Subsampling Sparse Graphons Under Minimal Assumptions
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We establish a general theory for subsampling network data generated by the sparse graphon model. In contrast to previous work for network data, we demonstrate validity under minimal assumptions; the main requirement is weak convergence of the functional of interest. We study the properties of two subsampling procedures: vertex subsampling, and p-subsampling. For the first procedure, we prove validity under the mild condition that the number of subsampled vertices is o(n). For the second, we establish validity under analogous conditions on the expected subsample size. For both procedures, we also establish conditions under which uniform validity holds. Under appropriate sparsity conditions, we also derive limiting distributions for the nonzero eigenvalues of the adjacency matrix of a low rank sparse graphon. Our weak convergence result immediately yields the validity of subsampling for the nonzero eigenvalues under suitable assumptions.
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