{
  "abstract": "This paper presents a Lagrangian framework to solve Wasserstein Hamiltonian flows (WHF) under two-point boundary conditions on large finite graphs. Characterized by a coupled system of discrete master equation and Hamilton-Jacobi (HJ) equation on the probability simplex, this formulation defines the macroscopic WHF. While classical density-based solvers can track this flow, they face severe memory and computational bottlenecks as the graph scale grows. To bypass these limitations, we introduce the stochastic Hamiltonian process (SHP), a sample-based mathematical framework that tracks individual particle dynamics rather than the full node-wise density. We parameterize the time-dependent potential using neural network, yielding the Neural SHP solver. Numerical experiments on synthetic graphs demonstrate the scalability of our method relative to density-based baselines. Furthermore, we validate Neural SHP across two large-scale real-world networks, a PubMed citation network and a massive road network, demonstrating its robust efficiency for graph partition and routing problems on complex graph structures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40497",
  "author_total": 3,
  "authors": [
    {
      "name": "Yaxin Feng"
    },
    {
      "name": "Yang Xiang"
    },
    {
      "name": "Haomin Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9w7k1bq5djwnj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40497",
  "fulltext_url": null,
  "id": "9w7k1bq5djwnj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40497",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40497",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9w7k1bq5djwnj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2610.40497",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40497"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Parameterized Stochastic Hamiltonian Process on Graphs with Two-Point Boundary Conditions",
  "title_text": "A Parameterized Stochastic Hamiltonian Process on Graphs with Two-Point Boundary Conditions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/9w7k1bq5djwnj",
  "venue": "arXiv",
  "versions": []
}