Trading off Accuracy for Speedup: Multiplier Bootstraps for Subgraph Counts

We propose a new class of multiplier bootstraps for count functionals. We consider bootstrap procedures with linear and quadratic weights. These correspond to the first and second-order terms of the Hoeffding decomposition of the bootstrapped statistic arising from the multiplier bootstrap, respectively. We show that the quadratic bootstrap procedure achieves higher-order correctness for appropriately sparse graphs. The linear bootstrap procedure requires fewer estimated network statistics, leading to improved accuracy over its higher-order correct counterpart in sparser regimes. To improve the computational properties of the linear bootstrap further, we consider fast sketching methods to conduct approximate subgraph counting and establish consistency of the resulting bootstrap procedure. We complement our theoretical results with a simulation study and real data analysis and verify that our procedure offers state-of-the-art performance for several functionals.