Smooth Isotropic Covariance Functions on Metric Graphs via Polyharmonic Resistance Distances
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.
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BibTeX
@misc{filosi2026smooth,
title = {{Smooth Isotropic Covariance Functions on Metric Graphs via Polyharmonic Resistance Distances}},
author = {Tobia Filosi and Emilio Porcu and Claudio Agostinelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33519},
eprint = {2609.33519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33519},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Smooth Isotropic Covariance Functions on Metric Graphs via Polyharmonic Resistance Distances
AU - Tobia Filosi
AU - Emilio Porcu
AU - Claudio Agostinelli
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33519
UR - https://arxiv.org/abs/2609.33519
AB - 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.
ER -