Community detection is a fundamental problem in complex network data analysis. Though many methods have been proposed, most existing methods require the number of communities to be the known parameter, which is not in practice. In this paper, we propose a novel goodness-of-fit test for the stochastic block model. The test statistic is based on the linear spectral of the adjacency matrix. Under the null hypothesis, we prove that the linear spectral statistic converges in distribution to $N(0,1)$. Some recent results in generalized Wigner matrices are used to prove the main theorems. Numerical experiments and real world data examples illustrate that our proposed linear spectral statistic has good performance.

翻译：社区检测是复杂网络数据分析中的基本问题。尽管已有诸多方法被提出，但现有方法大多需要预先设定社区数量作为已知参数，而这在实际中难以实现。本文针对随机块模型提出了一种新型拟合优度检验方法。该检验统计量基于邻接矩阵的线性谱特征。在原假设成立条件下，我们证明该线性谱统计量依分布收敛于$N(0,1)$。证明主要定理时采用了广义Wigner矩阵领域的最新研究成果。数值实验和实际数据案例表明，本文提出的线性谱统计量具有良好的表现性能。