A clustering procedure based on the comparison between the k nearest neighbors graph and the minimal spanning tree

González-Barrios, José M.; Quiróz, Adolfo J.

doi:10.1016/s0167-7152(02)00421-2

Cited by 39 publications

(17 citation statements)

References 13 publications

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“…This greatly improved the earlier bounds due to Xue and Kumar [11] and Gonzáles-Barrios and Quiroz [4]. More recently, we proved [2] that there is a critical value c such that for c 1 < c and k ≤ c 1 log n, G n,k is not connected whp, and that for c 2 > c and k ≥ c 2 log n, G n,k is connected whp.…”

Section: Introductionsupporting

confidence: 71%

Highly connected random geometric graphs

Balister

Bollobás

Sarkar

et al. 2009

Discrete Applied Mathematics

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a b s t r a c tLet P be a Poisson process of intensity 1 in a square S n of area n. We construct a random geometric graph G n,k by joining each point of P to its k nearest neighbours. For many applications it is desirable that G n,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of G n,k to our previous work on the connectivity of G n,k . Roughly speaking, we show that for s = o(log n), the threshold (in k) for s-connectivity is asymptotically the same as that for connectivity, so that, as we increase k, G n,k becomes s-connected very shortly after it becomes connected.

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Section: Introductionsupporting

confidence: 71%

Highly connected random geometric graphs

Balister

Bollobás

Sarkar

et al. 2009

Discrete Applied Mathematics

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“…Xue and Kumar [62] were the first to publish bounds on c l and c u : they obtained c l ≥ 0.074 and c u ≤ 5.1774, although a bound of c u ≤ 3.8597 can be read out of earlier work of Gonzáles-Barrios and Quiroz [33]. Subsequently, Wan and Yi [60] showed that c u ≤ e and Xue and Kumar [63] improved their bound to c u ≤ 1/ log 2.…”

Section: Connectivitymentioning

confidence: 99%

Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs

Balister

Sarkar

Bollobás

2008

Bolyai Society Mathematical Studies

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“…Using a result of Penrose [6], Xue and Kumar [10] showed that c u ≤ 5.1774. A bound 618 • SGSA V. FALGAS-RAVRY AND M. WALTERS of c u ≤ 2 log((4π/3 + √ 3/2)/(π + 3 √ 3/4)) ≈ 3.8597 can also be read out of earlier work by Gonzáles-Barrios and Quiroz [5].…”

Section: Introductionmentioning

confidence: 89%

Sharpness in the k-Nearest-Neighbours Random Geometric Graph Model

Falgas–Ravry

Walters

2012

Advances in Applied Probability

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Let Sn, k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn, k is connected is at least ε, then the probability that Sn, k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability Sn,k(n)+⌊C'slog logn⌋ is s-connected (i.e. remains connected after the deletion of any s − 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

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A clustering procedure based on the comparison between the k nearest neighbors graph and the minimal spanning tree

Cited by 39 publications

References 13 publications

Highly connected random geometric graphs

Highly connected random geometric graphs

Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs

Sharpness in the k-Nearest-Neighbours Random Geometric Graph Model

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