This paper studies the joint community detection and phase synchronization problem on the \textit{stochastic block model with relative phase}, where each node is associated with an unknown phase angle. This problem, with a variety of real-world applications, aims to recover the cluster structure and associated phase angles simultaneously. We show this problem exhibits a \textit{``multi-frequency''} structure by closely examining its maximum likelihood estimation (MLE) formulation, whereas existing methods are not originated from this perspective. To this end, two simple yet efficient algorithms that leverage the MLE formulation and benefit from the information across multiple frequencies are proposed. The former is a spectral method based on the novel multi-frequency column-pivoted QR factorization. The factorization applied to the top eigenvectors of the observation matrix provides key information about the cluster structure and associated phase angles. The second approach is an iterative multi-frequency generalized power method, where each iteration updates the estimation in a matrix-multiplication-then-projection manner. Numerical experiments show that our proposed algorithms significantly improve the ability of exactly recovering the cluster structure and the accuracy of the estimated phase angles, compared to state-of-the-art algorithms.

翻译：本文研究了基于相对相位的随机块模型上的联合社区检测与相位同步问题，其中每个节点关联一个未知相位角。该问题具有多种实际应用，旨在同时恢复聚类结构及其对应的相位角。通过深入分析其最大似然估计（MLE）的数学表达式，我们发现该问题呈现出“多频”结构，而现有方法并非源自这一视角。为此，我们提出了两种简单高效的算法，这些算法充分利用了MLE公式，并从多频信息中获益。第一种是基于新型多频列主元QR分解的谱方法：通过对观测矩阵的顶部特征向量进行该分解，可以获取关于聚类结构和相关相位角的关键信息。第二种是迭代式多频广义幂法，每次迭代通过“矩阵乘法-投影”方式更新估计。数值实验表明，与现有最先进算法相比，我们提出的算法显著提升了聚类结构的准确恢复能力以及相位角估计的精度。