Geometric and spectral properties of directed graphs under a lower Ricci curvature bound
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Calculus of Variations
Geometric and spectral properties of directed graphs under a lower Ricci curvature bound Ryunosuke Ozawa1 · Yohei Sakurai1 · Taiki Yamada2 Received: 5 November 2019 / Accepted: 30 June 2020 / Published online: 13 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization for strongly connected directed graphs by using the mean transition probability kernel which appears in the formulation of the Chung Laplacian. We conclude several geometric and spectral properties under a lower Ricci curvature bound extending previous results in the undirected case. Mathematics Subject Classification Primary 05C20 · 05C12 · 05C81 · 53C21 · 53C23
1 Introduction Ricci curvature is one of the most fundamental objects in Riemannian geometry. Based on a geometric observation on (smooth) Riemannian manifolds, Ollivier [31] has introduced the coarse Ricci curvature for (non-smooth) metric spaces by means of the Wasserstein distance which is an essential tool in optimal transport theory. Modifying the formulation in [31], Lin-Lu-Yau [25] have defined the Ricci curvature for undirected graphs. It is well-known that a lower Ricci curvature bound of Lin-Lu-Yau [25] implies various geometric and analytic properties (see e.g., [7,9,20,25,28,32], and so on).
Communicated by J. Jost.
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Yohei Sakurai [email protected] Ryunosuke Ozawa [email protected] Taiki Yamada [email protected]
1
Advanced Institute for Materials Research (AIMR), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
2
Research Institute for Humanity and Nature, 457-4 Motoyama, Kamigamo, Kita-ku, Kyoto 603-8047, Japan
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There have been some attempts to generalize the Ricci curvature of Lin-Lu-Yau [25] for directed graphs. The third author [40] has firstly proposed a generalization of their Ricci curvature (see Remark 3.7 for its precise definition). 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. We are now concerned with the following question: What is the suitable generalization of the Ricci curvature of Lin-Lu-Yau [25] for strongly connected directed graphs? In this paper, we provide a new Ricci curvature for such directed graphs, examine its basic properties, and conclude several geometric and analytic properties under a lower Ricci curvature bound. Our formulation is as follows (more precisely, see Sect. 2 and Subsection 3.1): Let (V , μ) denote a simple, strongly connected, finite weighted directed graph, where V is the vertex set, and μ : V ×V → [0, ∞) is the (non-symmetric) edge weight. For the
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