Characterizations of Forman curvature

Jost, Jürgen; Münch, Florentin

doi:10.48550/arxiv.2110.04554

Cited by 5 publications

(10 citation statements)

References 24 publications

(35 reference statements)

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“…Example 5. β, ζ and ξ-walks give rise to well-defined continuous-time generalized Ollivier-Ricci curvatures. So, in particular, we recover the well-definition results in [15,17,13].…”

supporting

confidence: 76%

“…Special cases of walks that our results apply to, include θ-walks; in particular, if one uses the restricted cases of β, ζ and ξ-walks (these are time-affine and are also called lazy walks in the literature), one retrieves the known constructions in the literature [15,17,13,18]; see § 6.2.2 for further details and definitions.…”

Section: Definition 13 a Continuous-time Random Walk µ εmentioning

confidence: 99%

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Discrete Ollivier-Ricci curvature

Fathi¹,

Lakzian²

2022

Preprint

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We analyze both continuous and discrete-time Ollivier-Ricci curvatures of locallyfinite weighted graphs G equipped with a given distance "d" (w.r.t. which G is metrically complete) and for general random walks. We show the continuous-time Ollivier-Ricci curvature is well-defined for a large class of Markovian and non-Markovian random walks and provide a criterion for existence of continuous-time Ollivier-Ricci curvature; the said results generalize the previous rather limited constructions in the literature.In addition, important properties of both discrete-time and continuous-time Ollivier-Ricci curvatures are obtained including -to name a few -Lipschitz continuity, concavity properties, piece-wise regularity (piece-wise linearity in the case of linear walks) for the discrete-time Ollivier-Ricci as well as Lipschitz continuity and limit-free formulation for the continuous-time Ollivier-Ricci. these properties were previously known only for very specific distances and very specific random walks. As an application of Lipschitz continuity, we obtain existence and uniqueness of generalized continuous-time Ollivier-Ricci curvature flows.Along the way, we obtain -by optimizing McMullen's upper bounds -a sharp upper bound estimate on the number of vertices of a convex polytope in terms of number of its facets and the ambient dimension, which might be of independent interest in convex geometry. The said upper bound allows us to bound the number of polynomial pieces of the discrete-time Ollivier-Ricci curvature as a function of time in the time-polynomial random walk. The limit-free formulation we establish allows us to define an operator theoretic Ollivier-Ricci curvature which is a nonlinear concave functional on suitable operator spaces.

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“…Example 5. β, ζ and ξ-walks give rise to well-defined continuous-time generalized Ollivier-Ricci curvatures. So, in particular, we recover the well-definition results in [15,17,13].…”

supporting

confidence: 76%

Section: Definition 13 a Continuous-time Random Walk µ εmentioning

confidence: 99%

Discrete Ollivier-Ricci curvature

Fathi¹,

Lakzian²

2022

Preprint

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“…with equality for cut links, where we stress that these are OR and FR curvatures with respect to particular data: ω as metric for OR and w = 1 as weight for FR. Apart from being an argument for the interpretation of the link resistance curvature as a discrete curvature, this result is relevant in the context of other works that investigate the relation between both curvatures [38,65,59].…”

Section: Discussionmentioning

confidence: 95%

“…In [26], Robin Forman introduced a new notion of curvature for CW complexes, a class of discrete/combinatorial spaces that includes graphs. This so-called Forman-Ricci (FR) curvature is defined by generalizing the definition of the classical Ricci curvature in terms of the Bochner Laplacian of a manifold to a definition for discrete spaces based on an analogous Bochner Laplacian on these spaces, see also [38]. The FR curvature is expressed in terms of local combinatorial data around a considered point, and Sreejith et al [64] translated Forman's general definition to graphs as…”

Section: Resistance Curvature and Forman-ricci Curvaturementioning

confidence: 99%

Discrete curvature on graphs from the effective resistance

Devriendt¹,

Lambiotte²

2022

Preprint

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This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to calculate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.

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“…In [30], Robin Forman introduced a new notion of curvature for CW complexes, a class of discrete/combinatorial spaces that includes graphs. This so-called FR curvature is defined by generalizing the definition of the classical Ricci curvature in terms of the Bochner Laplacian of a manifold to a definition for discrete spaces based on an analogous Bochner Laplacian on these spaces, see also [45]. The FR curvature is expressed in terms of local combinatorial data around a considered point, and Sreejith et al [74] translated Forman's general definition to graphs as…”

Section: Resistance Curvature and Forman-ricci Curvaturementioning

confidence: 99%

Discrete curvature on graphs from the effective resistance*

Devriendt

Lambiotte

2022

J. Phys. Complex.

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This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to approximate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.

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Characterizations of Forman curvature

Cited by 5 publications

References 24 publications

Discrete Ollivier-Ricci curvature

Discrete Ollivier-Ricci curvature

Discrete curvature on graphs from the effective resistance

Discrete curvature on graphs from the effective resistance*

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