Spectral algorithms are some of the main tools in optimization and inference problems on graphs. Typically, the graph is encoded as a matrix and eigenvectors and eigenvalues of the matrix are then used to solve the given graph problem. Spectral algorithms have been successfully used for graph partitioning, hidden clique recovery and graph coloring. In this paper, we study the power of spectral algorithms using two matrices in a graph partitioning problem. We use two different matrices resulting from two different encodings of the same graph and then combine the spectral information coming from these two matrices. We analyze a two-matrix spectral algorithm for the problem of identifying latent community structure in large random graphs. In particular, we consider the problem of recovering community assignments exactly in the censored stochastic block model, where each edge status is revealed independently with some probability. We show that spectral algorithms based on two matrices are optimal and succeed in recovering communities up to the information theoretic threshold. On the other hand, we show that for most choices of the parameters, any spectral algorithm based on one matrix is suboptimal. This is in contrast to our prior works (2022a, 2022b) which showed that for the symmetric Stochastic Block Model and the Planted Dense Subgraph problem, a spectral algorithm based on one matrix achieves the information theoretic threshold. We additionally provide more general geometric conditions for the (sub)-optimality of spectral algorithms.

翻译：谱算法是处理图上优化与推理问题的主要工具之一。通常，图被编码为矩阵，随后利用该矩阵的特征向量与特征值来解决给定的图问题。谱算法已成功应用于图划分、隐藏团恢复和图着色等领域。本文研究了在图划分问题中使用双矩阵的谱算法的能力。我们利用同一图的两种不同编码得到两个不同的矩阵，然后结合这两个矩阵所蕴含的谱信息。针对大型随机图中潜在社区结构的识别问题，我们分析了一种双矩阵谱算法。特别地，我们考虑了在删失随机块模型中精确恢复社区分配的问题，其中每条边的状态以一定概率独立地揭示。研究表明，基于双矩阵的谱算法是最优的，并能够在达到信息理论阈值时成功恢复社区。另一方面，我们发现对于大多数参数选择，基于单矩阵的任何谱算法都是次优的。这与我们先前的工作（2022a, 2022b）形成对比，后者表明，对于对称随机块模型和种植稠密子图问题，基于单矩阵的谱算法能够达到信息理论阈值。此外，我们还提供了关于谱算法（次）最优性的更一般的几何条件。