Ollivier Ricci Curvature of Directed Hypergraphs

Abstract

We develop a definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier's definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices.

Geometric and spectral properties of directed graphs under a lower Ricci curvature bound

“…In the undirected case (i.e., the weight µ is symmetric), our Ricci curvature coincides with that of Lin-Lu-Yau [24]. We also note that the third author [39] and Eidi-Jost [15] have used different probability measures from ν ε…”

“…x to the outer one − → ν ε y . On the other hand, Eidi-Jost [15] considered the in-out type Ricci curvature, and used them for the study of directed hypergraphs (see Definition 3.2 in [15]). In our setting, their in-out type Ricci curvature can be formulated as follows:…”

“…It seems that we can also consider the following out-in type Ricci curvature − → ← − κ (x, y), and the in-in type Ricci curvature ← − ← − κ (x, y) defined as follows (cf. Section 8 in [15]):…”

“…He computed it for some concrete examples, and given several estimates. Eidi-Jost [15] have recently introduced another formulation (see Remark 3.7). They have applied it to the study of directed hypergraphs.…”

Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

“…We now give the definition for Ollivier curvature which has been introduced in [42,43] and generalized in [1,12,21,29,38]. For a reversible graph G = (V, q) and f ∈ C(V ), and x = y ∈ V , we define…”