Lin--Lu--Yau Ricci Curvature of Digraphs via Optimal Transport Couplings

Abstract

In this paper, we study the Lin--Lu--Yau Ricci curvature of strongly connected locally finite digraphs through an explicit optimal-coupling construction. For an arc of a digraph, we derive a computable curvature formula by constructing a coupling between the probability measures at its tail and head, and by proving its optimality using a suitable $1$-Lipschitz function. The formula is not only effective for direct computation, but also unifies several known results: in particular, it recovers the Lin--Lu--Yau Ricci curvature formula for Cayley graphs of Right-Angled Artin--Coxeter Hybrid groups as a special case and gives shorter proofs of curvature results arising from matching-type conditions. We then characterize arcs with zero Ricci curvature through perfect distance matching and perfect distance partitions. We further prove that, under suitable assumptions, such arc curvature in directed Cayley graphs increases when an inverse generator or a new generator is added to the generating set. As applications, we compute the curvature of directed Cayley graphs of dihedral groups and generalized quaternion groups, including $Γ(D_n,\{a,b\})$, $Γ(Q_{4m},\{a,b\})$, $Γ(Q_{4m},\{a,a^{-1},b\})$ and $Γ(Q_{4m},\{a,b,b^{-1}\})$. Finally, we provide an algorithm for computing the Lin--Lu--Yau Ricci curvature of Cayley graphs of finitely generated groups with prescribed generating sets, together with complete curvature tables for several important families of finite groups.