On the size of a random sphere of influence graph

“…Remark 16. The proof shows that ρ k must be a bijection with equality in (13). Furthermore, if X k takes at most two values conditioned on X 1 , .…”

Section: Weakening the Independence Assumptionmentioning

confidence: 92%

On the Method of Typical Bounded Differences

Warnke¹

2015

Combinator. Probab. Comp.

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Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f (X) is a function of independent random variables X = (X1, . . . , Xn). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |fWhile this is easy to check, the main disadvantage is that it considers worst-case changes c k , which often makes the resulting bounds too weak to be useful.In this paper we prove a variant of the bounded differences inequality which can be used to establish concentration of functions f (X) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations why concentration should hold. Indeed, given an event Γ that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where Γ occurs. The point is that the resulting typical changes c k are often much smaller than the worst case ones.To illustrate its application we consider the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdős.

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“…и be the conditional probability given that X y X s t , X y X t , t , t 1 2 2 1 3 2 3 4 < < s t , and X y X s t . Then, by symmetry, 3 1 4 4…”

Section: žmentioning

confidence: 99%

On the variance of the random sphere of influence graph

Hitczenko

Janson

Yukich

1999

Random Struct. Alg.

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ABSTRACT:We show that the variance of the number of edges in the random sphere of influence graph built on n i.i.d. sites which are uniformly distributed over the unit cube in R d , grows linearly with n. This is then used to establish a central limit theorem for the number of edges in the random sphere of influence graph built on a Poisson number of sites. Some related proximity graphs are discussed as well.

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On the size of a random sphere of influence graph

Cited by 15 publications

References 7 publications

On a sphere of influence graph in a one-dimensional space

On a sphere of influence graph in a one-dimensional space

On the Method of Typical Bounded Differences

On the variance of the random sphere of influence graph

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