The estimation of functional networks through functional covariance and graphical models have recently attracted increasing attention in settings with high dimensional functional data, where the number of functional variables p is comparable to, and maybe larger than, the number of subjects. In this paper, we first reframe the functional covariance model estimation as a tuning-free problem of simultaneously testing p(p-1)/2 hypotheses for cross-covariance functions. Our procedure begins by constructing a Hilbert-Schmidt-norm-based test statistic for each pair, and employs normal quantile transformations for all test statistics, upon which a multiple testing step is proposed. We then explore the multiple testing procedure under a general error-contamination framework and establish that our procedure can control false discoveries asymptotically. Additionally, we demonstrate that our proposed methods for two concrete examples: the functional covariance model with partial observations and, importantly, the more challenging functional graphical model, can be seamlessly integrated into the general error-contamination framework, and, with verifiable conditions, achieve theoretical guarantees on effective false discovery control. Finally, we showcase the superiority of our proposals through extensive simulations and functional connectivity analysis of two neuroimaging datasets.

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