We consider a time-ordered sequence of networks stemming from stochastic block models where nodes gradually change memberships over time and no network at any single time point contains sufficient signal strength to recover its community structure. To estimate the time-varying community structure, we develop KD-SoS (kernel debiased sum-of-square), a method performing spectral clustering after a debiased sum-of-squared aggregation of adjacency matrices. Our theory demonstrates via a novel bias-variance decomposition that KD-SoS achieves consistent community detection of each network even when heterophilic networks do not require smoothness in the time-varying dynamics of between-community connectivities. We also prove the identifiability of aligning community structures across time based on how rapidly nodes change communities, and develop a data-adaptive bandwidth tuning procedure for KD-SoS. We demonstrate the utility and advantages of KD-SoS through simulations and a novel analysis of the time-varying dynamics in gene coordination in the human developing brain system.

翻译：我们考虑一个由随机块模型生成的时间有序网络序列，其中节点随时间逐步改变隶属关系，且任何单个时间点的网络均不具备足够的信号强度来恢复其社区结构。为估计时变社区结构，我们提出了KD-SoS（核去偏平方和）方法——该方法在对邻接矩阵进行去偏平方和聚合后执行谱聚类。我们的理论通过新颖的偏差-方差分解表明，即使异配网络不要求社区间连接性的时变动态具有平滑性，KD-SoS仍能实现对每个网络社区结构的一致检测。我们还基于节点改变社区的速度，证明了跨时间社区结构对齐的可识别性，并开发了一种适用于KD-SoS的数据自适应带宽选择程序。通过仿真实验及对人类发育脑系统中基因协调的时变动态分析，我们展示了KD-SoS的实用性与优势。