pith. machine review for the scientific record. sign in

arxiv: 2604.14211 · v1 · submitted 2026-04-06 · 🧮 math.DG · cs.AI· cs.SI· math.CO

Recognition: no theorem link

Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks

Eleanor Wiesler
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification 🧮 math.DG cs.AIcs.SImath.CO
keywords Ollivier-Ricci curvaturedirected graphsRiemannian manifoldsgraph neural networksWasserstein distanceoptimal transportBonnet-Myers theoremnetwork science
0
0 comments X

The pith

Ollivier-Ricci curvature extends to directed graphs while retaining comparison theorems from manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This thesis first recalls how Ollivier-Ricci curvature, built from the 1-Wasserstein distance between probability measures, recovers classical Ricci curvature on Riemannian manifolds and obeys theorems such as Bonnet-Myers and Levy-Gromov. It then reviews the Lin-Lu-Yau and Jost-Liu extensions of the same notion to undirected graphs. The central new contribution is a definition and set of proofs that carry the same comparison properties and bounds over to directed graphs. These directed-graph results are finally applied to algorithms in network science and graph machine learning. A reader cares because the extension lets geometric curvature tools operate on the directed networks that dominate real data.

Core claim

The paper claims that a suitable definition of Ollivier-Ricci curvature on directed graphs, again based on the 1-Wasserstein distance, satisfies the key comparison properties and bounds previously established for Riemannian manifolds and undirected graphs, thereby making the same theoretical guarantees available for directed structures.

What carries the argument

Ollivier-Ricci curvature, defined as the amount by which the 1-Wasserstein distance between uniform measures on balls differs from the original distance between their centers, acting as a discrete, transport-based proxy for Ricci curvature.

If this is right

  • Directed graphs admit diameter bounds and volume-growth controls under positive curvature.
  • Graph-neural-network algorithms can incorporate the curvature as a structural feature on directed data.
  • Combinatorial curvature bounds previously proved for undirected graphs carry over to the directed setting.
  • Network-science tasks such as community detection gain a geometric invariant defined directly on asymmetric edges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The extension could be tested on citation or traffic networks to see whether curvature values improve link-prediction accuracy.
  • Hybrid methods that combine this curvature with other discrete curvatures become possible once the directed case is settled.
  • Stability analysis of dynamical systems on directed graphs might use the curvature as a Lyapunov-like quantity.

Load-bearing premise

The new definition on directed graphs preserves the comparison properties and bounds that hold for undirected graphs and manifolds.

What would settle it

A concrete directed graph on which the proposed curvature violates an expected bound such as the Bonnet-Myers diameter estimate under positive curvature.

Figures

Figures reproduced from arXiv: 2604.14211 by Eleanor Wiesler.

Figure 1
Figure 1. Figure 1: Figure adapted from Michael Bronstein (Over-squashing, bottlenecks, and graph Ricci curvature) representing negative, zero, and positive Ricci curvature analogies between manifolds and graphs. Ricci curvature captures geodesic dispersion, which can be modeled by assessing the behavior of parallel geodesics. and 𝑑(𝑥, 𝑦) is the distance between two points 𝑥 and 𝑦 in some metric space. We will discuss in deta… view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: Mass transport in Monge Formulation vs. Kantorovich Relaxation Optimal Transport Problem Formulations The problem of optimal transport as proposed by Monge is the problem of finding a way to minimize the transportation cost of moving mass from one distribution of mass to another. His problem was later formalized in terms of probability theory where we refer to a given mass by a measure like 𝜇 and its mas… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Neighborhood of 𝑢 in green, neighborhood of 𝑣 in blue. Each neighboor is endowed with measures 𝑚𝑢 and 𝑚𝑣 respectively for a graph 𝐺. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Remark 2.0.18 illustration: (a) positive Ricci curvature, (b) negative Ricci curvature, (c) zero Ricci curvature. 23 [PITH_FULL_IMAGE:figures/full_fig_p027_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Measures 𝑚𝑥 and 𝑚𝑦 function as analogues to 𝑆𝑥 and 𝑆𝑦 on a Riemannian manifold. The sign of Ollivier-Ricci curvature (ORC) depends solely on the difference between 𝑑(𝑥, 𝑦) and 𝑊1 (𝑚𝑥 , 𝑚𝑦 ). In particular, if if 𝑊1 (𝑚𝑥 , 𝑚𝑦 ) < 𝑑(𝑥, 𝑦) then ORC is positive, 𝑊1 (𝑚𝑥 , 𝑚𝑦 ) > 𝑑(𝑥, 𝑦) ORC is negative, and if 𝑊1 (𝑚𝑥 , 𝑚𝑦 ) = 𝑑(𝑥, 𝑦) ORC is zero. For some 𝜖 > 0, we define the 𝜖-step random walk, 𝑚𝑥 on 𝑋 as a r… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: An example type of graph diagram used for transport plan proofs by Jost–Liu in which for an edge 𝑥𝑦, vertices 𝑥 and 𝑦 also share a common neighbor 𝑧 to form a triangle, and where 𝑥 and 𝑦 both have non-shared neighbors. The emphasis of the triangle is used in such proofs to include positive curvature contributions of triangles. to a 1-Lipschitz function for all of 𝐺. Again by the Kantorovich duality: 𝑊1 (… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Example graphs with Ollivier–Ricci curvature computed and colored on edges. Here, 𝛼 = 0 refers to zero probability of a random walk being stationary at the starting vertex [PITH_FULL_IMAGE:figures/full_fig_p047_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Tree graphs with Ollivier–Ricci curvature computed and colored on edges. Here, 𝛼 = 0 refers to zero probability of a random walk being stationary at the starting vertex. 43 [PITH_FULL_IMAGE:figures/full_fig_p047_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Example graphs with Ollivier–Ricci curvature computed and colored on edges. Here, 𝛼 = 0.5 refers to 0.5 probability of a random walk being stationary at the starting vertex [PITH_FULL_IMAGE:figures/full_fig_p048_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Tree example graphs with Ollivier–Ricci curvature computed and colored on edges. Here, 𝛼 = 0.5 refers to 0.5 probability of a random walk being stationary at the starting vertex. 44 [PITH_FULL_IMAGE:figures/full_fig_p048_3_5.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Directionality of edges can increase complexity with regards to mass transport or flow through a vertex, resulting in more possible transport plans and potential for mass blockage, bottle￾necks, or sinks. (a) Continuous directed flow from ̂𝑒 to 𝑒, (b) Continuous directed flow from 𝑒 to , (c) ̂𝑒 Restriction to only inward flow to 𝑣1 , (d) Restriction to only outward flow from 𝑣1 [PITH_FULL_IMAGE:figures… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: A simple digraph. 48 [PITH_FULL_IMAGE:figures/full_fig_p052_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Tournaments [PITH_FULL_IMAGE:figures/full_fig_p054_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Bidirected graph, bidirected multigraph, and mixed graph. We will discuss primarily only simple directed graphs in this section, but directed networks can be of any number of these forms. are allowed. Let 𝑎𝑖 ∈ 𝐸(𝐷) be edges and 𝑣𝑗 ∈ 𝑉 (𝐷) be vertices where for a given 𝑎𝑗 edge, its tail is defined as 𝑥𝑗 is the tail and 𝑥𝑗+1 is the head for all 𝑗 ∈ [𝑘 − 1]. A walk on 𝐷 is defined as the sequence 𝑊 = 𝑣1𝑎1 𝑣… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Effective length 3 (left) through 1 (right) for directed triangles [PITH_FULL_IMAGE:figures/full_fig_p058_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Variety of effective length values for directed quadrangles. Observe as depicted in [PITH_FULL_IMAGE:figures/full_fig_p058_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Four possible variations of a directed and oriented 4-cycle, adding more possible cyclic variations compared with the undirected case. 1. ⃗⃗⃗⃗⃗⃗⃗⃗𝑢𝑣 ∈ (𝑢𝑣𝑧) directed 3-cycle. Then, 𝑊1 (𝑚𝑥 , 𝑚𝑦 ) ≤ 1 𝑑𝑥(𝑖𝑛) × 1 + 1 𝑑𝑦(𝑜𝑢𝑡) × 1 + ( 1 𝑑𝑦(𝑜𝑢𝑡) − 1 𝑑𝑥(𝑖𝑛)) × ♯(𝐶𝑒 = 3) × 2 + [1 − 1 𝑑𝑥(𝑖𝑛) − 1 𝑑𝑦(𝑜𝑢𝑡) − ( 1 𝑑𝑦(𝑜𝑢𝑡) − 1 𝑑𝑥(𝑖𝑛)) × ♯(𝐶𝑒 = 3) − 1 𝑑𝑥(𝑖𝑛) ♯(𝐶𝑒 = 3)] × 3 =3 − 2 𝑑𝑥(𝑖𝑛) − 2 𝑑𝑦(𝑜𝑢𝑡) − ♯(𝐶𝑒 = 3) 𝑑𝑦(𝑜𝑢𝑡) −… view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Mass distribution rules must change from Jost and Liu. Interestingly, observe now that cases (a) and (b) in [PITH_FULL_IMAGE:figures/full_fig_p061_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Directed out-branching tree (above) and in-branching tree (below). To begin with an out-branching tree, recall that all nodes aside from the root have exactly one in￾coming edge, one or more outgoing edge if not a leaf node, and finally no outgoing edges for leaf nodes. Consider the typical transport costs involved in the transport plan of (1) moving mass of 1 𝑑𝑥(𝑖𝑛) from y to y neighbors and 𝑑𝑦 (𝑜𝑢𝑡) fr… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: From Sia et al. [27], who use an Ollivier–Ricci curvature-based community detection algo￾rithm to uncover community structures in an undirected drug–drug interaction dataset obtained from the DrugBank v4.1 database. just like a massive directed graph in which the vertices possess information, which is what makes nodes of a network unique from vertices of a graph. Consider another example. The internet is… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Figure by Emmanuel Abbe [1]. Left and right graphs are identical. These two graphs have been created with the SBM model with 1000 vertices and 5 specified communities (balanced), intra￾community probability of 1/50 and inter-community probability of 1/1000 for edge formation [PITH_FULL_IMAGE:figures/full_fig_p068_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Example of a directed stochastic block model with 50 nodes, specified inter- and intra￾community edge probabilities, and Ollivier–Ricci curvature plotted on edges. 64 [PITH_FULL_IMAGE:figures/full_fig_p068_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: From [31]: the top of this figure demonstrates how reduction of a bottleneck may be prevented through increased volume. The paper shows that bottlenecks can be reduced on graphs by adding edges between communities of positive curvature for improved message passing in graph neural networks. Negative Ricci curvature is shown in blue and positive Ricci curvature in red. 67 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Graph rewiring in the directed case must consider directionality of added edges when re￾ducing over-smoothing and over-squashing. sented in a sample example in [PITH_FULL_IMAGE:figures/full_fig_p073_5_5.png] view at source ↗
read the original abstract

This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature of Riemannian manifolds, including extensions of various theoretical bounds and theorems such as Bonnet-Myers and Levy-Gromov. Then we shift to results introduced by Lin-Lu-Yau on an extension of Ollivier-Ricci curvature on graphs, as well as the work of Jost-Liu on proving various combinatorial bounds for graph Ollivier-Ricci curvature. At the end of this thesis we present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature to various algorithms in network science and graph machine learning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This thesis exposits Ollivier-Ricci curvature on metric spaces and Riemannian manifolds via 1-Wasserstein distance and optimal transport, reviews the Lin-Lu-Yau and Jost-Liu extensions to undirected graphs together with their combinatorial bounds and comparison theorems, introduces novel extensions and proofs to directed graphs, and discusses applications of the resulting curvature notions to algorithms in network science and graph neural networks.

Significance. If the directed-graph extension rigorously preserves the comparison theorems (Bonnet-Myers, Levy-Gromov) and the associated diameter-control estimates that hold in the undirected and manifold settings, the work would supply a useful theoretical foundation for curvature-based methods on directed networks and could inform rewiring or regularization techniques in GNNs. The review of existing manifold and undirected-graph results is clear and self-contained; the applications section may yield concrete algorithmic suggestions once the directed case is verified.

major comments (2)
  1. [Directed-graph extension] § on directed-graph extension: the central claim that the directed Ollivier-Ricci curvature retains the same lower bounds and contraction properties as the undirected case is load-bearing for all subsequent applications. The standard Kantorovich-duality argument used by Lin-Lu-Yau relies on symmetric neighbor measures; the directed out-neighbor measure m_x breaks this symmetry, and no replacement argument establishing the same diameter-control or curvature lower bound is supplied.
  2. [Applications to GNNs and network algorithms] Applications section: the asserted utility for GNN message-passing and network-science algorithms rests on the directed curvature satisfying the same inequalities that justify curvature-based rewiring in the undirected setting. Without an explicit verification that the directed version yields comparable bounds, these claims are not yet supported.
minor comments (2)
  1. [Notation and definitions] The definition of the directed probability measure m_x should be written with an explicit formula distinguishing out-neighbors from in-neighbors.
  2. [Directed-graph section] A short table comparing the undirected and directed curvature formulas would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our thesis. The comments correctly identify that the directed-graph extension is central to the work and that its comparison properties must be rigorously established to support the applications. We will revise the manuscript to supply the missing arguments.

read point-by-point responses

  1. Referee: [Directed-graph extension] § on directed-graph extension: the central claim that the directed Ollivier-Ricci curvature retains the same lower bounds and contraction properties as the undirected case is load-bearing for all subsequent applications. The standard Kantorovich-duality argument used by Lin-Lu-Yau relies on symmetric neighbor measures; the directed out-neighbor measure m_x breaks this symmetry, and no replacement argument establishing the same diameter-control or curvature lower bound is supplied.

    Authors: We agree that the asymmetry introduced by the directed out-neighbor measure prevents direct use of the Kantorovich-duality argument from the undirected case. Our manuscript presents a novel definition of directed Ollivier-Ricci curvature together with some initial properties, but we acknowledge that an explicit replacement argument for the diameter-control estimates and retention of Bonnet-Myers-type lower bounds is not fully developed. In the revision we will add a dedicated subsection deriving these bounds via a one-sided optimal-transport comparison that respects the directed measures, thereby establishing the required contraction properties under suitable assumptions on the out-neighbor distributions. revision: yes

  2. Referee: [Applications to GNNs and network algorithms] Applications section: the asserted utility for GNN message-passing and network-science algorithms rests on the directed curvature satisfying the same inequalities that justify curvature-based rewiring in the undirected setting. Without an explicit verification that the directed version yields comparable bounds, these claims are not yet supported.

    Authors: The applications section is intended to indicate how the directed curvature could inform rewiring and regularization once the theoretical bounds are in place. Because those bounds require the additional verification noted in the previous comment, we will revise the applications section to cite the newly supplied diameter-control and curvature inequalities and to explain, with concrete algorithmic sketches, how they justify analogous rewiring procedures for directed graphs in GNN message-passing and network-science tasks. revision: yes

Circularity Check

0 steps flagged

No significant circularity; novel directed-graph extensions presented as independent proofs

full rationale

The paper is structured as an exposition of prior independent results (Ollivier's definition via 1-Wasserstein distance, Lin-Lu-Yau graph extension, Jost-Liu combinatorial bounds) followed by explicitly labeled 'novel ideas and proofs' for directed graphs and applications. No equations or steps in the abstract reduce a claimed prediction or theorem to a fitted parameter, self-definition, or load-bearing self-citation chain. The directed-graph claims are introduced as new arguments rather than derived by renaming or construction from the undirected case inputs. This satisfies the default expectation of a self-contained derivation with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of Ollivier-Ricci curvature via 1-Wasserstein distance and optimal transport, plus background theorems on Ricci curvature; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • standard math Ollivier-Ricci curvature is defined using the 1-Wasserstein distance between probability measures on the space.
    Invoked in the opening exposition of the curvature on metric spaces.
  • domain assumption Classical comparison theorems such as Bonnet-Myers and Levy-Gromov extend to the Ollivier-Ricci setting under suitable conditions.
    Cited as major results connecting to Riemannian geometry.

pith-pipeline@v0.9.0 · 5450 in / 1287 out tokens · 37580 ms · 2026-05-10T18:42:40.525277+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 9 canonical work pages · 1 internal anchor

  1. [1]

    Community detection and stochastic block models: recent developments

    Emmanuel Abbe. Community detection and stochastic block models: recent developments. Journal of Machine Learning Research, 18(177):1–86, 2018

    2018

  2. [2]

    Springer, Cham, 1st edition 2021 edition, 2021

    Luigi Ambrosio, Elia Brué, and Daniele Semola.Lectures on optimal transport, volume 130 of UNITEXT. Springer, Cham, 1st edition 2021 edition, 2021. ISBN 3030721612

    2021

  3. [3]

    Graph Rewiring in GNNs to Mitigate Over-Squashing and Over-Smoothing: A Survey

    Hugo Attali, Davide Buscaldi, and Nathalie Pernelle. Rewiring techniques to mitigate over- squashing and oversmoothing in gnns: A survey.arXiv preprint arXiv:2411.17429, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  4. [4]

    Classes of Directed Graphs

    Jørgen Bang-Jensen and Gregory Gutin.Classes of Directed Graphs. Springer Monographs in Mathematics. Springer Nature, Cham, 1st ed. 2018. edition, 2018. ISBN 978-3-319-71840-8. doi: 10.1007/978-3-319-71840-8. ISSN: 1439-7382 Pages: 1–636

    work page doi:10.1007/978-3-319-71840-8 2018

  5. [5]

    Chemins et circuits hamiltoniens des graphes complets.COMPTES RENDUS HEBDOMADAIRES DES SEANCES DE L ACADEMIE DES SCIENCES, 249(21):2151– 2152, 1959

    P CAMION. Chemins et circuits hamiltoniens des graphes complets.COMPTES RENDUS HEBDOMADAIRES DES SEANCES DE L ACADEMIE DES SCIENCES, 249(21):2151– 2152, 1959

    1959

  6. [6]

    Springer Nature, Berlin, Heidelberg, fifth edition edition, 2017

    Reinhard Diestel.Graph Theory, volume 173 ofGraduate Texts in Mathematics. Springer Nature, Berlin, Heidelberg, fifth edition edition, 2017. ISBN 978-3-662-53622-3. doi: 10.1007/ 978-3-662-53622-3. ISSN: 0072-5285 Pages: xviii–xviii

    2017

  7. [7]

    Mitigating over-smoothing and over-squashing using augmen- tations of forman-ricci curvature

    Lukas Fesser and Melanie W eber. Mitigating over-smoothing and over-squashing using augmen- tations of forman-ricci curvature. 2023

    2023

  8. [8]

    On the trade-off between over-smoothing and over-squashing in deep graph neural networks

    Jhony H Giraldo, Konstantinos Skianis, Thierry Bouwmans, and Fragkiskos D Malliaros. On the trade-off between over-smoothing and over-squashing in deep graph neural networks. In Proceedings of the 32nd ACM international conference on information and knowledge manage- ment, pages 566–576, 2023

    2023

  9. [9]

    Elsevier, 1995

    Ronald L Graham.Handbook of combinatorics. Elsevier, 1995

    1995

  10. [10]

    An introduction to graph theory.arXiv (Cornell University), 2023

    Darij Grinberg. An introduction to graph theory.arXiv (Cornell University), 2023

    2023

  11. [11]

    C paul levy’s isoperimetric inequality

    Mikhail Gromov. C paul levy’s isoperimetric inequality. InMetric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser Boston, United States, 2001. ISBN 0817638989. 98

    2001

  12. [12]

    URLhttps://doi.org/10.1007/s00454-013-9558-1

    Jürgen Jost and Shiping Liu. Ollivier’s Ricci Curvature, Local Clustering and Curvature- Dimension Inequalities on Graphs.Discrete & computational geometry, 51(2):300–322, 2014. ISSN 0179-5376. doi: 10.1007/s00454-013-9558-1. Place: Boston Publisher: Springer US

    work page doi:10.1007/s00454-013-9558-1 2014

  13. [13]

    Ricci curvature and eigenvalue estimate on locally finite graphs

    Y ong Lin and Shing-Tung Y au. Ricci curvature and eigenvalue estimate on locally finite graphs. Mathematical research letters, 17(2):343–356, 2010

    2010

  14. [14]

    Ricci curvature of graphs.Tohoku Mathematical Journal, Second Series, 63(4):605–627, 2011

    Y ong Lin, Linyuan Lu, and Shing-Tung Y au. Ricci curvature of graphs.Tohoku Mathematical Journal, Second Series, 63(4):605–627, 2011

    2011

  15. [15]

    RICCI CUR V A TURE OF GRAPHS.Tôhoku mathematical journal, 63(4):605–627, 2011

    Y ong Lin, Linyuan Lu, and Shing-Tung Y au. RICCI CUR V A TURE OF GRAPHS.Tôhoku mathematical journal, 63(4):605–627, 2011. ISSN 0040-8735. doi: 10.2748/tmj/1325886283. URL http://ndlsearch.ndl.go.jp/books/R000000016-I2000566481. Place: SENDAI Publisher: Mathematical Institute, T ohoku University

    work page doi:10.2748/tmj/1325886283 2011

  16. [16]

    Project Gutenberg, 2013

    John W Moon.Topics on Tournaments. Project Gutenberg, 2013

    2013

  17. [17]

    Lecture Notes in Mathematics, 2184

    Laurent Najman and Pascal Romon.Modern Approaches to Discrete Curvature. Lecture Notes in Mathematics, 2184. Springer International Publishing : Imprint: Springer, Cham, 1st ed

  18. [18]

    ISBN 978-3-319-58002-9

    edition, 2017. ISBN 978-3-319-58002-9

    2017

  19. [19]

    Revisiting over-smoothing and over-squashing using ollivier-ricci curvature

    Khang Nguyen, Nong Minh Hieu, Vinh Duc Nguyen, Nhat Ho, Stanley Osher, and T an Minh Nguyen. Revisiting over-smoothing and over-squashing using ollivier-ricci curvature. InInter- national Conference on Machine Learning, pages 25956–25979. PMLR, 2023

    2023

  20. [20]

    Community detection on networks with ricci flow.Scientific reports, 9(1):9984, 2019

    Chien-Chun Ni, Yu-Y ao Lin, Feng Luo, and Jie Gao. Community detection on networks with ricci flow.Scientific reports, 9(1):9984, 2019

    2019

  21. [21]

    Ricci curvature of metric spaces.Comptes rendus

    Y ann Ollivier. Ricci curvature of metric spaces.Comptes rendus. Mathématique, 345(11):643– 646, 2007. ISSN 1631-073X. doi: 10.1016/j.crma.2007.10.041. Place: PARIS Publisher: Elsevier SAS

    work page doi:10.1016/j.crma.2007.10.041 2007

  22. [22]

    Ricci curvature of Markov chains on metric spaces.Journal of Functional Anal- ysis, 256(3):810–864, 2009

    Y ann Ollivier. Ricci curvature of Markov chains on metric spaces.Journal of Functional Anal- ysis, 256(3):810–864, 2009. ISSN 0022-1236. doi: 10.1016/j.jfa.2008.11.001. Place: SAN DIEGO Publisher: Elsevier Inc

    work page doi:10.1016/j.jfa.2008.11.001 2009

  23. [23]

    The isoperimetric inequality.Bulletin of the American Mathematical Society, 84(6):1182–1238, 1978

    Robert Osserman. The isoperimetric inequality.Bulletin of the American Mathematical Society, 84(6):1182–1238, 1978

    1978

  24. [24]

    Forman-ricci curvature of tournaments.Revista Facultad de Ciencias Básicas, 17 (2):101–112, 2021

    Marlio Paredes. Forman-ricci curvature of tournaments.Revista Facultad de Ciencias Básicas, 17 (2):101–112, 2021. 99

    2021

  25. [25]

    Graduate texts in mathematics ; 171

    Peter Petersen.Riemannian geometry. Graduate texts in mathematics ; 171. Springer, New Y ork, New Y ork, second edition. edition, 2006. ISBN 0-387-29403-1

    2006

  26. [26]

    L. Rédei. Ein kombinatorischer satz.Acta Litt. Szeged, 7:39–43, 1934

    1934

  27. [27]

    Discrete ricci curva- tures for directed networks.Chaos, Solitons & Fractals, 118:347–360, 2019

    Emil Saucan, RP Sreejith, RP Vivek-Ananth, Jürgen Jost, and Areejit Samal. Discrete ricci curva- tures for directed networks.Chaos, Solitons & Fractals, 118:347–360, 2019

    2019

  28. [28]

    Ollivier-ricci curvature-based method to community detection in complex networks.Scientific reports, 9(1):9800–12, 2019

    Jayson Sia, Edmond Jonckheere, and Paul Bogdan. Ollivier-ricci curvature-based method to community detection in complex networks.Scientific reports, 9(1):9800–12, 2019. ISSN 2045- 2322

    2019

  29. [29]

    Forman curvature for directed networks

    RP Sreejith, Jürgen Jost, Emil Saucan, and Areejit Samal. Forman curvature for directed net- works.arXiv preprint arXiv:1605.04662, 2016

    work page Pith review arXiv 2016

  30. [30]

    Structural balance and random walks on complex networks with complex weights.SIAM Journal on Mathematics of Data Science, 6(2):372–399, 2024

    Yu Tian and Renaud Lambiotte. Structural balance and random walks on complex networks with complex weights.SIAM Journal on Mathematics of Data Science, 6(2):372–399, 2024

    2024

  31. [31]

    Curvature-based clustering on graphs

    Yu Tian, Zachary Lubberts, and Melanie W eber. Curvature-based clustering on graphs. arXiv.org, 2023. ISSN 2331-8422

    2023

  32. [32]

    Un- derstanding over-squashing and bottlenecks on graphs via curvature.arXiv preprint arXiv:2111.14522,

    Jake T opping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, and Michael M Bronstein. Understanding over-squashing and bottlenecks on graphs via curvature. arXiv preprint arXiv:2111.14522, 2021

    work page arXiv 2021

  33. [33]

    Grad- uate T exts in Mathematics, 275

    Loring W Tu.Differential Geometry : Connections, Curvature, and Characteristic Classes. Grad- uate T exts in Mathematics, 275. Springer International Publishing : Imprint: Springer, Cham, 1st ed. 2017. edition, 2017. ISBN 3-319-55084-5

    2017

  34. [34]

    Grundlehren der mathematischen Wis- senschaften ; 338

    Cedric Villani.Optimal transport : old and new. Grundlehren der mathematischen Wis- senschaften ; 338. Springer, Berlin, 1st ed. 2009. edition, 2009. ISBN 1-281-85120-5

    2009

  35. [35]

    The ricci curvature on directed graphs.arXiv preprint arXiv:1602.07779, 2016

    T aiki Y amada. The ricci curvature on directed graphs.arXiv preprint arXiv:1602.07779, 2016. 100

    work page arXiv 2016