Parametric Bootstrap for Fixed Edge-Probability Network Models

This paper studies parametric bootstrap methods for network data, with the goal of quantifying the uncertainty of network statistics of interest. While existing network resampling methods primarily focus on count statistics under node-exchangeable graphon models, we consider more general network statistics, including local statistics, under the Chung-Lu model without assuming node exchangeability. We show that the natural network parametric bootstrap, which first estimates the network-generating model and then draws bootstrap samples from the estimated model, generally suffers from bootstrap bias. As a general remedy, we show that a two-level bootstrap procedure provably reduces this bias. This extends the classical idea of the iterative bootstrap to the network setting, where the number of parameters grows with the network size. Moreover, for many network statistics, the second-level bootstrap provides a way to construct confidence intervals with higher accuracy. As a by-product of this analysis, we also obtain a central limit theorem for subgraph counts under the inhomogeneous Erdos-R\'enyi model, which may be of independent interest.