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Idleness Functions for Ollivier-Ricci Curvature on Hypergraphs
Abstract
Let ℋ=(V,E) be a locally finite simple hypergraph, equip V with the hyperpath metric, and consider the equal-edges random walk \cite{CoupetteEtAl2023}. For adjacent vertices x and y, we prove that the idleness function α↦καℋ(x,y) is piecewise affine and with no more than three affine pieces. A separate mass-balance argument gives linearity on [1/2,1] for every locally finite simple hypergraph and, consequently, a limit-free expression for the Lin--Lu--Yau curvature. In the r-uniform linear case, the hypergraph walk agrees exactly with the simple random walk on its 2-section. This reduction transfers the sharp endpoint intervals of Bourne, Cushing, Liu, Münch, and Peyerimhoff \cite{BourneEtAl2018}.
Related
BibTeX
@misc{xia2026idleness,
title = {{Idleness Functions for Ollivier-Ricci Curvature on Hypergraphs}},
author = {Qing Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01970},
eprint = {2608.01970},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01970},
}
RIS
TY - UNPB
TI - Idleness Functions for Ollivier-Ricci Curvature on Hypergraphs
AU - Qing Xia
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2608.01970
UR - https://arxiv.org/abs/2608.01970
AB - Let ℋ=(V,E) be a locally finite simple hypergraph, equip V with the hyperpath metric, and consider the equal-edges random walk \cite{CoupetteEtAl2023}. For adjacent vertices x and y, we prove that the idleness function α↦κ_α^ℋ(x,y) is piecewise affine and with no more than three affine pieces. A separate mass-balance argument gives linearity on [1/2,1] for every locally finite simple hypergraph and, consequently, a limit-free expression for the Lin--Lu--Yau curvature. In the r-uniform linear case, the hypergraph walk agrees exactly with the simple random walk on its 2-section. This reduction transfers the sharp endpoint intervals of Bourne, Cushing, Liu, Münch, and Peyerimhoff \cite{BourneEtAl2018}.
ER -