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Stretch and Diameter in Random Geometric Graphs

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Abstract

Consider the random geometric graph \(G = G(n,r_n,f)\) consisting of n nodes independently distributed in \(S = \left[ -\frac{1}{2},\frac{1}{2}\right] ^2\) each according to a density \(f({\cdot })\). Two nodes u and v are joined by an edge if the Euclidean distance between them is less than \(r_n.\) Let \(C_G\) denote the component of G containing the largest number of nodes and denote \(\text {diam}(C_G)\) to be its diameter. Let s and t be the nodes of G closest to \(\left( -\frac{1}{2},\frac{1}{2}\right) \) and \(\left( \frac{1}{2},\frac{1}{2}\right) ,\) respectively and let \(d_G(s,t)\) denote their graph distance. We prove that the normalized diameter \(\frac{r_n}{\sqrt{2}} \text {diam}(C_G)\) and the stretch \(r_nd_G(s,t)\) both converge to one in probability if \(nr_n^2 \rightarrow \infty \) as \(n \rightarrow \infty \).

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References

  1. Bollobas, B., Riordan, O.: Percolation. Academic, Berlin (2006)

    Book  MATH  Google Scholar 

  2. Bradonjic, M., Elsasser, R., Friedrich, T., Sauerwald, T., Stauffer, A.: Efficient broadcast on random geometric graphs. In: SODA 2010, pp. 1412–1421 (1984)

  3. Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ellis, R.B., Martin, J., Yan, C.: Random geometric graph diameter in the unit ball. Algorithmica 47, 421–438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Franceschetti, M., Dousse, O., Tse, D.N.C., Thiran, P.: Closing gap in the capacity of wireless networks via percolation theory. IEEE Trans. Inform. Theory 53, 1009–1018 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Friedrich, T., Sauerwald, T., Stauffer, A.: Diameter and broadcast time of random geometric graphs in arbitrary dimensions. In: ISAAC 2011, pp. 190–199 (2011)

  7. Ganesan, G.: Size of the giant component in a random geometric graph. Ann. Inst. Henri Poincare 49, 1130–1140 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ganesan, G.: First passage percolation with nonidentical passage times (2014). http://arxiv.org/abs/1409.2602

  9. Ganesan, G.: Outermost boundaries for star and plus connected components in percolation (2015). http://arxiv.org/abs/1508.06443

  10. Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity in wireless networks. In: McEneany, W.M., Yin, G., Zhang, Q. (eds.) Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming, pp.547–566. Birkhäuser, Boston, MA (1998)

  11. Muthukrishnan, S., Pandurangan, G.: The bin-covering technique for thresholding random geometric graph properties. In: Proc. SODA 2005, pp. 989–998 (2005)

  12. Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  13. Sarkar, A.: Some problems in continuum percolation. Ph.D. Thesis, ISI Delhi (1996)

Download references

Acknowledgements

I thank Professors Rahul Roy, Thomas Mountford, Federico Camia and the referees for crucial comments that led to an improvement of the paper. I also thank Professors Rahul Roy, Thomas Mountford, Federico Camia and the National Board for Higher Mathematics for my fellowships.

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Correspondence to Ghurumuruhan Ganesan.

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Ganesan, G. Stretch and Diameter in Random Geometric Graphs. Algorithmica 80, 300–330 (2018). https://doi.org/10.1007/s00453-016-0253-5

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