), 509–527.

    Article  MathSciNet  MATH  Google Scholar 

  • Petrović, S., Rinaldo, A., & Fienberg, S. E. (2010). Algebraic statistics for a directed random graph model with reciprocation. In M. Viana & H. Wynn (Eds.), Algebraic methods in statistics and probability, Volume II. Contemporary Mathematics, Providence, RI: American Mathematical Society, 516, 261–283.

    Google Scholar 

  • Petrović, S., Rinaldo, A., & Fienberg, S. E. (2011). Maximum likelihood estimation in network models. Manuscript submitted for publication.

    Google Scholar 

  • Pistone, G., Riccomagno, E., & Wynn, H. P. (2001). Algebraic statistics: Computational commutative algebra in statistics. New York, NY: Chapman & Hall.

    MATH  Google Scholar 

  • Rinaldo, A. (2006). Computing maximum likelihood estimates in log-linear models (Technical Rep. No. 835). Pittsburgh, PA: Carnegie Mellon University, Department of Statistics.

    Google Scholar 

  • Rinaldo, A., Fienberg, S. E., & Zhou, Y. (2009). On the geometry of discrete exponential families with application to exponential random graph models. Electronic Journal of Statistics, 3, 446–484.

    Article  MathSciNet  Google Scholar 

  • Strauss, D., & Ikeda, M. (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85, 204–212.

    Article  MathSciNet  Google Scholar 

  • Sturmfels, B. (1996). Gröbner bases and convex polytopes (University Lecture Series (Vol. 8)). Providence, RI: American Mathematical Society.

    Google Scholar 

  • Travers, J., & Milgram, S. (1969). An experimental study of the small world problem. Sociometry, 32, 425–443.

    Article  Google Scholar 

  • Villarreal, R. (2001). Monographs and textbooks in pure and applied mathematics: Vol. 8. Monomial algebras. New York, NY: Marcel Dekker.

    Google Scholar 

  • Wasserman, S., & Pattison, P. (1996). Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and \( {p^* } \). Psychometrika, 61, 401–425.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

To Paul Holland, whose work on contingency tables and network models continues to provide us with research ideas. This paper continues the exploration of their connections.

This research was supported in part by NSF grant DMS-0631589 and a grant from the Pennsylvania Department of Health through the Commonwealth Universal Research Enhancement Program. We have received valuable comment from a number of colleagues when we presented preliminary versions of this paper at a series of workshops. We are especially grateful to two reviewers for helpful comments on an earlier draft of the paper.

Author information

Authors and Affiliations

  1. Department of Statistics, Carnegie Mellon University, 132G Baker Hall, Pittsburgh, PA, 15213, USA

    Stephen E. Fienberg

Authors
  1. Stephen E. Fienberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sonja Petrović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alessandro Rinaldo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen E. Fienberg .

Editor information

Editors and Affiliations

  1. , Research and Development, Educational Testing Service, MS 12T, Rosedale Road, Princeton, 08541, New Jersey, USA

    Neil J. Dorans

  2. , Research and Development, Educational Testing Service, MS 12T, Rosedale Road, Princeton, 08541, New Jersey, USA

    Sandip Sinharay

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this paper

Cite this paper

Fienberg, S.E., Petrović, S., Rinaldo, A. (2011). Algebraic Statistics for p 1 Random Graph Models: Markov Bases and Their Uses. In: Dorans, N., Sinharay, S. (eds) Looking Back. Lecture Notes in Statistics(), vol 202. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9389-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation