Abstract
We consider that a network is an observation, and a collection of observed networks forms a sample. In this setting, we provide methods to test whether all observations in a network sample are drawn from a specified model. We achieve this by deriving the joint asymptotic properties of average subgraph counts as the number of observed networks increases but the number of nodes in each network remains finite. In doing so, we do not require that each observed network contains the same number of nodes, or is drawn from the same distribution. Our results yield joint confidence regions for subgraph counts, and therefore methods for testing whether the observations in a network sample are drawn from: a specified distribution, a specified model, or from the same model as another network sample. We present simulation experiments and an illustrative example on a sample of brain networks where we find that highly creative individuals’ brains present significantly more short cycles than found in less creative people. Supplementary materials for this article are available online.
Highlights
We show that subgraph counts are flexible and powerful statistics for testing distributional properties of networks, when more than one network is observed
We present a central limit theorem as well as practical methods to build confidence regions for the subgraph counts observed in a network sample G = (G1, . . . , GN)
We provide the tools to perform statistical inference on a network sample using subgraph counts
Summary
We show that subgraph counts are flexible and powerful statistics for testing distributional properties of networks, when more than one network is observed. While other network comparison techniques use embeddings (Asta and Shalizi 2014; Gao and Lafferty 2017; Tang, Athreya, Sussman, Lyzinski, Priebe, et al 2017; Wang, Zhang, and Dunson 2019), using subgraph counts presents three key advantages: first, if the Gi’s are generated by a blockmodel (Hoff, Raftery, and Handcock 2012)—one of the most popular random network models to date—and for some families of subgraphs, the embedding is one-to-one This result is known as the finite forcibility of a family of graphs We conclude with an analysis of connectome data, and with a discussion
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