Continuum Percolation

Meester, Ronald; Roy, Rahul

doi:10.1017/cbo9780511895357

Cited by 841 publications

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“…We denote the resulting graph by RCM(λ, f ). For a formal description of the RCM, see [10]. It is well known that provided 0 < ∞ 0 rf (r) dr < ∞, the RCM has a critical density value λ f ∈ (0, ∞), in fact with .…”

Section: Statement and Discussion Of Resultsmentioning

confidence: 99%

“…Such graphs are not covered by previous results because they are not finitely transitive; since their node degrees are not bounded, the group action defined by their automorphisms has infinitely many orbits. They are of particular interest in the context of communication networks, and are treated extensively in the books by Franceschetti and Meester [5], Meester and Roy [10], and Penrose [14].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Statement and Discussion Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strict Inequalities of Critical Values in Continuum Percolation

2011

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“…They, however, have the disadvantage that through the use of a deterministic connection algorithm, many structural characteristics, observed in real-world networks, cannot be modelled [13]. Methods used in the field of random graphs have the advantage that vertices can be connected with a certain probability, which depends both on the types of vertices and the distance between them [14]. They, however, have the disadvantage that assumptions of complete independence among possible edges are largely untenable in practice [15].…”

Section: Edge Creationmentioning

confidence: 99%

Generation of Spatially Embedded Random Networks to Model Complex Transportation Networks

Hackl

Adey

2016

14th International Probabilistic Workshop

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Random networks are increasingly used to analyse complex transportation networks, such as airline routes, roads and rail networks. So far, this research has been focused on describing the properties of the networks with the help of random networks, often without considering their spatial properties. In this article, a methodology is proposed to create random networks conserving their spatial properties. The produced random networks are not intended to be an accurate model of the real-world network being investigated, but are to be used to gain insight into the functioning of the network taking into consideration its spatial properties, which has potential to be useful in many types of analysis, e.g. estimating the network related risk. The proposed methodology combines a spatial non-homogeneous point process for vertex creation, which accounts for the spatial distribution of vertices, considering clustering effects of the network and a hybrid connection model for the edge creation. To illustrate the ability of the proposed methodology to be used to gain insight into a real world network, it is used to estimate standard structural statistics for part of the Swiss road network, and these are then compared with the known values.

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“…In this case, indeed, there is no standard notion of network topology we can rely on for building the connectivity matrix. This task gets even more challenging if we focus on networks operated in the subconnectivity regime [13], in which no giant component exists.…”

Section: Epidemic Spreading In Highly Partitioned Mobile Networkmentioning

confidence: 99%

Eigenvector Centrality in Highly Partitioned Mobile Networks: Principles and Applications

Carreras¹,

Miorandi²,

Canright

et al. 2007

Advances in Biologically Inspired Information Systems

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Summary. In this chapter we introduce a model for analyzing the spread of epidemics in a disconnected mobile network. The work is based on an extension, to a dynamic setting, of the eigenvector centrality principle introduced by two of the authors for the case of static networks. The extension builds on a new definition of connectivity matrix for a highly partitioned mobile system, where the connectivity between a pair of nodes is defined as the number of contacts taking place over a finite time window. The connectivity matrix is then used to evaluate the eigenvector centrality of the various nodes. Numerical results from real-world traces are presented and discussed. The applicability of the proposed approach to select on-line message forwarders is also addressed.

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Continuum Percolation

Cited by 841 publications

References 0 publications

Strict Inequalities of Critical Values in Continuum Percolation

Strict Inequalities of Critical Values in Continuum Percolation

Generation of Spatially Embedded Random Networks to Model Complex Transportation Networks

Eigenvector Centrality in Highly Partitioned Mobile Networks: Principles and Applications

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