In the study of sparse stochastic block model (SBM) one needs to analyze a distributional recursion, known as belief propagation (BP) on a tree. 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. Here, we analyze broadcasting of $q$-ary spins on a Galton-Watson tree with expected offspring degree $d$ and Potts channels with second-largest eigenvalue $\lambda$. We allow for the intermediate vertices to be observed through noisy channels (side information) We prove 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 $q,d,\lambda$. 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 formula and optimal recovery algorithms for the $q$-community SBM in the corresponding ranges. For $q\ge 4$, Sly (2011); Mossel et al. (2022) shows that there exist choices of $q,d,\lambda$ below Kesten-Stigum (i.e. $d\lambda^2 < 1$) but reconstruction is possible. Somewhat surprisingly, we show that in such regimes BP uniqueness \textit{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，需要分析一种被称为树上置信传播（BP）的分布递归。该递归不动点的唯一性蕴含着SBM的多项结论，包括SBM（Mossel等, 2016）及含边信息SBM（Mossel和Xu, 2016）的最优恢复算法，以及SBM互信息公式（Abbe等, 2021）。二社区情形对应伊辛模型，Yu和Polyanskiy（2022）证明了所有情形下的唯一性。本文分析Galton-Watson树上$q$元自旋的广播过程（期望后代度数为$d$，Potts通道的第二大特征值为$\lambda$），并允许中间顶点通过噪声通道（边信息）被观测。我们证明：当存在某个与$q,d,\lambda$无关的绝对常数$C>0$使得$d\lambda^2 \ge 1 + C \max\{\lambda, q^{-1}\}\log q$时，无论有无边信息，BP唯一性均成立。对于大$q$和$\lambda = o(1/\log q)$，该条件渐近达到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的互信息公式和最优恢复算法。对于$q\ge4$，Sly（2011）；Mossel等（2022）指出存在低于Kesten-Stigum（即$d\lambda^2 < 1$）但重构仍可能的$(q,d,\lambda)$选择。令人意外的是，我们证明在此类参数区域中，至少当存在弱边信息时，BP唯一性\textit{不成立}。本文的技术工具是首次建立的$q$元对称通道理论，该理论推广了经典且广泛使用的BMS（二进制无记忆对称）通道的信息论刻画。