Modularity is a popular metric for quantifying the degree of community structure within a network. The distribution of the largest eigenvalue of a network's edge weight or adjacency matrix is well studied and is frequently used as a substitute for modularity when performing statistical inference. However, we show that the largest eigenvalue and modularity are asymptotically uncorrelated, which suggests the need for inference directly on modularity itself when the network size is large. To this end, we derive the asymptotic distributions of modularity in the case where the network's edge weight matrix belongs to the Gaussian Orthogonal Ensemble, and study the statistical power of the corresponding test for community structure under some alternative model. We empirically explore universality extensions of the limiting distribution and demonstrate the accuracy of these asymptotic distributions through type I error simulations. We also compare the empirical powers of the modularity based tests with some existing methods. Our method is then used to test for the presence of community structure in two real data applications.

中文翻译：

---

加权有符号网络中模块化的渐近分布

模块化是用于量化网络内社区结构程度的流行指标。网络边缘权重或邻接矩阵的最大特征值的分布得到了很好的研究，并且在执行统计推理时经常用作模块化的替代品。然而，我们表明最大特征值和模块化是渐近不相关的，这表明当网络规模很大时需要直接对模块化本身进行推理。为此，我们推导出了网络边缘权重矩阵属于高斯正交集合的情况下模块化的渐近分布，并研究了在某些替代模型下社区结构的相应检验的统计功效。我们凭经验探索了极限分布的普遍性扩展，并通过 I 类错误模拟证明了这些渐近分布的准确性。我们还将基于模块化的测试的经验功效与一些现有方法进行了比较。然后使用我们的方法来测试两个真实数据应用程序中社区结构的存在。