Smooth Isotropic Covariance Functions on Metric Graphs via Polyharmonic Resistance Distances

2026-09-27

AI Summary

This study addresses the challenge of simultaneously satisfying smoothness and vertex continuity constraints for Gaussian processes defined on metric graphs. To this end, it proposes a novel class of polyharmonic distances that unifies spectral constructions with variational representations. By integrating composition techniques involving completely monotone functions to accommodate graph topologies, the method constructs isotropic stochastic processes that rigorously satisfy Kirchhoff conditions. The primary contribution lies in establishing a theoretical framework for covariance functions that jointly encode prescribed smoothness and network physical laws. This framework enables mean-square differentiability of arbitrary finite order while strictly enforcing physical constraints at vertices, thereby bridging the gap between regularity requirements and structural fidelity in Gaussian process modeling on metric graphs.

Abstract

Metric graphs are generalisations of linear networks and provide a natural framework for the definition of continuously-indexed Gaussian processes. We define a new class of distances on these topologies, termed polyharmonic distances, which unify and extend the spectral construction underlying the effective resistance distance and the biharmonic one. We give both a spectral and a variational characterisation. Furthermore, we show an explicit class of stochastic processes whose variograms coincide with the squared polyharmonic distances. Finally, we show how these metrics can be composed with suitable completely monotonic functions to define isotropic processes having any prescribed finite-order mean-square differentiability along the edges and satisfying the Kirchhoff conditions up to order one at the vertices.

Problem

• metric graphs
• covariance functions
• Gaussian processes
• polyharmonic distances
• isotropic processes

Innovation

• Metric Graphs
• Polyharmonic Distances
• Covariance Functions
• Gaussian Processes
• Kirchhoff Conditions