We propose a functional stochastic block model whose vertices involve functional data information. This new model extends the classic stochastic block model with vector-valued nodal information, and finds applications in real-world networks whose nodal information could be functional curves. Examples include international trade data in which a network vertex (country) is associated with the annual or quarterly GDP over certain time period, and MyFitnessPal data in which a network vertex (MyFitnessPal user) is associated with daily calorie information measured over certain time period. Two statistical tasks will be jointly executed. First, we will detect community structures of the network vertices assisted by the functional nodal information. Second, we propose computationally efficient variational test to examine the significance of the functional nodal information. We show that the community detection algorithms achieve weak and strong consistency, and the variational test is asymptotically chi-square with diverging degrees of freedom. As a byproduct, we propose pointwise confidence intervals for the slop function of the functional nodal information. Our methods are examined through both simulated and real datasets.

翻译：我们提出了一种顶点包含函数型数据信息的函数型随机块模型。该新模型扩展了具有向量值节点信息的经典随机块模型，适用于节点信息为函数曲线的现实网络。例如国际贸易数据中网络顶点（国家）与特定时间段内年度或季度GDP相关联，以及MyFitnessPal数据中网络顶点（MyFitnessPal用户）与特定时间段内每日卡路里信息相关联。本模型将同步执行两项统计任务：首先，在函数型节点信息的辅助下检测网络顶点的社区结构；其次，我们提出计算高效的变分检验来验证函数型节点信息的统计显著性。我们证明社区检测算法具有弱相合性与强相合性，且变分检验服从自由度发散的渐近卡方分布。作为衍生成果，我们构建了函数型节点信息斜率函数的逐点置信区间。通过模拟数据与真实数据集验证了所提方法的有效性。