Abstract

Functional regression is nowadays widely used to analyze data in which each observation takes the form of a curve or function rather than a scalar or vector. Traditional methods, however, generally assume independent observations, an assumption that often fails in modern applications involving network, structured data. In these cases, relationships among units, such as those in social networks, transportation systems, or brain connectivity networks, have a strong impact on functional responses. Ignoring these dependencies can result in biased estimates, lower predictive accuracy, and underestimated uncertainty.

To address these issues, we introduce a Network-Weighted Functional Regression (NWFR) model that explicitly incorporates network structure by assigning weights to functional predictors based on their connections within the network. This framework allows the model to leverage information from neighboring nodes, effectively capturing dependencies and correlations induced by the underlying graph. Building on this approach, we focus on quantifying predictive uncertainty through a functional conformal prediction procedure. Conformal prediction enables the construction of distribution-free prediction intervals with guaranteed finite-sample coverage, providing a robust way to assess uncertainty in complex functional regression settings. By adapting conformal inference to network-weighted functional data, our method yields intervals that remain reliable in the presence of network-induced dependence.

We evaluate the proposed approach through extensive empirical studies. Simulation results show that explicitly accounting for network structure significantly improves point prediction accuracy compared to conventional functional regression methods that ignore connectivity. Moreover, the conformal prediction intervals produced by our method achieve the desired coverage levels while being consistently narrower and more informative than those obtained from simpler baseline approaches. Applications to real-world datasets further demonstrate the practical value of incorporating network information into functional regression models.