Spectrally positive Bakry-Émery Ricci curvature on graphs

Münch, Florentin; Rose, Christian

doi:10.1016/j.matpur.2020.03.008

Cited by 7 publications

(1 citation statement)

References 23 publications

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“…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].…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Bakry-Émery curvature on graphs as an eigenvalue problem

et al. 2022

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In this paper, we reformulate the Bakry-Émery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.

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Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Bakry-Émery curvature on graphs as an eigenvalue problem

et al. 2022

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Stability of metric measure spaces with integral Ricci curvature bounds

Ketterer

2021

Journal of Functional Analysis

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Bakry–Émery–Ricci curvature: an alternative network geometry measure in the expanding toolbox of graph Ricci curvatures

Mondal,

Samal,

Münch

et al. 2024

Journal of Complex Networks

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The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. Various types of such Ricci curvatures capture different aspects of network geometry. In the present work, we investigate Bakry–Émery–Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Furthermore, the curvature distributions are different for different types of model networks. We observe a high positive correlation between Bakry–Émery–Ricci and both Forman–Ricci and Ollivier–Ricci curvature, and in particular with the augmented version of Forman–Ricci curvature while comparing them for both model and real-world networks. We investigate the correlation of Bakry–Émery–Ricci curvature with degree, clustering coefficient and vertex centrality measures. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. Additionally, for Forman–Ricci, Augmented Forman–Ricci and Ollivier–Ricci curvature, we compare the robustness of the networks by comparing the sum of the incident edges and the minimum of the incident edges as vertex measures and find that the sum identifies vertices that are important for maintaining the connectivity of the network. The computational time for Bakry–Émery–Ricci curvature is shorter than that required for Ollivier–Ricci curvature but higher than for Augmented Forman–Ricci curvature. We therefore conclude that for empirical network analysis, the latter is the tool of choice.

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Spectrally positive Bakry-Émery Ricci curvature on graphs

Cited by 7 publications

References 23 publications

Bakry-Émery curvature on graphs as an eigenvalue problem

Bakry-Émery curvature on graphs as an eigenvalue problem

Stability of metric measure spaces with integral Ricci curvature bounds

Bakry–Émery–Ricci curvature: an alternative network geometry measure in the expanding toolbox of graph Ricci curvatures

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