Higher-Order Correct Multiplier Bootstraps for Count Functionals of Networks

Subgraph counts play a central role in both graph limit theory and network data analysis. In recent years, substantial progress has been made in the area of uncertainty quantification for these functionals; several procedures are now known to be consistent for the problem. In this paper, we propose a new class of multiplier bootstraps for count functionals. We show that a bootstrap procedure with a multiplicative weights exhibits higher-order correctness under appropriate sparsity conditions. Since this bootstrap is computationally expensive, we propose linear and quadratic approximations to the multiplier bootstrap, which correspond to the first and second-order Hayek projections of an approximating U-statistic, respectively. We show that the quadratic bootstrap procedure achieves higher-order correctness under analogous conditions to the multiplicative bootstrap while having much better computational properties. We complement our theoretical results with a simulation study and verify that our procedure offers state-of-the-art performance for several functionals.