Threshold functions

Bollobás, B.; Thomason, Andrew

doi:10.1007/bf02579198

Cited by 245 publications

(218 citation statements)

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“…We remark that, by a general result of Bollobás and Thomason [16], the event G n,p H = K n has a threshold, i.e., if p ≪ p c (n, H) then the probability of percolation is o(1), and if p ≫ p c (n, H) then it is 1 − o(1). Moreover, a general result of Friedgut [24,Theorem 1.4], combined with Theorem 2, below, shows that this event has a sharp threshold 1 when H = K 4 , and we expect this to hold for all K r .…”

Section: Introductionmentioning

confidence: 82%

Graph bootstrap percolation

Balogh

Bollobás

Morris

2012

Random Struct Algorithms

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Abstract. Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph H, and a set G ⊂ E(K n ) of initially 'infected' edges, we infect, at each time step, a new edge e if there is a copy of H in K n such that e is the only not-yet infected edge of H. We say that G percolates in the H-bootstrap process if eventually every edge of K n is infected. The extremal questions for this model, when H is the complete graph K r , were solved (independently) by Alon, Kalai and Frankl almost thirty years ago. In this paper we study the random questions, and determine the critical probability p c (n, K r ) for the K r -process up to a poly-logarithmic factor. In the case r = 4 we prove a stronger result, and determine the threshold for p c (n, K 4 ).

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Section: Introductionmentioning

confidence: 82%

Graph bootstrap percolation

Balogh

Bollobás

Morris

2012

Random Struct Algorithms

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“…If the ratio δ/p c is bounded away from zero, we will say that properties have a coarse threshold. (Bollobás and Thomason [6] showed that this ratio is bounded from above.) From [14] a coarse threshold for a graph property can only happen for small enough p, i.e.…”

Section: Introduction and Definitionsmentioning

confidence: 95%

“…One of the first observations made about random graphs by Erdös and Rényi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollobás and Thomason proved the existence of threshold functions for all monotone set properties ( [6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval.…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Sharp thresholds of graph properties, and the $k$-sat problem

Friedgut

Bourgain

1999

J. Amer. Math. Soc.

543

430

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“…It is observed, as theoretically expected, that as the radius of emission increases the angle of emission necessary to ensure the same probability of success decreases. As colisions are taken into account one observe on Figures 4,5,6 that increasing the angle of emissions increases initially the chance of success up to a maximal value and afterwards the chance of succes decreases. On these figures one can also observe that for large value of the emission angle, the performances are better with small radius of emission.…”

Section: Methodsmentioning

confidence: 98%

“…This is a particular situation for which there are general applicable theoretical results. In particular, sharp threshold is expected as the number of sensors increases [11,12,5,30]. This means that if the number of sensors is large enough we should observe that, as the angle of emission increases, the probability of success suddenly changes from small values (close to 0) to larger values (close to 1).…”

Section: Methodsmentioning

confidence: 99%

Numerical Estimation of the Impact of Interferences on the Localization Problem in Sensor Networks

Bouget

Leone

Rolim

2006

Experimental Algorithms

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Abstract.In this paper we numerically analyze the impact of interferences in wireless networks. More particularly, the impact on the probability of success of a localization algorithm. This problem is particularly relevant in the context of sensor networks. Actually, our numerical experiments provide information about this algorithm, even when we do not consider interferences. Indeed, in this case, sharp threshold is observed on the probability of success of the algorithm. Moreover, our numerical computations show that the main harmful interferences are the ones occurring between sensors which get localized at the same time and send in turn their own location. This is demonstrated by varying the time span of the random waiting time before the emissions. We then observe that the longer the waiting time the closer the curves are to the ones obtained without considering interferences. Hence, this proves to be an efficient way of reducing the impact of interferences on the localization algorithm. Moreover, our numerical experiments demonstrate that among the sectors of disk with same area, the one with the smaller radius of emission and larger angle of emission is the more appropriate to the localization algorithm.

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Threshold functions

Cited by 245 publications

References 1 publication

Graph bootstrap percolation

Graph bootstrap percolation

Sharp thresholds of graph properties, and the $k$-sat problem

Numerical Estimation of the Impact of Interferences on the Localization Problem in Sensor Networks

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