Advertisement

Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels

  • Wenjian Liu
  • Ning NingEmail author
Article
  • 10 Downloads

Abstract

In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs et al. (in: 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS, IEEE Computer Society, 2006), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees.

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 Sébastien Roch for his inspiring discussions and reading of some proofs in the first version of this paper. We truly appreciate the warm encouragements on finishing this complete version and helpful discussions on this topic as well as future extensions, from colleagues at the 2017 and 2018 Columbia-Princeton Probability Day, 2017 Northeast Probability Seminar, 2018 Frontier Probability Days, 2017 and 2018 Finger Lakes Probability Seminar, and 2017 and 2018 Seminar on Stochastic Processes.

References

  1. 1.
    Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131(3), 311–340 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernussou, J., Abatut, J.-L.: Point mapping stability. Pergamon (1977)Google Scholar
  3. 3.
    Bhamidi, S., Rajagopal, R., Roch, S.: Network delay inference from additive metrics. Random Struct. Algorithms 37(2), 176–203 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borgs, C., Chayes, J., Mossel, E., Roch, S.: The Kesten–Stigum reconstruction bound is tight for roughly symmetric binary channels. In: 47th Annual IEEE Symposium on Foundations of Computer Science, 2006. FOCS’06, pp. 518–530. IEEE Computer Society, Berkeley, CA (2006)Google Scholar
  6. 6.
    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
  7. 7.
    Daskalakis, Constantinos, 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
  8. 8.
    Evans, W., Kenyon, C., Peres, Y., Schulman, L.J.: Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10, 410–433 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions, vol. 9. Walter de Gruyter, Berlin (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kesten, H., Stigum, B.P.: Additional limit theorems for indecomposable multidimensional Galton–Watson processes. Ann. Math. Stat. 37(6), 1463–1481 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kesten, H., Stigum, B.P.: Limit theorems for decomposable multi-dimensional Galton–Watson processes. J. Math. Anal. Appl. 17(2), 309–338 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mézard, M., Montanari, A.: Reconstruction on trees and spin glass transition. J. Stat. Phys. 124(6), 1317–1350 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mossel, E.: Reconstruction on trees: beating the second eigenvalue. Ann. Appl. Probab. 11, 285–300 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mossel, E.: Phase transitions in phylogeny. Trans. Am. Math. Soc. 356(6), 2379–2404 (2004a)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mossel, E.: Survey-information flow on trees. DIMACS Ser. Discret. Math. Theor. Comput. Sci. 63, 155–170 (2004b)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mossel, E., Neeman, J., Sly, A.: Stochastic block models and reconstruction. arXiv preprint arXiv:1202.1499 (2012)
  19. 19.
    Mossel, E., Neeman, J., Sly, A.: A proof of the block model threshold conjecture. Combinatorica 38, 1–44 (2013)MathSciNetGoogle Scholar
  20. 20.
    Neeman, J., Netrapalli, P.: Non-reconstructability in the stochastic block model. arXiv preprint arXiv:1404.6304 (2014)
  21. 21.
    Roch, S.: A short proof that phylogenetic tree reconstruction by maximum likelihood is hard. IEEE/ACM Trans. Comput. Biol. Bioinform. 3(1), 92–94 (2006)CrossRefGoogle Scholar
  22. 22.
    Sly, A.: Reconstruction for the Potts model. Ann. Probab. 39, 1365–1406 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tetali, P., Vera, J.C., Vigoda, E., Yang, L.: Phase transition for the mixing time of the Glauber dynamics for coloring regular trees. Ann. Appl. Probab. 22, 2210–2239 (2012)MathSciNetCrossRefzbMATHGoogle 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