Advertisement

Journal of Statistical Physics

, Volume 177, Issue 3, pp 438–467 | Cite as

Information Reconstruction on an Infinite Tree for a \(4\times 4\)-State Asymmetric Model with Community Effects

  • Wenjian Liu
  • Ning NingEmail author
Article
  • 43 Downloads

Abstract

The reconstruction problem on an infinite tree, is to collect and analyze massive data samples at the nth level of the tree to identify whether there is non-vanishing information of the root, as n goes to infinity. This problem has wide applications in various fields such as biology, information theory, and statistical physics. Although it has been studied in numerous contexts, the existing literature with rigorous reconstruction thresholds established are very limited. In this paper, we study the noise channel in terms of a \(4\times 4\)-state asymmetric transition matrix with community effects. By means of refined analyses of moment recursion, in-depth concentration estimates, and thorough investigations on an asymptotic 4-dimensional nonlinear second order dynamical system, we give the first rigorous result on asymmetric noisy channels with community effects by providing the exact conditions that the reconstruction bound is not tight, which yields the hybrid-hard phase in corresponding stochastic block models.

Keywords

Kesten–Stigum reconstruction bound Markov random fields on trees Distributional recursion Nonlinear dynamical system 

Mathematics Subject Classification

60K35 82B26 82B20 

Notes

Acknowledgements

We give special thanks to the journal editor and two anonymous reviewers who provided us with many constructive and helpful comments. We also give special thanks to Joseph Felsenstein and Sébastien Roch for conversations on this subject.

References

  1. 1.
    Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Prob. Theory Relat. Fields 131(3), 311–340 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhamidi, S., Rajagopal, R., Roch, S.: Network delay inference from additive metrics. Random Struct. Algorithms 37(2), 176–203 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of the limiting Gibbs state for the Ising model on the bethe lattice. J. Stat. Phys. 79(1–2), 473–482 (1995)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Borgs, C., Chayes, J., Mossel, E., Roch, S.: The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels. In Foundations of Computer Science, 2006. FOCS’06. IEEE Symposium on 47th Annual IEEE, pp 518–530 (2006)Google Scholar
  5. 5.
    Chayes, J.T., Chayes, L., Sethna, J.P., Thouless, D.J.: A mean field spin glass with short-range interactions. Commun. Math. Phys. 106(1), 41–89 (1986)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Daskalakis, C., Mossel, E., Roch, S.: Optimal phylogenetic reconstruction. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pp 159–168. ACM (2006)Google Scholar
  7. 7.
    Derrida, B., Bray, A.J., Godreche, C.: Non-trivial exponents in the zero temperature dynamics of the 1d ising and potts models. J. Phys. A 27(11), L357 (1994)ADSCrossRefGoogle Scholar
  8. 8.
    Dhar, D.: The relaxation to equilibrium in one-dimensional potts models. J. Indian Inst. Sci. 75(3), 297 (2013)MathSciNetGoogle Scholar
  9. 9.
    Evans, W., Kenyon, C., Peres, Y., Schulman, L.J.: Broadcasting on trees and the Ising model. Ann. Appl. Prob. 10, 410–433 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Felsenstein, J.: Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Mol. Evol. 17(6), 368–376 (1981)ADSCrossRefGoogle Scholar
  11. 11.
    Felsenstein, J.: Inferring Phylogenies, vol. 2. Sinauer associates, Sunderland (2004)Google Scholar
  12. 12.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions, vol. 9. Walter de Gruyter, New York (2011)CrossRefGoogle Scholar
  13. 13.
    Giuliani, A., Mastropietro, V.: Universal finite size corrections and the central charge in non-solvable ising models. Commun. Math. Phys. 324(1), 179–214 (2013)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Giuliani, A., Seiringer, R.: Periodic striped ground states in ising models with competing interactions. Commun. Math. Phys. 347(3), 983–1007 (2016)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kesten, H., Stigum, B.P.: Additional limit theorems for indecomposable multidimensional galton-watson processes. Ann. Math. Stat. 37(6), 1463–1481 (1966)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kesten, H., Stigum, B.P.: Limit theorems for decomposable multi-dimensional galton-watson processes. J. Math. Anal. Appl. 17(2), 309–338 (1967)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kimura, M.: A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J. Mol. Evol. 16(2), 111–120 (1980)ADSCrossRefGoogle Scholar
  18. 18.
    Liu, W., Ning, N.: Large degree asymptotics and the reconstruction threshold of the asymmetric binary channels. J. Stat. Phys. 174(6), 1161–1188 (2019)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Liu, W., Jammalamadaka, S.R., Ning, N.: The tightness of the Kesten-Stigum reconstruction bound of symmetric model with multiple mutations. J. Stat. Phys. 170(3), 617–641 (2018)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Lupo, C., Parisi, G., Ricci-Tersenghi, F.: the random field xy model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism. arXiv:1902.07132 (2019)
  21. 21.
    Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, colorings, and other models on trees. Random Struct. Algorithms 31(2), 134–172 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mézard, M., Montanari, A.: Reconstruction on trees and spin glass transition. J. Stat. Phys. 124(6), 1317–1350 (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Mossel, E.: Reconstruction on trees: beating the second eigenvalue. Ann. Appl. Prob. 11, 285–300 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mossel, E.: Phase transitions in phylogeny. Trans. Am. Math. Soc. 356(6), 2379–2404 (2004a)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mossel, E.: Survey: information flow on trees. DIMACS Ser. Discret. Math. Theor. Comput. Sci. 63, 155–170 (2004b)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mossel, E.: Deep learning and hierarchal generative models. arXiv:1612.09057 (2016)
  27. 27.
    Mossel, E., Neeman, J., Sly, A.: A proof of the block model threshold conjecture. Combinatorica 38, 1–44 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mossel, E., Neeman, J., Sly, A.: Belief propagation, robust reconstruction and optimal recovery of block models. In Conference on Learning Theory, pp. 356–370 (2014)Google Scholar
  29. 29.
    Neeman, J., Netrapalli, P.: Non-reconstructability in the stochastic block model. arXiv:1404.6304 (2014)
  30. 30.
    Ricci-Tersenghi, F., Semerjian, G., Zdeborová, L.: Typology of phase transitions in bayesian inference problems. Phys. Rev. E 99(4), 042109 (2019)ADSCrossRefGoogle Scholar
  31. 31.
    Roch, S.: A short proof that phylogenetic tree reconstruction by maximum likelihood is hard. IEEE/ACM Trans. Comput. Biol. Bioinf. 3(1), 92–94 (2006)CrossRefGoogle Scholar
  32. 32.
    Sly, Allan: Reconstruction for the Potts model. Ann. Prob. 39, 1365–1406 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tetali, P., Vera, J.C., Vigoda, E., Yang, L.: L Phase transition for the mixing time of the glauber dynamics for coloring regular trees. Ann. Appl. Prob. 22, 2210–2239 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Queensborough Community CollegeCity University of New YorkNew YorkUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations