Differentially Private Exponential Random Graphs

  • Vishesh Karwa
  • Aleksandra B. Slavković
  • Pavel Krivitsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8744)

Abstract

We propose methods to release and analyze synthetic graphs in order to protect privacy of individual relationships captured by the social network. Proposed techniques aim at fitting and estimating a wide class of exponential random graph models (ERGMs) in a differentially private manner, and thus offer rigorous privacy guarantees. More specifically, we use the randomized response mechanism to release networks under ε-edge differential privacy. To maintain utility for statistical inference, treating the original graph as missing, we propose a way to use likelihood based inference and Markov chain Monte Carlo (MCMC) techniques to fit ERGMs to the produced synthetic networks. We demonstrate the usefulness of the proposed techniques on a real data example.

Keywords

Exponential random graphs edge differential privacy missing data synthetic graphs 

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References

  1. 1.
    Backstrom, L., Dwork, C., Kleinberg, J.: Wherefore art thou r3579x?: anonymized social networks, hidden patterns, and structural steganography. In: Proceedings of the 16th International Conference on World Wide Web, pp. 181–190. ACM (2007)Google Scholar
  2. 2.
    Barak, B., Chaudhuri, K., Dwork, C., Kale, S., McSherry, F., Talwar, K.: Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In: Proceedings of the Twenty-sixth ACM SIGMOD SIGACT-SIGART Symposium on Principles of Database Systems, pp. 273–282. ACM (2007)Google Scholar
  3. 3.
    Bearman, P.S., Moody, J., Stovel, K.: Chains of affection: The structure of adolescent romantic and sexual networks. American Journal of Sociology 110(1), 44–91 (2004)CrossRefGoogle Scholar
  4. 4.
    Besag, J.: Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological), 192–236 (1974)Google Scholar
  5. 5.
    Carroll, R.J., Ruppert, D., Stefanski, L.A., Crainiceanu, C.M.: Measurement error in nonlinear models: a modern perspective. CRC Press (2012)Google Scholar
  6. 6.
    Chaudhuri: Randomized Response: Theory and Techniques (Statistics: A Series of Textbooks and Monographs). CRC Press (September 1987)Google Scholar
  7. 7.
    Dwork, C., Kenthapadi, K., McSherry, F., Mironov, I., Naor, M.: Our data, ourselves: Privacy via distributed noise generation. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 486–503. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Dwork, C., McSherry, F., Nissim, K., Smith, A.: Calibrating noise to sensitivity in private data analysis. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 265–284. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Engström, A., Norén, P.: Polytopes from subgraph statistics. arXiv preprint arXiv:1011.3552 (2010)Google Scholar
  10. 10.
    Fienberg, S.E., Slavković, A.B.: Data Privacy and Confidentiality. International Encyclopedia of Statistical Science, pp. 342–345. Springer (2010)Google Scholar
  11. 11.
    Frank, O., Strauss, D.: Markov graphs. Journal of the American Statistical Association 81(395), 832–842 (1986)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fuller, W.A.: Measurement error models, vol. 305. John Wiley & Sons (2009)Google Scholar
  13. 13.
    Ganta, S.R., Kasiviswanathan, S.P., Smith, A.: Composition attacks and auxiliary information in data privacy. In: Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 265–273. ACM (2008)Google Scholar
  14. 14.
    Geyer, C.J., Thompson, E.A.: Constrained monte carlo maximum likelihood for dependent data (with discussion). Journal of the Royal Statistical Society. Series B. Methodological 54(3), 657–699 (1992)MathSciNetGoogle Scholar
  15. 15.
    Goldenberg, A., Zheng, A.X., Fienberg, S.E., Airoldi, E.M.: A survey of statistical network models. Foundations and Trends® in Machine Learning 2(2), 129–233 (2010)CrossRefGoogle Scholar
  16. 16.
    Goodreau, S.M., Kitts, J.A., Morris, M.: Birds of a feather, or friend of a friend? using exponential random graph models to investigate adolescent social networks. Demography 46(1), 103–125 (2009)CrossRefGoogle Scholar
  17. 17.
    Handcock, M.S.: Statistical models for social networks: Inference and degeneracy. Dynamic Social Network Modeling and Analysis 126, 229–252 (2003)Google Scholar
  18. 18.
    Handcock, M.S., Gile, K.J., et al.: Modeling social networks from sampled data. The Annals of Applied Statistics 4(1), 5–25 (2010)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Hay, M., Li, C., Miklau, G., Jensen, D.: Accurate estimation of the degree distribution of private networks. In: Ninth IEEE International Conference on Data Mining, ICDM 2009, pp. 169–178. IEEE (2009)Google Scholar
  20. 20.
    Hout, A., Heijden, P.G.M.: Randomized response, statistical disclosure control and misclassificatio: a review. International Statistical Review 70(2), 269–288 (2002)CrossRefMATHGoogle Scholar
  21. 21.
    Hunter, D.R., Goodreau, S.M., Handcock, M.S.: Goodness of fit of social network models. Journal of the American Statistical Association 103(481), 248–258 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hunter, D.R., Handcock, M.S., Butts, C.T., Goodreau, S.M., Morris, M.: ergm: A package to fit, simulate and diagnose exponential-family models for networks. Journal of Statistical Software 24(3), nihpa54860 (2008)Google Scholar
  23. 23.
    Karwa, V., Raskhodnikova, S., Smith, A., Yaroslavtsev, G.: Private analysis of graph structure. Proceedings of the VLDB Endowment 4(11) (2011)Google Scholar
  24. 24.
    Karwa, V., Slavkovic, A.: Differentially private synthetic graphs. arXiv preprint arXiv:1205.4697 (2012)Google Scholar
  25. 25.
    Karwa, V., Slavković, A.B.: Differentially private graphical degree sequences and synthetic graphs. In: Domingo-Ferrer, J., Tinnirello, I. (eds.) PSD 2012. LNCS, vol. 7556, pp. 273–285. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  26. 26.
    Michell, L., Amos, A.: Girls, pecking order and smoking. Social Science & Medicine 44(12), 1861–1869 (1997)CrossRefGoogle Scholar
  27. 27.
    Morris, M., Handcock, M.S., Hunter, D.R.: Specification of exponential-family random graph models: terms and computational aspects. Journal of Statistical Software 24(4), 1548 (2008)Google Scholar
  28. 28.
    Narayanan, A., Shmatikov, V.: De-anonymizing social networks. In: 2009 30th IEEE Symposium on Security and Privacy, pp. 173–187. IEEE (2009)Google Scholar
  29. 29.
    Nissim, K., Raskhodnikova, S., Smith, A.: Smooth sensitivity and sampling in private data analysis. In: STOC, pp. 75–84. ACM (2007)Google Scholar
  30. 30.
    Pearson, M., Michell, L.: Smoke rings: social network analysis of friendship groups, smoking and drug-taking. Drugs: Education, Prevention and Policy 7(1), 21–37 (2000)Google Scholar
  31. 31.
    R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2014)Google Scholar
  32. 32.
    Robins, G., Pattison, P., Kalish, Y., Lusher, D.: An introduction to exponential random graph models for social networks. Social Networks 29(2), 173–191 (2007)CrossRefGoogle Scholar
  33. 33.
    Siena. Description excerpt of 50 girls from “teenage friends and lifestyle study” data, http://www.stats.ox.ac.uk/~snijders/siena/s50_data.htm/ (May 2014)
  34. 34.
    Snijders, T.A.B.: Markov chain monte carlo estimation of exponential random graph models. Journal of Social Structure 3(2), 1–40 (2002)MathSciNetGoogle Scholar
  35. 35.
    Woo, Y.M.J., Slavković, A.B.: Logistic regression with variables subject to post randomization method. In: Domingo-Ferrer, J., Tinnirello, I. (eds.) PSD 2012. LNCS, vol. 7556, pp. 116–130. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Vishesh Karwa
    • 1
  • Aleksandra B. Slavković
    • 1
  • Pavel Krivitsky
    • 2
  1. 1.Department of StatisticsThe Pennsylvania State UniversityUSA
  2. 2.School of Mathematics and Applied Statistics of University of WollongongAustralia

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