We develop a novel method of inference for network-dependent high-dimensional random vectors. Dependence is characterized via a functional dependence measure based on graph distance, allowing the approximation theory to capture the interaction between the decay of dependence and the growth of network neighborhoods. We establish Gaussian approximation results for the maximum norm under finite-moment and sub-Weibull conditions, providing explicit conditions under which the dimension may increase with the network size. We also propose a high-dimensional network HAC covariance estimator and establish its convergence properties, yielding a feasible procedure for simultaneous inference. Simulation studies demonstrate favorable finite-sample performance of the proposed method. We apply the procedure to study how spillover effects vary with an index of network homophily by constructing confidence bands for the conditional spillover-effect function. The application reveals heterogeneity and local significance that would be obscured by conventional low-dimensional inference.

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