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 $r\ge 3$ 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.

arXiv comment30 pages. Major revision incorporating the idleness-function results of arXiv:2608.01970 into a unified paper. Corrected the attribution and discussion of the Tian-Zhao random walk and clarified its relation to the ORCHID equal-edges walk. Several citation inaccuracies were also corrected. Main results unchanged