Ollivier--Ricci Idleness Functions and Edge-Connectivity of Hypergraphs
Abstract
We give a local geometric criterion ensuring that the edge-connectivity of a hypergraph equals its minimum incidence degree: every locally finite connected r-uniform linear hypergraph with r≥3 and nonnegative Lin--Lu--Yau curvature has this property. Among the various extensions of Ollivier--Ricci curvature to hypergraphs, we work with the equal-edges random walk on hypergraphs \cite{CoupetteEtAl2023}. Moreover, we show that both uniformity and linearity are essential: if either assumption is removed, there exist positively curved hypergraphs for which the gap between minimum incidence degree and edge-connectivity is arbitrarily large. For arbitrary locally finite simple hypergraphs, we also determine the dependence on idleness completely: every idleness function is piecewise affine with at most three affine pieces and is affine on the universal interval [1/2,1]. These results extend the corresponding theory for graphs \cite{BourneEtAl2018}. They also provide two useful tools below: the $2$-section reduction underlying the edge-connectivity argument and a limit-free formula used in the sharpness constructions.
Related
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BibTeX
@misc{xia2026ollivierricci,
title = {{Ollivier--Ricci Idleness Functions and Edge-Connectivity of Hypergraphs}},
author = {Qing Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06029},
eprint = {2608.06029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06029},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ollivier--Ricci Idleness Functions and Edge-Connectivity of Hypergraphs
AU - Qing Xia
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2608.06029
UR - https://arxiv.org/abs/2608.06029
AB - We give a local geometric criterion ensuring that the edge-connectivity of a hypergraph equals its minimum incidence degree: every locally finite connected r-uniform linear hypergraph with r≥3 and nonnegative Lin--Lu--Yau curvature has this property. Among the various extensions of Ollivier--Ricci curvature to hypergraphs, we work with the equal-edges random walk on hypergraphs \cite{CoupetteEtAl2023}. Moreover, we show that both uniformity and linearity are essential: if either assumption is removed, there exist positively curved hypergraphs for which the gap between minimum incidence degree and edge-connectivity is arbitrarily large. For arbitrary locally finite simple hypergraphs, we also determine the dependence on idleness completely: every idleness function is piecewise affine with at most three affine pieces and is affine on the universal interval [1/2,1]. These results extend the corresponding theory for graphs \cite{BourneEtAl2018}. They also provide two useful tools below: the $2$-section reduction underlying the edge-connectivity argument and a limit-free formula used in the sharpness constructions.
ER -