Spatially adaptive neighborhood smoothing for directed sparse graphons

We propose an adaptive neighborhood smoothing estimator for recovering the row-wise connection probabilities in directed sparse graphon models. While standard neighborhood smoothers typically rely on a fixed local bandwidth (e.g., k -nearest neighbors), our approach constructs neighborhoods using a global distance threshold on the outgoing connection profiles derived from the common out-neighbors AA^ AA ⊤ similarity kernel. This global thresholding mechanism yields a spatially adaptive neighborhood size: nodes in homogeneous regions of the latent space automatically aggregate information from a larger number of neighbors, significantly reducing variance, while nodes in complex or transitional regions maintain a minimal neighborhood size to control bias. We establish the first fully non-asymptotic row-wise risk bound for directed sparse graphons, capturing the favorable bias-variance trade-off induced by spatial adaptation. The resulting estimator naturally handles network asymmetry and is shown to outperform fixed-bandwidth benchmarks in simulations. A real-data application and analysis of the C. elegans connectome is presented.