Volume Growth, Curvature, and Buser-Type Inequalities in Graphs

Abstract

We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature.Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, λ 2 of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the second eigenvalue. We also describe a method for proving Buser's Inequality in graphs, particularly under a lower bound assumption on curvature.1 on average, the distances between points in B(x, r) and B(y, r) will be closer than their counterparts under the parallel transport. Ollivier observed that the average distance can be replaced by the L 1 -Wasserstein distance between uniform distributions on B(x, r) and B(y, r), and this metric is used in definition of the so-called Ollivier curvature, which can be used to recover the manifold's Ricci curvature (up to a factor) [42].Ollivier used this concept to help define the discrete Ricci curvature [42]. The metric balls B(x, r) and B(y, r) can also be defined on a graph where r is a non-negative integer and x and y are vertices of the graph. Then the L 1 -Wasserstein distance between the balls B(x, r) and B(y, r) determines a notion of curvature on the graph.While definitions of Ollivier curvature can be applied to any metric measure space, arguably its most fruitful use has been to define curvature in graphs with the graph distance and counting measure, for example [9,16,29]. That will also be our focus in this work: A well-known fact due to Bishop is that a Riemannian manifold with a lower bound on its Ricci curvature will have the volume growth of its metric balls controlled by this lower bound [12]. Under many notions of discrete curvature it is unclear whether such a volume growth bound exists. In this work we will present a volume growth that is interesting for regular graphs with a negative lower bound on Ollivier curvature.We will also briefly discuss the CDE ′ curvature, which was created by Bauer, Jost, and Liu [10]. The CDE ′ inequality is a modification of the CD inequality of Bakry-Émery, which is a discrete generalization of the Bochner formula from Riemannian geometry. Those authors demonstrated a version of the Li-Yau gradient estimate for graphs under the CDE ′ curvature. This is a result that does not have any known analogue in the setting of Ollivier curvature.Volume growth est...

Inner-Outer Curvatures, Ricci-Ollivier Curvature and Volume Growth of Graphs

“…By contrast, the positivity of the Ollivier Ricci curvature implies bounds on the diameter of a graph and therefore of its volume under assumptions on the degree ( [2,15,19]). In addition, the Ollivier curvature has shown to have a role in obtaining spectral estimates ( [1,2]).…”

Maximal diameter theorem for directed graphs of positive Ricci curvature

“…Recently, there are some attempts to introduce the notion of the Ricci curvature for discrete spaces, and the Ricci curvature for (undirected) graphs introduced by Lin-Lu-Yau [11] is one of the well-studied objects (see also a pioneering work of Ollivier [14]). It is well-known that a lower Ricci curvature bound of them leads us to various geometric and analytic consequences (see e.g., [1], [2], [3], [7], [9], [11], [13], [16], and so on). In [11], they have provided a discrete analogue of Bonnet-Myers theorem in Riemannian geometry.…”

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

“…Modifying the formulation in [31], Lin-Lu-Yau [24] have defined the Ricci curvature for undirected graphs. It is well-known that a lower Ricci curvature bound of Lin-Lu-Yau [24] implies various geometric and analytic properties (see e.g., [7], [9], [20], [24], [28], [32], and so on).…”