Geometric and spectral properties of directed graphs under a lower Ricci curvature bound

Ozawa, Ryunosuke; Sakurai, Yohei; Yamada, Taiki

doi:10.48550/arxiv.1909.07715

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Two Notions Of Ricci-flatness For Graphs1

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2020

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“…One way to define Ricci-flatness is to modify the notion of Ricci curvature for metric spaces in the sense of [Oll] The classification was addressed in [LLY] (cf. also [CKLLLY,CKLLLY2,OSY]). For vertices x, y ∈ V , and any real value α ∈ [0, 1], define the probability distribution µ α…”

Section: Two Notions Of Ricci-flatness For Graphsmentioning

confidence: 92%

Graph Laplacians, Riemannian Manifolds and their Machine-Learning

He¹,

Yau²

2020

Preprint

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Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics. We apply some of the latest techniques in data science such as supervised and unsupervised machine-learning and topological data analysis to the Wolfram database of some 8000 finite graphs in light of studying these correspondences. Encouragingly, we find that neural classifiers, regressors and networks can perform, with high efficiently and accuracy, a multitude of tasks ranging from recognizing graph Ricci-flatness, to predicting the spectral gap, to detecting the presence of Hamiltonian cycles, etc.

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Section: Two Notions Of Ricci-flatness For Graphsmentioning

confidence: 92%