Multivariate Inference of Network Moments by Subsampling

Network moments--rescaled counts of motifs such as stars and triangles--are fundamental summaries of network structure, widely used in goodness-of-fit testing, model selection, and network comparison. While the univariate distribution of a single network moment can be approximated by subsampling, the consistency of subsampling for their {\it joint} distribution has remained unestablished. In this paper, we prove that node subsampling provides an asymptotically accurate approximation of the joint distribution of multiple network moments under a general sparse graphon model. The theoretical analysis requires a careful characterization of the dependence structure among network moments and the corresponding multivariate asymptotic convergence, going substantially beyond existing univariate results. Building on this foundation, we address a practically important open problem: two-sample testing between unmatchable networks with unequal edge densities. We propose a novel subsampling-based procedure that combines {\it sparsification} with a {\it sample-splitting} strategy. This yields the first subsampling-based inferential procedure valid for this setting, to our knowledge. We demonstrate the utility of multivariate subsampling inference through simulation studies and a real data application comparing coexpression networks of core and non-core genes in a study of parallel adaptation in Trinidadian guppies, where joint and conditional moment distributions reveal a structural difference that no marginal test can detect.