Network dependence is widely regarded as a fundamental obstacle to inference: outcomes linked through a graph are not independent, the graph is rarely observed in full, and models for how dependence propagates are easily misspecified. We show that for a broad class of responses generated through a random network, the obstacle is smaller than it appears. We study a heterogeneous spatial autoregression on a sparse graphon, in which each node's response is a weighted average of its neighbors' responses with a node-specific, nonparametric coefficient plus a node-specific input, and we prove that sample means and smooth functions of vector means are asymptotically Gaussian down to average degree nearly logarithmic in the population size, allowing unbounded graphons with hubs and heavy-tailed inputs. The limiting variance interpolates exactly between two regimes according to the sampling fraction. When the observed nodes are a vanishing fraction of the population, the variance is the marginal variance of the response, the ordinary sample variance is consistent, and the usual $t$-interval is valid with no knowledge of the network. The proof averages over the graph rather than conditioning on it, isolating the behavior of the response through an exact identity for resolvents of Markov kernels and controlling the network noise by an Efron-Stein inequality; neither step requires the walk-by-walk covariance bounds of fixed-network analyses. Consequences for regression with node-level and network-derived covariates are derived.

翻译：暂无翻译