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The coverage holes of the largest component of random geometric graph

Abstract

In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage hole in the cube. In addition, we give an exponentially decaying tail bound for the probability that a line with length s do not intersect with the coverage of the infinite component of continuum percolation. These results have applications in communication networks and especially in wireless ad-hoc sensor networks.

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References

  1. [1]

    Balister, P., Bollobas, B., Sarkar, A., Walters, M. Sentry Selection in Wireless Networks. Adv. in Appl. Probab, 42: 1–25 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  2. [2]

    Balister, P., Bollobas, B., Sarkar, A. Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs. Handbook of Large-Scale Random Networks, Springer-Verlag, 2009

    Google Scholar 

  3. [3]

    Balister, P., Bollobas, B., Sarkar, A., Kumar, S. Reliable Density Estimates for Coverage and Connectivity in Thin Strips of Finite Length. ACM MobiCom, 2007

    Google Scholar 

  4. [4]

    Balister, P., Zheng, Z., Kumar, S., Sinha, P. Trap Coverage: Allowing Coverage Holes of Bounded Diameter in Wireless Sensor Networks. Proc. of IEEE INFOCOM, 2009

    Google Scholar 

  5. [5]

    Dousse, O. Asymptotic Properties of Wireless Multi-hop Networks. EPFL Ph.D. Thesis, No. 3310, 2005

  6. [6]

    Dousse, O., Tavoularis, C., Thiran, P. Delay of Intrusion Detection inWireless Sensor Networks. MobiHoc, 2006

    Google Scholar 

  7. [7]

    Franceschetti, M., Meester, R. Random Networks for Communication: from Statistical Physics to Information Systems, Cambridge University Press. New York, 2007

    Google Scholar 

  8. [8]

    Grimmett, G. Percolation, 2nd ed. Springer-Verlag, Berlin, 1999

    MATH  Book  Google Scholar 

  9. [9]

    Haenggi, M., Andrews, J.G., Baccelli, F., Dousse, O., Franceschetti, M. Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks. IEEE Journal on Selected Areas in Communications, 27(7), 2009

    Google Scholar 

  10. [10]

    Kumar, S., Lai, Th.., Arora, A. Barrier Coverage with Wireless Sensors. ACM MobiCom, 2005

    Google Scholar 

  11. [11]

    Kumar, S., Lai, Th.., Balogh, J. On k-Coverage in a Mostly Sleeping Sensor Network. ACM MobiCom, 2004

    Google Scholar 

  12. [12]

    Meester, R., Roy, R. Continuum Percolation. Cambridge University Press, New York, 1996

    MATH  Book  Google Scholar 

  13. [13]

    Penrose, M. Random Geometric Graphs. Oxford University Press, New York, 2003

    MATH  Book  Google Scholar 

  14. [14]

    Peres, Y., Sinclair, A., Sousi, P., Stauffer, A. Mobile Geometric Graphs: Detection, Coverage and Percolation. Proc. 22nd ACM-SIAM SODA, 412–428, 2011

    Google Scholar 

  15. [15]

    Sarkar, A. Co-existence of the Occupied and Vacant Phase in Boolean Models in Three or More Dimensions. Adv. in Appl. Probab, 29: 878–889 (1997)

    MATH  MathSciNet  Article  Google Scholar 

  16. [16]

    Yang, R., Yao, C.L., Guo, T.D. The Coverage of The Largest Component in Random Geometric Graphs with Applications in Sensor Networks (in Chinese). Acta Mathematicae Applicate Sinica, 32(6): 1027–1035 (2009)

    MATH  MathSciNet  Google Scholar 

  17. [17]

    Yao, C.L., Chen, G., Guo, T.D. Large Deviations for the Graph Distance in Supercritical Continuum Percolation. Journal of Applied Probability, 48(1): 154–172 (2011)

    MATH  MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Tian-de Guo.

Additional information

Supported by the National Natural Science Foundation of China (No. 71271204) and Knowledge Innovation Program of the Chinese Academy of Sciences (No. kjcx-yw-s7).

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Yao, Cl., Guo, Td. The coverage holes of the largest component of random geometric graph. Acta Math. Appl. Sin. Engl. Ser. 31, 855–862 (2015). https://doi.org/10.1007/s10255-015-0515-z

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Keywords

  • random geometric graph
  • continuum percolation
  • wireless sensor networks
  • coverage

2000 MR Subject Classification

  • 60K35
  • 82B43