The diameter of long‐range percolation clusters on finite cycles

Benjamini, Itaı; Berger, Noam

doi:10.1002/rsa.1022

Cited by 93 publications

(184 citation statements)

References 16 publications

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“…Benjamini and Berger [11] studied this problem in one dimension, modifying the model so that the graph is finite (restricted to the integers {1, 2, . .…”

Section: Further Results On Small-world Network and Decentralized Sementioning

confidence: 99%

“…As a result of this work, we know that the diameter of the graph changes qualitatively at the "critical values" α = d and α = 2d. In particular, with high probability, the diameter is constant when α < d (due in essence to a result of [12]), is proportional to log n/ log log n when α = d [20], is polylogarithmic in n when d < α < 2d (with an essentially tight bound provided in [13]), and is lower-bounded by a polynomial in n when α > 2d [11], [20]. The case α = 2d is largely open, and conjectured to have diameter polynomial in n with high probability [11], [13].…”

Section: Further Results On Small-world Network and Decentralized Sementioning

confidence: 99%

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Complex networks and decentralized search algorithms

Kleinberg¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

151

194

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Abstract. The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized.We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks. Mathematics Subject Classification (2000). Primary 68R10; Secondary 05C80, 91D30.

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“…Benjamini and Berger [11] studied this problem in one dimension, modifying the model so that the graph is finite (restricted to the integers {1, 2, . .…”

Section: Further Results On Small-world Network and Decentralized Sementioning

confidence: 99%

Section: Further Results On Small-world Network and Decentralized Sementioning

confidence: 99%

Complex networks and decentralized search algorithms

Kleinberg¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

151

194

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“…Here we conjecture: Note that, according to this conjecture, in d = 1, the interval α ∈ (0, 2) of "interesting" exponents is larger than the interval for which an infinite connected component may occur even without the "help" of nearest neighbor connections. On the other hand, in dimensions d ≥ 3, the interval conjectured for stable convergence is strictly smaller than that of "genuine" long-range percolation behavior, as defined, e.g., in terms of the scaling of graph distance with Euclidean distance; cf [4,5,8].…”

Section: B Some Questions and Conjecturesmentioning

confidence: 97%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

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193

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Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

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“…The generalization to cases where the dynamical process takes place in lattices where at least some of the infections are LR was considered [17][18][19]. Further interest in the study of LR percolation was also triggered by papers giving some more exact results and its realization on finite graphs [20,21].…”

Section: Introductionmentioning

confidence: 99%

One-dimensional long-range percolation: A numerical study

et al. 2017

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In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r 1+σ , where r is the distance length between distinct sites. We introduce and test an order N Monte Carlo algorithm and we determine as a function of σ the critical value Cc at which percolation occurs. The critical exponents in the range 0 < σ < 1 are reported and compared with mean-field and ε-expansion results. Our analysis is in agreement, up to a numerical precision ≈ 10 −3 , with the mean field result for the anomalous dimension η = 2 − σ, showing that there is no correction to η due to correlation effects.

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The diameter of long‐range percolation clusters on finite cycles

Abstract: Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle /N are given.

Cited by 93 publications

References 16 publications

Complex networks and decentralized search algorithms

Complex networks and decentralized search algorithms

Quenched invariance principle for simple random walk on percolation clusters

One-dimensional long-range percolation: A numerical study

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