In the study of sparse stochastic block models (SBMs) one often needs to analyze a distributional recursion, known as the belief propagation (BP) recursion. Uniqueness of the fixed point of this recursion implies several results about the SBM, including optimal recovery algorithms for SBM (Mossel et al. (2016)) and SBM with side information (Mossel and Xu (2016)), and a formula for SBM mutual information (Abbe et al. (2021)). The 2-community case corresponds to an Ising model, for which Yu and Polyanskiy (2022) established uniqueness for all cases. In this paper we analyze the $q$-ary Potts model, i.e., broadcasting of $q$-ary spins on a Galton-Watson tree with expected offspring degree $d$ through Potts channels with second-largest eigenvalue $\lambda$. We allow the intermediate vertices to be observed through noisy channels (side information). We prove that BP uniqueness holds with and without side information when $d\lambda^2 \ge 1 + C \max\{\lambda, q^{-1}\}\log q$ for some absolute constant $C>0$ independent of $q,\lambda,d$. For large $q$ and $\lambda = o(1/\log q)$, this is asymptotically achieving the Kesten-Stigum threshold $d\lambda^2=1$. These results imply mutual information formulas and optimal recovery algorithms for the $q$-community SBM in the corresponding ranges. For $q\ge 4$, Sly (2011); Mossel et al. (2022) showed that there exist choices of $q,\lambda,d$ below Kesten-Stigum (i.e. $d\lambda^2 < 1$) but reconstruction is possible. Somewhat surprisingly, we show that in such regimes BP uniqueness does not hold at least in the presence of weak side information. Our technical tool is a theory of $q$-ary symmetric channels, that we initiate here, generalizing the classical and widely-utilized information-theoretic characterization of BMS (binary memoryless symmetric) channels.

翻译：在稀疏随机块模型（SBM）的研究中，常需分析一种称为置信传播（BP）递归的分布递归。该递归的不动点唯一性蕴含了SBM的若干结论，包括SBM（Mossel等，2016）及带辅助信息的SBM（Mossel与Xu，2016）的最优恢复算法，以及SBM互信息公式（Abbe等，2021）。双社区情形对应于伊辛模型，Yu与Polyanskiy（2022）已证明所有此类情形的唯一性。本文分析$q$元Potts模型，即通过第二特征值$\lambda$的Potts信道，在期望后代度为$d$的Galton-Watson树上广播$q$元自旋。我们允许中间顶点通过噪声信道（辅助信息）被观测。证明当$d\lambda^2 \ge 1 + C \max\{\lambda, q^{-1}\}\log q$时（其中$C>0$为与$q,\lambda,d$无关的绝对常数），BP唯一性在有/无辅助信息情形下均成立。对于大$q$且$\lambda = o(1/\log q)$，此结果渐近达到Kesten-Stigum阈值$d\lambda^2=1$。这些结论在相应参数范围内蕴含了$q$社区SBM的互信息公式与最优恢复算法。对于$q\ge 4$，Sly（2011）；Mossel等（2022）证明存在低于Kesten-Stigum阈值（即$d\lambda^2 < 1$）的$q,\lambda,d$参数选择，但重构仍然可能。略显意外的是，我们证明在此类区域中，至少当存在弱辅助信息时，BP唯一性不成立。我们的技术工具是本文首创的$q$元对称信道理论，其推广了经典且广泛应用的BMS（二元无记忆对称）信道的信息论刻画。