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Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality

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Abstract

Conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only. Since many substantively important networks are weighted, this limitation is especially problematic. We extend the existing exponential framework by proposing a generic common distribution for the edge weights. Minimal assumptions are placed on the distribution, that is, it is non-degenerate and supported on the unit interval. By doing so, we recognize the essential properties associated with near-degeneracy and universality in edge-weighted exponential random graphs.

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References

  1. Frank, O., Strauss, D.: Markov graphs. J. Am. Stat. Assoc. 81, 832–842 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Newman, M.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  4. Besag, J.: Statistical analysis of non-lattice data. J. R. Stat. Soc. Ser. D. Stat. 24, 179–195 (1975)

    Google Scholar 

  5. Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M.: New specifications for exponential random graph models. Sociol. Methodol. 36, 99–153 (2006)

    Article  Google Scholar 

  6. Rinaldo, A., Fienberg, S.E., Zhou, Y.: On the geometry of discrete exponential families with application to exponential random graph models. Electron. J. Stat. 3, 446–484 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fienberg, S.E.: Introduction to papers on the modeling and analysis of network data. Ann. Appl. Stat. 4, 1–4 (2010)

    Article  MathSciNet  Google Scholar 

  8. Fienberg, S.E.: Introduction to papers on the modeling and analysis of network data II. Ann. Appl. Stat. 4, 533–534 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chatterjee, S., Varadhan, S.R.S.: The large deviation principle for the Erdős-Rényi random graph. Eur. J. Comb. 32, 1000–1017 (2011)

    Article  MATH  Google Scholar 

  10. Chatterjee, S., Diaconis, P.: Estimating and understanding exponential random graph models. Ann. Stat. 41, 2428–2461 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Radin, C., Yin, M.: Phase transitions in exponential random graphs. Ann. Appl. Probab. 23, 2458–2471 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lubetzky, E., Zhao, Y.: On replica symmetry of large deviations in random graphs. Random Struct. Algorithms 47, 109–146 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lubetzky, E., Zhao, Y.: On the variational problem for upper tails in sparse random graphs. Random Struct. Algorithms 50, 420–436 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Radin, C., Sadun, L.: Phase transitions in a complex network. J. Phys. A: Math. Theor. 46, 305002 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Radin, C., Sadun, L.: Singularities in the entropy of asymptotically large simple graphs. J. Stat. Phys. 158, 853–865 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Radin, C., Ren, K., Sadun, L.: The asymptotics of large constrained graphs. J. Phys. A: Math. Theor. 47, 175001 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Kenyon, R., Radin, C., Ren, K., Sadun, L.: Multipodal structure and phase transitions in large constrained graphs. arXiv: 1405.0599 (2014)

  18. Yin, M.: Critical phenomena in exponential random graphs. J. Stat. Phys. 153, 1008–1021 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Kenyon, R., Yin, M.: On the asymptotics of constrained exponential random graphs. J. Appl. Probab. 54, 165–180 (2017)

    Article  MathSciNet  Google Scholar 

  20. Aristoff, D., Zhu, L.: Asymptotic structure and singularities in constrained directed graphs. Stoch. Process. Appl. 125, 4154–4177 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chatterjee, S., Dembo, A.: Nonlinear large deviations. Adv. Math. 299, 396–450 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Borgs, C., Chayes, J., Lovasz, L., Sos, V.T., Vesztergombi, K.: Counting graph homomorphisms. In: Klazar, M., Kratochvil, J., Loebl, M., Thomas, R., Valtr, P. (eds.) Topics in Discrete Mathematics, vol. 26, pp. 315–371. Springer, Berlin (2006)

    Chapter  Google Scholar 

  23. Borgs, C., Chayes, J.T., Lovasz, L., Sos, V.T., Vesztergombi, K.: Convergent sequences of dense graphs I. Subgraph frequencies, metric properties and testing. Adv. Math. 219, 1801–1851 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12, 1454–1508 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lovász, L.: Large Networks and Graph Limits. American Mathematical Society, Providence (2012)

    Book  MATH  Google Scholar 

  26. Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory Ser. B. 96, 933–957 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Aldous, D.: Representations for partially exchangeable arrays of random variables. J. Multivar. Anal. 11, 581–598 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hoover, D.: Row-column exchangeability and a generalized model for probability. In: Koch, G., Spizzichino, F. (eds.) Exchangeability in Probability and Statistics, pp. 281–291. North-Holland, Amsterdam (1982)

    Google Scholar 

  29. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6, 1–13 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Aldous, D., Steele, J.M.: The objective method: probabilistic combinatorial optimization and local weak convergence. In: Kesten, H. (ed.) Probability on Discrete Structures, pp. 1–72. Springer, Berlin (2004)

    Google Scholar 

  31. Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12, 1454–1508 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lyons, R.: Asymptotic enumeration of spanning trees. Comb. Probab. Comput. 14, 491–522 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Borgs, C., Chayes, J., Cohn, H., Zhao, Y.: An \(L^{p}\) theory of sparse graph convergence I. Limits, sparse random graph models, and power law distributions. arXiv:1401.2906 (2014)

  34. Borgs, C., Chayes, J., Cohn, H., Zhao, Y.: An \(L^{p}\) theory of sparse graph convergence II. LD convergence, quotients, and right convergence. arXiv:1408.0744 (2014)

  35. Cranmer, S.J., Desmarais, B.A.: Inferential network analysis with exponential random graph models. Pol. Anal. 19, 66–86 (2011)

    Article  Google Scholar 

  36. Yin, M.: Phase transitions in edge-weighted exponential random graphs. arXiv:1607.04084 (2016)

  37. Handcock, M.S.: Assessing degeneracy in statistical models of social networks. Working Paper 39, Center for Statistics and the Social Sciences, Univ. Washington, Seattle, WA (2003)

  38. Yin, M.: A detailed investigation into near degenerate exponential random graphs. J. Stat. Phys. 164, 241–253 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Brown, L.D.: Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Lecture Notes-Monograph Series, Volume 9. Institute of Mathematical Statistics, Hayward (1986)

  40. Bhamidi, S., Bresler, G., Sly, A.: Mixing time of exponential random graphs. Ann. Appl. Probab. 21, 2146–2170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mukherjee, S.: Consistence estimation in the two star exponential random graph model. arXiv:1310.4526 (2013)

  42. Zia, R.K.P., Redish, E.F., McKay, S.: Making sense of the Legendre transform. Am. J. Phys. 77, 614–622 (2009)

    Article  ADS  Google Scholar 

  43. 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. J. Stat. Softw. 24, 1–29 (2008)

    Article  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referees for the invaluable suggestions that greatly improved the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    Ryan DeMuse, Danielle Larcomb & Mei Yin

Authors
  1. Ryan DeMuse
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  2. Danielle Larcomb
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  3. Mei Yin
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Correspondence to Mei Yin.

Additional information

Mei Yin’s research was partially supported by NSF Grant DMS-1308333.

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DeMuse, R., Larcomb, D. & Yin, M. Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality. J Stat Phys 171, 127–144 (2018). https://doi.org/10.1007/s10955-018-1991-3

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  • DOI: https://doi.org/10.1007/s10955-018-1991-3

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