Learning the community structure of a large-scale graph is a fundamental problem in machine learning, computer science and statistics. We study the problem of exactly recovering the communities in a graph generated from the Stochastic Block Model (SBM) in the Massively Parallel Computation (MPC) model. Specifically, given $kn$ vertices that are partitioned into $k$ equal-sized clusters (i.e., each has size $n$), a graph on these $kn$ vertices is randomly generated such that each pair of vertices is connected with probability~$p$ if they are in the same cluster and with probability $q$ if not, where $p > q > 0$. We give MPC algorithms for the SBM in the (very general) \emph{$s$-space MPC model}, where each machine has memory $s=\Omega(\log n)$. Under the condition that $\frac{p-q}{\sqrt{p}}\geq \tilde{\Omega}(k^{\frac12}n^{-\frac12+\frac{1}{2(r-1)}})$ for any integer $r\in [3,O(\log n)]$, our first algorithm exactly recovers all the $k$ clusters in $O(kr\log_s n)$ rounds using $\tilde{O}(m)$ total space, or in $O(r\log_s n)$ rounds using $\tilde{O}(km)$ total space. If $\frac{p-q}{\sqrt{p}}\geq \tilde{\Omega}(k^{\frac34}n^{-\frac14})$, our second algorithm achieves $O(\log_s n)$ rounds and $\tilde{O}(m)$ total space complexity. Both algorithms significantly improve upon a recent result of Cohen-Addad et al. [PODC'22], who gave algorithms that only work in the \emph{sublinear space MPC model}, where each machine has local memory~$s=O(n^{\delta})$ for some constant $\delta>0$, with a much stronger condition on $p,q,k$. Our algorithms are based on collecting the $r$-step neighborhood of each vertex and comparing the difference of some statistical information generated from the local neighborhoods for each pair of vertices. To implement the clustering algorithms in parallel, we present efficient approaches for implementing some basic graph operations in the $s$-space MPC model.

翻译：学习大规模图的社区结构是机器学习、计算机科学和统计学中的一个基本问题。我们研究在随机块模型（SBM）生成的图中，于大规模并行计算（MPC）模型下精确恢复社区的问题。具体来说，给定被划分为k个大小相等的聚类（每个聚类大小为n）的kn个顶点，这些kn个顶点上随机生成一个图，使得同一聚类中的每对顶点以概率p连接，不同聚类的顶点以概率q连接，其中p > q > 0。我们在（非常通用的）s空间MPC模型中给出SBM的MPC算法，其中每台机器内存为s = Ω(log n)。在条件\(\frac{p-q}{\sqrt{p}}\geq \tilde{\Omega}(k^{\frac12}n^{-\frac12+\frac{1}{2(r-1)}})\)（对任意整数r ∈ [3, O(log n)]）下，我们的第一个算法使用\(\tilde{O}(m)\)总空间在O(kr log_s n)轮中精确恢复所有k个聚类，或使用\(\tilde{O}(km)\)总空间在O(r log_s n)轮中完成。若\(\frac{p-q}{\sqrt{p}}\geq \tilde{\Omega}(k^{\frac34}n^{-\frac14})\)，第二个算法实现O(log_s n)轮和\(\tilde{O}(m)\)总空间复杂度。两个算法均显著改进了Cohen-Addad等人[PODC'22]的最新结果，后者仅在子线性空间MPC模型（每台机器本地内存s = O(n^δ)，δ为某正常数）中工作，且对p、q、k有更强条件。我们的算法基于收集每个顶点的r步邻域，并通过比较每对顶点局部邻域生成的统计信息差异来实现。为并行实现聚类算法，我们提出了在s空间MPC模型中高效实现基本图操作的方法。