Large scale Ricci curvature on graphs
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Calculus of Variations
Large scale Ricci curvature on graphs Mark Kempton1 · Gabor Lippner2 · Florentin Münch3 Received: 26 April 2020 / Accepted: 31 July 2020 / Published online: 13 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We define a hybrid between Ollivier and Bakry-Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient estimates similar to the ones obtained from Bakry-Emery curvature allowing us to prove Harnack and Buser inequalities. Mathematics Subject Classification 05C50 · 35K05
1 Introduction The analogy between graphs and Riemannian manifolds, via their Laplace operators, has a long history [1,7,8,10,12,17,18,27]. Originally, this analogy was mostly exploited on the level of spectrum and eigenfunctions. Recently, however, various notions of curvature for discrete spaces, and graphs in particular, have received growing attention [2–6,9,11,13–16,19–25]. Unfortunately, it seems, there is no single best notion of curvature in the discrete setting. Instead, one typically chooses a standard consequence of curvature bounded from below in the manifold setting, and uses that as the definition in the discrete setting. So far the most popular methods were based on optimal transport (Ollivier-Ricci curvature) or on Bochner’s identity and curvature-dimension inequalities (Bakry-Emery curvature). While Bakry-Emery curvature conditions imply many interesting properties, the examples even where the simplest C D(0, ∞) condition is known to hold are very limited. On the other hand, Ollivier curvature
Communicated by J. Jost.
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Gabor Lippner [email protected] Mark Kempton [email protected] Florentin Münch [email protected]
1
Brigham Young University, Provo, UT, USA
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Northeastern University, Boston, MA, USA
3
Max Planck Institute, Leipzig, Germany
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conditions, while allowing for more examples, are not known to imply many of the usual and important consequences that hold on manifolds. The quest is then to find conditions that are strong enough to imply interesting theorems, but not too restrictive to allow many interesting examples. Our view is that the extreme and rigid locality (that curvature depends only on the 2-step neighborhood of a node) is one of the major roadblocks in finding interesting examples. For instance, all previously studied curvature variants will assign negative curvature to the hexagonal lattice, since it is locally tree-like up to two steps. However, it “should” clearly have non-negative curvature as it is a planar tiling. In this paper we introduce three related, but slightly different, new notions of curvature. These are hybrids between Ollivier and Bakry-Emery curvatures. Their common, and perhaps most important, feature is that they allow a scaling paramete
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