High-Dimensional Random Geometric Graphs and their Clique Number

Devroye, Luc; György, András; Lugosi, Gábor; Udina, Frederic

doi:10.1214/ejp.v16-967

Cited by 54 publications

(87 citation statements)

References 8 publications

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“…In probability theory, models such as

G (n, p, d)

have been studied for a long time in the low‐dimensional regime, see, e.g., . The high‐dimensional setting was first investigated recently in . In this paper it was observed that with n fixed and

d \to \infty, G (n, p, d)

converges in total variation to G ( n, p ).…”

Section: Introductionmentioning

confidence: 99%

“…In spite of previous works, the high‐dimensional

G (n, p, d)

remains mysterious in many ways. Essentially, in the dense regime, the only graph parameter which is well understood is the clique number, due to the results of , while in the sparse case basically nothing is known. One of the technical contributions of the present paper is to compute rather precisely the probability of a triangle in the sparse case.…”

Section: Introductionmentioning

confidence: 99%

“…To put it differently, we are interested in studying the total variation distance between these two models, denoted by

TV (G (n, p), G (n, p, d))

. Recall that in it was proved that, if n is fixed and

d \to \infty

, then

TV (G (n, p), G (n, p, d)) \to 0.

…”

Section: Introductionmentioning

confidence: 99%

“…which, by (13), is bounded from below by a universal constant. Together with (47), this concludes the proof.…”

mentioning

confidence: 99%

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Testing for high‐dimensional geometry in random graphs

Bubeck

Ding

Eldan

et al. 2016

Random Struct Algorithms

105

236

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We study the problem of detecting the presence of an underlying high‐dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdős‐Rényi random graph G(n, p). Under the alternative, the graph is generated from the G(n,p,d) model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere double-struckSd−1, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., p is a constant), we propose a near‐optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in G(n,p,d). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 503–532, 2016

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“…In probability theory, models such as

G (n, p, d)

have been studied for a long time in the low‐dimensional regime, see, e.g., . The high‐dimensional setting was first investigated recently in . In this paper it was observed that with n fixed and

d \to \infty, G (n, p, d)

converges in total variation to G ( n, p ).…”

Section: Introductionmentioning

confidence: 99%

“…In spite of previous works, the high‐dimensional

G (n, p, d)

Section: Introductionmentioning

confidence: 99%

“…To put it differently, we are interested in studying the total variation distance between these two models, denoted by

TV (G (n, p), G (n, p, d))

. Recall that in it was proved that, if n is fixed and

d \to \infty

, then

TV (G (n, p), G (n, p, d)) \to 0.

…”

Section: Introductionmentioning

confidence: 99%

“…which, by (13), is bounded from below by a universal constant. Together with (47), this concludes the proof.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Testing for high‐dimensional geometry in random graphs

Bubeck

Ding

Eldan

et al. 2016

Random Struct Algorithms

105

236

View full text Add to dashboard Cite

show abstract

“…The only random geometric graphs we consider here are those described above, where the points are independently and uniformly distributed over a square in the plane. See [28] for more general models of random geometric graphs, and see [9] in particular for models in high dimensions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Clique Colourings of Geometric Graphs

McDiarmid

Mitsche

Prałat

2018

Electron. J. Combin.

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A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given n points x 1 , . . . , x n in the plane, and a threshold r > 0, the corresponding geometric graph has vertex set {v 1 , . . . , v n }, and distinct v i and v j are adjacent when the Euclidean distance between x i and x j is at most r. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph G(n, r) in the plane, where n random points are independently and uniformly distributed in a suitable square. We see that as r increases from 0, with high probability the clique chromatic number is 1 for very small r, then 2 for small r, then at least 3 for larger r, and finally drops back to 2.Date: 30 June 2017, v3. 1991 Mathematics Subject Classification. 05C80, 05C15, 05C35.

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