Variational Nonparametric Inference in Stochastic Block Models with Functional Covariates

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 a certain time period, and MyFitnessPal data in which a network vertex (MyFitnessPal user) is associated with daily calorie information measured over a 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 a 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 slope function of the functional nodal information. Our methods are examined through both simulated and real datasets. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.