  We establish a general theory for subsampling network data generated by the
sparse graphon model. In contrast to previous work for networks, we demonstrate
validity under minimal assumptions; the main requirement is weak convergence of
the functional of interest. We study the properties of two procedures: vertex
subsampling and $p$-subsampling. For the first, 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. Furthermore, under appropriate sparsity conditions, we
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.
