This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this paper contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

翻译：本文重新审视了网络模块度的经典概念及其在图数据分析中使用的谱松弛方法。我们提出并研究了若干模块度统计量变体，并针对具有节点社区结构的网络，在大规模网络极限下建立了这些统计量的渐近分布性质。我们的工作有助于检验网络差异，且可与随机块模型随机图的现有理论保证结合使用。这些结果得益于近期对大规模网络邻接矩阵低秩截断研究的进展。我们提供了验证性模拟研究及实际数据分析，涉及神经科学领域关于精神病（特别是精神分裂症）的网络研究。总体而言，本文为当前有限的基于模块度的网络分析统计推断文献做出了贡献。本文的补充材料可在线获取。