Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-28T02:52:57.559Z Has data issue: false hasContentIssue false
  • English
  • Français

Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

Part of: Equilibrium statistical mechanics Graph theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

C. Hirsch ,
D. Neuhäuser  and
V. Schmidt
C. Hirsch*
Affiliation:
Ulm University
D. Neuhäuser*
Affiliation:
Ulm University
V. Schmidt*
Affiliation:
Ulm University
*
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝd is an open problem for dimension d>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). For n=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of Gn(X) holds for all n≥2, all dimensions d≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.

Keywords

Minimal spanning forestdescending chainβ-skeletonpoint processcontinuum percolation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C80: Random graphs 05C10: Planar graphs; geometric and topological aspects of graph theory 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 20 - 36
Copyright
© Applied Probability Trust 

References

Aldous, D. J. (2009). Which connected spatial networks on random points have linear route-lengths? Preprint. Available at http://arxiv.org/abs/0911.5296v1.Google Scholar
Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Prob. 12, 1454–1508.Google Scholar
Aldous, D. J. and Shun, J. (2010). Connected spatial networks over random points and a route-length statistic. Statist. Sci. 25, 275–288.Google Scholar
Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Prob. Theory Relat. Fields 92, 247–258.CrossRefGoogle Scholar
Aldous, D. and Steele, J. M. (2004). The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (Encyclopedia Math. Sci. 110), ed. Kesten, H., Springer, Berlin.Google Scholar
Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Prob. 23, 87–104.Google Scholar
Błaszczyszyn, B. and Yogeshwaran, D. (2014). On comparison of clustering properties of point processes. To appear in Adv. Appl. Prob. CrossRefGoogle Scholar
Daley, D. J. and Last, G. (2005). Descending chains, the lilypond model, and mutual-nearest-neighbour matching. Adv. Appl. Prob. 37, 604–628.Google Scholar
Gaiselmann, G. et al. (2013). Stochastic 3D modeling of La0.6Sr0.4CoO3-δ cathodes based on structural segmentation of FIB-SEM images. Computational Materials Sci. 67, 48–62.CrossRefGoogle Scholar
Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Commun. Prob. 8, 17–27.Google Scholar
Last, G. (2006). Stationary partitions and Palm probabilities. Adv. Appl. Prob. 38, 602–620.Google Scholar
Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47, 655–693.CrossRefGoogle Scholar
Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95.Google Scholar
Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Prob. 34, 1665–1692.Google Scholar
Neuhäuser, D., Hirsch, C., Gloaguen, C. and Schmidt, V. (2012). On the distribution of typical shortest-path lengths in connected random geometric graphs. Queueing Systems 71, 199–220.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar
Timár, Á. (2006). Ends in free minimal spanning forests. Ann. Prob. 34, 865–869.Google Scholar
Toussaint, G. T. (1980). The relative neighbourhood graph of a finite planar set. Pattern Recognition 12, 261–268.Google Scholar