In this article, we develop a nonparametric graphical model for multivariate random functions. Most existing graphical models are restricted by the assumptions of multivariate Gaussian or copula Gaussian distributions, which also imply linear relations among the random variables or functions on different nodes. We relax those assumptions by building our graphical model based on a new statistical object---the functional additive regression operator. By carrying out regression and neighborhood selection at the operator level, our method can capture nonlinear relations without requiring any distributional assumptions. Moreover, the method is built up using only one-dimensional kernel, and thus avoids the curse of dimensionality from which a fully nonparametric approach often suffers. This feature enables us to work with large-scale networks. We derive error bounds for the estimated regression operator and establish graph estimation consistency, while allowing the number of functions to diverge at the exponential rate of the sample size. We demonstrate the efficacy of our method by both simulations and analysis of an electroencephalography dataset.