Curvature

Rena Elkin; Jung Hun Oh; Ying L. Liu; Pier Selenica; Britta Weigelt; Jorge S. Reis-Filho; Dmitriy Zamarin; Joseph O. Deasy; Larry Norton; Arnold J. Levine; Allen R. Tannenbaum

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Curvature

RE Rena Elkin

JO Jung Hun Oh

YL Ying L. Liu

PS Pier Selenica

BW Britta Weigelt

JR Jorge S. Reis-Filho

DZ Dmitriy Zamarin

JD Joseph O. Deasy

LN Larry Norton

AL Arnold J. Levine

AT Allen R. Tannenbaum

This method is extracted from research article: NPJ Genom Med, Nov 2021

Geometric network analysis provides prognostic information in patients with high grade serous carcinoma of the ovary treated with immune checkpoint inhibitors

DOI: 10.1038/s41525-021-00259-9

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The interplay between OR curvature, network entropy, and functional robustness is linked by OMT and is rich in theory. We outline this now, beginning with the OR curvature⁶.

Based on the work of von Renesse and Sturm¹⁶, Ollivier extended the notion of Ricci curvature, defined on a Riemannian manifold, to discrete metric measure spaces⁶. Specifically, let $X$ be a metric measure space equipped with a distance d such that for each $x \in X$ , one is given a probability measure μ_x. The probability measure μ_x can be thought of as fuzzifying or blurring the point x. For two points $x, y \in X$ , OR curvature is defined as

where W₁ is the Wasserstein distance.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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