Spread‐out percolation in ℝd

Bollobás, Béla; Janson, Svante; Riordan, Oliver

doi:10.1002/rsa.20175

Cited by 13 publications

(17 citation statements)

References 9 publications

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“…Then ν 1 n p → µ κ in the space of finite measures on S with the weak topology, where µ κ is the measure on S defined by dµ κ = ρ κ dµ. In other words, for every µ-continuity set A, 20) where G n = G V (n, κ n ). Furthermore, if f : S → R is continuous µ-a.e.…”

Section: )mentioning

confidence: 99%

The phase transition in inhomogeneous random graphs

Bollobás

Janson

Riordan

2007

Random Struct Algorithms

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632

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Abstract. The 'classical' random graph models, in particular G(n, p), are 'homogeneous', in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying 'inhomogeneous' random graph models.One of the most studied properties of these new models is their 'robustness', or, equivalently, the 'phase transition' as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years.Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence.Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n, p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied.We show that, under one very weak assumption (that the expected number of edges is 'what it should be'), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze.We also consider other properties of the model, showing, for example, that when there is a giant component, it is 'stable': for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n).

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Section: )mentioning

confidence: 99%

The phase transition in inhomogeneous random graphs

Bollobás

Janson

Riordan

2007

Random Struct Algorithms

Self Cite

632

1,592

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“…In a lattice version of the spreading transformation, where one considers e.g. bond percolation on the vertices of Z d with range r becoming large, the critical value is known to approach its branching process limit (see [13] or [4]), but the convergence is not known to be monotone.…”

Section: Introductionmentioning

confidence: 99%

Strict Inequalities of Critical Values in Continuum Percolation

2011

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We consider the supercritical finite-range random connection model where the points x, y of a homogeneous planar Poisson process are connected with probability f (|y − x|) for a given f . Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality p site c > p bond c . We also show that reducing the connection function f strictly increases the critical Poisson intensity.Finally, we deduce that performing a spreading transformation on f (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) strictly reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.

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“…Hence, when it comes to proofs, we lose no generality by taking \documentclass{book} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{S}=\lbrack 0,1\rbrack \end{align*} \end{document} , but we usually prefer allowing an arbitrary type space, which is more flexible for applications. For example, as with the model in [10], type spaces such as \documentclass{book} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{S}=\lbrack 0,1\rbrack ^2 \end{align*} \end{document} are likely to be useful for geometric applications, as in [11].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Laser microdissection in translational and clinical research

et al. 2006

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Laser microdissection (LMD) is now a well established method for isolating individual cells or subcellular structures from a heterogeneous cell population. In recent years, cell, DNA, RNA, and protein based techniques has been successfully coupled to LMD and important information has been gathered through the analysis of the genome, transcriptome, and more recently the proteome of individual microdissected cells.The aims of this review are to summarize and compare the principles of different laser microdissection instruments and techniques, to discuss sample preparation procedures for microdissection, and to provide wide variety of examples of translational/clinical research applications of LMD. Novel techniques specifically developed for the improved isolation of stained cells, living cells, or rare cells are also discussed.LMD has become an indispensable tool in the preparation of homogenous samples for sophisticated cell or molecular assays. Despite major technological advances, the labor requirements of LMD are still relatively high. However, understanding the advantages and disadvantages of LMD technology and associated sample preparation procedures may aid in the earlier introduction of this method into the routine clinical diagnostics. © 2006 International Society for Analytical Cytology

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Spread‐out percolation in ℝ^d

Cited by 13 publications

References 9 publications

The phase transition in inhomogeneous random graphs

The phase transition in inhomogeneous random graphs

Strict Inequalities of Critical Values in Continuum Percolation

Laser microdissection in translational and clinical research

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