Phase Transition of the Reconstructability of a General Model with Different In-Community and Out-Community Mutations on an Infinite Tree

35 Pages Posted: 10 Oct 2019

See all articles by Wenjian Liu

Wenjian Liu

City University of New York (CUNY)

Ning Ning

University of Michigan at Ann Arbor

Date Written: September 30, 2019

Abstract

In this paper, we analyze the tree reconstruction problem, which is to determine whether symbols at the nth level of the tree provide information on the root as nā†’āˆž.This problem has wide applications in various fields such as biology, information theory, and statistical physics, and its close connections to the clustering problem in the setting of the stochastic block model (SBM) have been well established. Inspired by the recently proposed q1+q2 SBM, we extend the classical works on the Ising model (Borgs et al. [2006]) and the Potts model (Sly [2011]), by studying a general model which incorporates the characteristics of both Ising and Potts through different in-community and out-community transition probabilities, and rigorously establishing the exact conditions for reconstruction.

Keywords: Phase transition, Kesten-Stigum reconstruction bound, Markov random fields on trees, Distributional recursion, Nonlinear dynamical system

JEL Classification: 60K35, 82B26, 82B20

Suggested Citation

Liu, Wenjian and Ning, Ning, Phase Transition of the Reconstructability of a General Model with Different In-Community and Out-Community Mutations on an Infinite Tree (September 30, 2019). Available at SSRN: https://ssrn.com/abstract=3461707 or http://dx.doi.org/10.2139/ssrn.3461707

Wenjian Liu

City University of New York (CUNY)

695 Park Avenue
New York, NY 10021
United States

Ning Ning (Contact Author)

University of Michigan at Ann Arbor

500 S. State Street
Ann Arbor, MI 48109
United States

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