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

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 n→∞.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