Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature

Abstract

We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

Bakry-Émery curvature on graphs as an eigenvalue problem

“…In recent years, the discrete Bakry-Émery theory on graphs has become an active emerging research field. There are a growing number of articles investigating this theory, see e.g., [7,8,9,10,11,12,14,15,17,20,21,22,23,24,26,27,28,29,31,32,33,34,35,37,38,39,41,42,44,45,47,48,49,52]. Let us mention here important related works on non-linear discrete curvature dimension inequalities, see e.g., [4,13,18,19,43].…”

Spectrally positive Bakry-Émery Ricci curvature on graphs

“…Similarly to [LMP18,LMPR17], we apply a gradient estimate (Theorem 4.1) to conclude a diameter bound.…”

“…Recently, authors began to generalize the uniform bounds in the curvature dimension condition to obtain Bonnet-Myers type theorems under almost positivity assumptions with exceptions [LMPR17,Mün18]. A big gap in dealing with almost positive curvature is that it is not clear that Sobolev inequalities hold on graphs even in the case of uniformly positive curvature, such that techniques based on heat kernel estimates do not carry over immediately.…”

Coverings and the heat equation on graphs: Stochastic incompleteness, the Feller property, and uniform transience

“…Recently, there has been a tremendous interest in various notions of curvature, and especially of Ricci curvature, for graphs, see, for example, [4,8,10,15,17,24,[26][27][28][33][34][35]42,50,51,60,[63][64][65][66][67][68][69]71,[73][74][75][76][77][78][79][83][84][85]87,90,97]. In particular, two notions have been most prominently explored for finding analogues to results in the setting of Riemannian manifolds for graphs: the Bakry-Émery approach having its origins in [2] and the coarse Ollivier-Ricci curvature originating in the work [78].…”