{
  "abstract": "Isotropic Gaussian processes on networks have a covariance that is a function of a distance between locations, but valid constructions have so far required graphs with Euclidean edges and the resistance metric. We introduce a one-parameter family of metrics $d_κ$, $κ\\ge0$, on arbitrary compact metric graphs, including those with loops and parallel edges, defined as the variograms of first-order Whittle--Matérn fields, where $κ=0$ corresponds to an intrinsic limiting field. We show that these Whittle--Matérn metrics are invariant under insertion and removal of vertices of degree two, that they decrease monotonically in $κ$, and that they can be evaluated by sparse linear solves. The metric $d_0$ coincides with the resistance metric on graphs with Euclidean edges, and thus extends it to general compact metric graphs without any modification of the graph. For every completely monotone function $r$ and every $κ\\ge0$, we show that $r(d_κ)$ is a positive semidefinite function. This yields isotropic Gaussian processes on general compact metric graphs. For $κ>0$ the metric levels off beyond the screening length $κ^{-1}$, which can be used to induce long-range dependence in short-range covariance models, and $κ$ can be estimated from data together with the parameters of $r$. The models are compared on daily traffic maps from the street network of Paris and the motorway network around London. For both, $κ$ is estimated to be positive, $d_κ$ increases the maximised log-likelihood relative to $d_0$ for every covariance function, and it gives the best predictions at randomly held-out locations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35732",
  "author_total": 2,
  "authors": [
    {
      "name": "David Bolin"
    },
    {
      "name": "Alexandre B. Simas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fzf9297ebcfyj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35732",
  "fulltext_url": null,
  "id": "fzf9297ebcfyj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35732",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Gaussian Processes and Bayesian Inference"
    ]
  ],
  "openalex": "W7223855920",
  "pdf_url": "https://arxiv.org/pdf/2610.35732",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fzf9297ebcfyj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.35732",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35732"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Whittle--Matérn metrics and isotropic Gaussian processes on compact metric graphs",
  "title_text": "Whittle--Matérn metrics and isotropic Gaussian processes on compact metric graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/fzf9297ebcfyj",
  "venue": "arXiv",
  "versions": []
}