A Parameterized Stochastic Hamiltonian Process on Graphs with Two-Point Boundary Conditions
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.
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BibTeX
@misc{feng2026parameterized,
title = {{A Parameterized Stochastic Hamiltonian Process on Graphs with Two-Point Boundary Conditions}},
author = {Yaxin Feng and Yang Xiang and Haomin Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40497},
eprint = {2610.40497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40497},
}
RIS
TY - UNPB
TI - A Parameterized Stochastic Hamiltonian Process on Graphs with Two-Point Boundary Conditions
AU - Yaxin Feng
AU - Yang Xiang
AU - Haomin Zhou
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40497
UR - https://arxiv.org/abs/2610.40497
AB - 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.
ER -