We study the problem of $\textit{robust community recovery}$: efficiently recovering communities in sparse stochastic block models in the presence of adversarial corruptions. In the absence of adversarial corruptions, there are efficient algorithms when the $\textit{signal-to-noise ratio}$ exceeds the $\textit{Kesten--Stigum (KS) threshold}$, widely believed to be the computational threshold for this problem. The question we study is: does the computational threshold for robust community recovery also lie at the KS threshold? We answer this question affirmatively, providing an algorithm for robust community recovery for arbitrary stochastic block models on any constant number of communities, generalizing the work of Ding, d'Orsi, Nasser & Steurer on an efficient algorithm above the KS threshold in the case of $2$-community block models. There are three main ingredients to our work: (i) The Bethe Hessian of the graph is defined as $H_G(t) \triangleq (D_G-I)t^2 - A_Gt + I$ where $D_G$ is the diagonal matrix of degrees and $A_G$ is the adjacency matrix. Empirical work suggested that the Bethe Hessian for the stochastic block model has outlier eigenvectors corresponding to the communities right above the Kesten-Stigum threshold. We formally confirm the existence of outlier eigenvalues for the Bethe Hessian, by explicitly constructing outlier eigenvectors from the community vectors. (ii) We develop an algorithm for a variant of robust PCA on sparse matrices. Specifically, an algorithm to partially recover top eigenspaces from adversarially corrupted sparse matrices under mild delocalization constraints. (iii) A rounding algorithm to turn vector assignments of vertices into a community assignment, inspired by the algorithm of Charikar \& Wirth \cite{CW04} for $2$XOR.

翻译：我们研究$\textit{鲁棒社区恢复}$问题：在存在对抗性损坏的情况下，高效恢复稀疏随机块模型中的社区结构。在无对抗性损坏时，当$\textit{信噪比}$超过$\textit{Kesten--Stigum (KS)}$阈值时存在高效算法，该阈值被广泛认为是该问题的计算阈值。我们研究的问题是：鲁棒社区恢复的计算阈值是否也位于KS阈值处？我们对此问题给出肯定回答，提出了一种适用于任意常数个社区的任意随机块模型的鲁棒社区恢复算法，推广了Ding、d'Orsi、Nasser和Steurer在二社区块模型KS阈值以上高效算法的工作。我们的工作包含三个主要组成部分：(i) 图的Bethe Hessian定义为$H_G(t) \triangleq (D_G-I)t^2 - A_Gt + I$，其中$D_G$为度对角矩阵，$A_G$为邻接矩阵。经验研究表明，随机块模型的Bethe Hessian在Kesten-Stigum阈值正上方存在对应社区的外征特征向量。我们通过从社区向量显式构造外征特征向量，正式确认了Bethe Hessian外征特征值的存在性。(ii) 我们提出了一种针对稀疏矩阵的鲁棒PCA变体算法。具体而言，该算法能在温和的离域约束下，从受对抗性损坏的稀疏矩阵中部分恢复顶部特征空间。(iii) 受Charikar和Wirth在2XOR问题中算法启发，我们设计了一种将顶点向量分配转化为社区分配的舍入算法。