一种稀疏完全正松弛的模块最大化社区检测算法
A Sparse Completely Positive Relaxation of the Modularity Maximization for Community Detection
摘要：
在本文中，我们考虑的社区检测问题下的随机块模型（SBM）的假设或度相关的随机块模型（DCSBM）的假设。社区检测问题的模块度最大化公式一般是NP困难的。针对模块度最大化问题，提出了一种稀疏的低秩完全正松弛算法，并给出了一种有效的逐行（RBR）型块坐标下降（BCD）算法，证明了$\mathcal{O}(1/\sqrt{N})$收敛速度稳定点的速度，其中$N$为迭代次数.构造了一种快速舍入方案来从解中检索社区结构。非渐近的高概率界的错误分类率的建立，以证明我们的方法。我们进一步发展了一个异步并行RBR算法，以加快收敛速度。在人工网络和真实网络上的大量数值实验表明，该方法在聚类精度和数值效率方面都具有优势。我们的数值结果表明，新提出的方法是一个非常有竞争力的替代稀疏网络上的社区检测超过50万个节点。
英文摘要：
In this paper, we consider the community detection problem under either the stochastic block model (SBM) assumption or the degree-correlated stochastic block model (DCSBM) assumption. The modularity maximization formulation for the community detection problem is NP-hard in general. In this paper, we propose a sparse and low-rank completely positive relaxation for the modularity maximization problem, we then develop an efficient row-by-row (RBR) type block coordinate descent (BCD) algorithm to solve the relaxation and prove an $\mathcal{O}(1/\sqrt{N})$ convergence rate to a stationary point where $N$ is the number of iterations. A fast rounding scheme is constructed to retrieve the community structure from the solution. Non-asymptotic high probability bounds on the misclassification rate are established to justify our approach. We further develop an asynchronous parallel RBR algorithm to speed up the convergence. Extensive numerical experiments on both synthetic and real world networks show that the proposed approach enjoys advantages in both clustering accuracy and numerical efficiency. Our numerical results indicate that the newly proposed method is a quite competitive alternative for community detection on sparse networks with over 50 million nodes.
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