To characterize the community structure in network data, researchers have developed various block-type models, including the stochastic block model, the degree-corrected stochastic block model, the mixed membership block model, the degree-corrected mixed membership block model, and others. A critical step in applying these models effectively is determining the number of communities in the network. However, to the best of our knowledge, existing methods for estimating the number of network communities either rely on explicit model fitting or fail to simultaneously accommodate network sparsity and a diverging number of communities. In this paper, we propose a model-free spectral inference method based on eigengap ratios that addresses these challenges. The inference procedure is straightforward to compute, requires no parameter tuning, and can be applied to a wide range of block models without the need to estimate network distribution parameters. Furthermore, it is effective for both dense and sparse networks with a divergent number of communities. Technically, we show that the proposed spectral test statistic converges to a {function of the type-I Tracy-Widom distribution via the Airy kernel} under the null hypothesis, and that the test is asymptotically powerful under weak alternatives. Simulation studies on both dense and sparse networks demonstrate the efficacy of the proposed method. Three real-world examples are presented to illustrate the usefulness of the proposed test.

翻译：为刻画网络数据中的社区结构，研究者已发展出多种区块型模型，包括随机区块模型、度校正随机区块模型、混合隶属度区块模型、度校正混合隶属度区块模型等。有效应用这些模型的关键步骤在于确定网络中的社区数量。然而，据我们所知，现有估计网络社区数量的方法要么依赖于显式的模型拟合，要么无法同时适应网络稀疏性与社区数量的发散性。本文提出一种基于特征值间隙比值的无模型谱推断方法以应对这些挑战。该推断过程计算简便，无需参数调优，且可广泛应用于各类区块模型而无需估计网络分布参数。此外，该方法对具有发散社区数量的稠密与稀疏网络均具有效性。在技术上，我们证明了在原假设下所提出的谱检验统计量通过Airy核收敛于{I型Tracy-Widom分布的函数}，且在弱备择假设下该检验具有渐近有效性。针对稠密与稀疏网络的模拟研究验证了所提方法的有效性。本文通过三个实际案例展示了该检验的实用价值。