{
  "abstract": "Can the nodes of a random graph be placed in an auxiliary space $\\mathbb{R}^{M}$ so that its connectivity becomes local? The spring electric layout is a standard algorithm to try to find such local graph layouts; we study it as a statistical-physics problem by considering the Gibbs ensemble of spring-electric layouts. For graphs with independent edges we solve the model exactly at large number of nodes $N$ and fixed embedding dimension $M$. The node positions obey a spherical law: they are uniformly distributed in an $M$-dimensional ball; for $M=2$ this is closely related to Girko's circular law. Crucially, the ground state energy is flat across all layouts with that density: an optimal layout for one graph is asymptotically optimal for any other. Optimal layouts of i.i.d. graphs are therefore frustrated liquids, macroscopically constrained but microscopically unoptimizable. We discuss relations to graph-partitioning, Laplacian eigenmaps, and dimensionality reduction, and outline how this harsh result breaks down for correlated graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40965",
  "author_total": 2,
  "authors": [
    {
      "name": "Jakob Stubenrauch"
    },
    {
      "name": "Christian Keup"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f3rfb241sbpk4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40965",
  "fulltext_url": null,
  "id": "f3rfb241sbpk4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40965",
  "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",
      "cond-mat.dis-nn"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f3rfb241sbpk4.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:15Z",
      "native_id": "2610.40965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40965"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Spherical law for optimal layouts of random graphs",
  "title_text": "Spherical law for optimal layouts of random graphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/f3rfb241sbpk4",
  "venue": "arXiv",
  "versions": []
}