Variance and covariance of distributions on graphs

Devriendt, Karel; Martin-Gutierrez, Samuel; Lambiotte, Renaud

doi:10.48550/arxiv.2008.09155

Cited by 5 publications

(18 citation statements)

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“…In Appendix E, we furthermore show that the potential can be rewritten as tr( 12 ΩQΩ) = 2nσ 2 where σ 2 = 1 2 p T Ωp is a graph invariant that appears in several contexts in the study of effective resistances, see for instance [18] and [19,Appendix I]. The gradient expression (10) further adds to the theory of this invariant.…”

Section: Resistance Ricci Flowmentioning

confidence: 80%

“…where the range for t = D − D j is lower bounded by −r, which are the neighbours of i furthest away from the boundary, and upper bounded by min(r, D), which are the neighbours closest to the boundary. Introducing (19) for the function S retrieves the formula (8) for the expected resistance curvature in Section 4.3. Equation ( 21) can be integrated numerically for different values of D as shown in Figure 5.…”

Section: Appendixmentioning

confidence: 99%

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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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Section: Resistance Ricci Flowmentioning

confidence: 80%

Section: Appendixmentioning

confidence: 99%

Discrete curvature on graphs from the effective resistance

Devriendt¹,

Lambiotte²

2022

Preprint

Self Cite

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“…We furthermore believe that the vector r is an important algebraic object of a graph and worthy of a deeper study, e.g. as initiated in [35]. Remark: For non-hyperacute Simplices S, the matrix identity…”

Section: Effective Resistances and Fiedler's Identitymentioning

confidence: 99%

“…The interpretation of R and Sr as the respective circumradius and circumcenter of S are proven by Fiedler [9, Cor. 1.4.13], and in [35] using the identities Ωr = (r T Ωr)u and R 2 = 1 2 r T Ωr. While G, Q, S, Ω are mutually interchangeable, it is somehow natural to think of G and S as the combinatorial and geometric objects of interest, with Q and Ω as convenient and practical algebraic representations.…”

Section: Effective Resistances and Fiedler's Identitymentioning

confidence: 99%

Effective resistance is more than distance: Laplacians, Simplices and the Schur complement

Devriendt

2020

Preprint

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This article discusses a geometric perspective on the well-known fact in graph theory that the effective resistance is a metric on the nodes of a graph. The classical proofs of this fact make use of ideas from electrical circuits or random walks; here we describe an alternative approach which combines geometric (using simplices) and algebraic (using the Schur complement) ideas. These perspectives are unified in a matrix identity of Miroslav Fiedler, which beautifully summarizes a number of related ideas at the intersection of graphs, Laplacian matrices and simplices, with the metric property of the effective resistance as a prominent consequence.

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“…In section 4.2.2 we will show that expression ( 6) is closely related to the definition of OR curvature. In [21] we found that var(f ) := 1 2 f T Ωf can be interpreted as the variance of a distribution on a graph; this allows to rewrite (5) as…”

Section: Average Distance Characterizationmentioning

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.

show abstract

Variance and covariance of distributions on graphs

Cited by 5 publications

References 0 publications

Discrete curvature on graphs from the effective resistance

Discrete curvature on graphs from the effective resistance

Effective resistance is more than distance: Laplacians, Simplices and the Schur complement

Discrete curvature on graphs from the effective resistance*

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