scholarly journals Counting $k$ -Hop Paths in the Random Connection Model

2018 ◽  
Vol 17 (5) ◽  
pp. 3201-3210 ◽  
Author(s):  
Alexander P. Kartun-Giles ◽  
Sunwoo Kim
Keyword(s):  
Random Connection Model

Probability of k-Hop Connection under Random Connection Model

2010 ◽  
Vol 14 (11) ◽  
pp. 1023-1025 ◽  
Author(s):  
Guoqiang Mao ◽  
Zijie Zhang ◽  
Brian D.O. Anderson
Keyword(s):  
Random Connection Model

The random connection model in high dimensions

1997 ◽  
Vol 35 (2) ◽  
pp. 145-153 ◽  
Author(s):  
Ronald Meester ◽  
Mathew D. Penrose ◽  
Anish Sarkar
Keyword(s):  
High Dimensions ◽  
Random Connection Model

scholarly journals The Random Connection Model on the Torus

2014 ◽  
Vol 23 (5) ◽  
pp. 796-804
Author(s):  
LUC DEVROYE ◽  
NICOLAS FRAIMAN
Keyword(s):  
Wireless Networks ◽  
Random Graphs ◽  
High Probability ◽  
Random Connection Model

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two pointsxandyare connected with probabilityg(y−x), wheregis a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

scholarly journals On the Ornstein–Zernike equation for stationary cluster processes and the random connection model

2017 ◽  
Vol 49 (4) ◽  
pp. 1260-1287 ◽  
Author(s):  
Günter Last ◽  
Sebastian Ziesche
Keyword(s):  
Point Process ◽  
Cluster Model ◽  
Combinatorial Properties ◽  
Palm Calculus ◽  
Fourier Theory ◽  
Associated Pair ◽  
Special Case ◽  
Random Connection Model ◽  
Palm Probabilities ◽  
Cluster Processes

Abstract In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

scholarly journals Navigation in small-world networks: a scale-free continuum model

2006 ◽  
Vol 43 (04) ◽  
pp. 1173-1180 ◽  
Author(s):  
Massimo Franceschetti ◽  
Ronald Meester
Keyword(s):  
Social Sciences ◽  
Scale Invariance ◽  
Small World ◽  
Local Information ◽  
Engineering Systems ◽  
Small World Networks ◽  
Social Contacts ◽  
Scale Free ◽  
Free Model ◽  
Random Connection Model

The small-world phenomenon, the principle that we are all linked by a short chain of intermediate acquaintances, has been investigated in mathematics and social sciences. It has been shown to be pervasive both in nature and in engineering systems like the World Wide Web. Work of Jon Kleinberg has shown that people, using only local information, are very effective at finding short paths in a network of social contacts. In this paper we argue that the underlying key to finding short paths is scale invariance. In order to appreciate scale invariance we suggest a continuum setting, since true scale invariance happens at all scales, something which cannot be observed in a discrete model. We introduce a random-connection model that is related to continuum percolation, and we prove the existence of a unique scale-free model, among a large class of models, that allows the construction, with high probability, of short paths between pairs of points separated by any distance.

scholarly journals Cluster Expansions for GIBBS Point Processes

2019 ◽  
Vol 51 (4) ◽  
pp. 1129-1178 ◽  
Author(s):  
S. Jansen
Keyword(s):  
Correlation Functions ◽  
Point Processes ◽  
Branching Processes ◽  
Factorial Moment ◽  
Convergence Condition ◽  
Random Measures ◽  
Cluster Expansions ◽  
Gibbs Point Processes ◽  
Random Connection Model ◽  
Boolean Percolation

AbstractWe provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood–Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández, and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

scholarly journals Moments of k-hop counts in the random-connection model

2019 ◽  
Vol 56 (4) ◽  
pp. 1106-1121
Author(s):  
Nicolas Privault
Keyword(s):  
Point Process ◽  
Stochastic Integrals ◽  
Model Based ◽  
Random Connection Model

AbstractWe derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. The identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams if the multiparameter processes vanish on diagonals. As an application, we obtain general identities for the moments of k-hop counts in the random-connection model, which simplify the derivations available in the literature.

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