Small Worlds and Rapid Mixing with a Little More Randomness on Random Geometric Graphs

  • Gunes Ercal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6640)


We theoretically and experimentally analyze the process of adding sparse random links to a random wireless networks modeled as a random geometric graph. While this process has been previously proposed, we are the first to (i) theoretically consider sparse new-wiring (with asymptotically less than constant fraction of nodes allowed very small constant new wired edges), (ii) prove bounds for any new-wiring process upon random geometric graphs, and (iii) consider the effect of such sparse new-wiring upon the spectral gap of the resultant normalized Laplacian. In particular, we consider the following models of adding new wired edges: Divide the normalized space into bins of length \(k\frac{r}{2\sqrt 2} \times k\frac{r}{2\sqrt 2}\), given that the radius is on the order required to guarantee asymptotic connectivity. For each bin, choose a bin-leader. Let the G 1 new wiring be such that we form a random cubic graph amongst the bin-leaders and superimpose this upon the random geometric graph. Let the G 2 new wiring be such that we form a random 1-out graph amongst the bin-leaders and superimpose this upon the random geometric graph. We prove that the diameter for G 1 is O(k + log(n)) with high probability, and the diameter for G 2 is O(k log(n)) with high probability, both of which exponentially improve the \(\Theta(\sqrt \frac{n}{\log n})\) diameter of the random geometric graph, thus also inducing small-world characteristics as the high clustering remains unchanged. Further, we theoretically motivate and experimentally demonstrate that the spectral gap for both G 1 and G 2 are significantly greater than that of the original random geometric graph. These results further motivate future hybrid networks and advances in the use of directional antennas.


small world wireless networks spectral gap theory 


  1. 1.
    Milgram, S.: The small world problem. Psychology Today 2(1), 60–67 (1967)MathSciNetGoogle Scholar
  2. 2.
    Newman, M., Barabási, A.L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  3. 3.
    Bollobás, B.: Random Graphs. Academic Press, Orlando (1985)zbMATHGoogle Scholar
  4. 4.
    Watts, D., Strogatz, S.: Collective dynamics of small-world networks. Nature 393(6684), 440–442 (1998)CrossRefGoogle Scholar
  5. 5.
    Penrose, M.D.: Random Geometric Graphs. Oxford Studies in Probability, vol. 5. Oxford University Press, Oxford (2003)zbMATHCrossRefGoogle Scholar
  6. 6.
    Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity in wireless networks. In: Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming, pp. 547–566 (1998)Google Scholar
  7. 7.
    Helmy, A.: Small worlds in wireless networks. IEEE Communications Letters 7(10), 490–492 (2003)CrossRefGoogle Scholar
  8. 8.
    Cavalcanti, D., Agrawal, D., Kelner, J., Sadok, D.F.H.: Exploiting the small-world effect to increase connectivity in wireless ad hoc networks. In: de Souza, J.N., Dini, P., Lorenz, P. (eds.) ICT 2004. LNCS, vol. 3124, pp. 388–393. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Ye, X., Xu, L., Lin, L.: Small-world model based topology optimization in wireless sensor network. In: Proceedings of the 2008 International Symposium on Information Science and Engieering, vol. 01, pp. 102–106. IEEE Computer Society, Washington, DC (2008)CrossRefGoogle Scholar
  10. 10.
    Alon, N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Sinclair, A.: Improved bounds for mixing rates of markov chains and multicommodity flow. Combinatorics, Probability and Computing 1, 351–370 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Jerrum, M., Sinclair, A.: The markov chain monte carlo method: an approach to approximate counting and integration. In: Hochbaum, D. (ed.) Approximations for NP-hard Problems, pp. 482–520. PWS Publishing, Boston (1997)Google Scholar
  13. 13.
    Kempe, D., Dobra, A., Gehrke, J.: Gossip-based computation of aggregate information. In: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 482–491 (2003)Google Scholar
  14. 14.
    Servetto, S.D., Barrenechea, G.: Constrained random walks on random graphs: routing algorithms for large scale wireless sensor networks. In: Proc. of the 1st Int. Workshop on Wireless Sensor Networks and Applications, pp. 12–21 (2002)Google Scholar
  15. 15.
    Gkantsidis, C., Mihail, M., Saberi, A.: Random walks in peer-to-peer networks. In: Proc. 23 Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM (2004)Google Scholar
  16. 16.
    Friedman, J., Kahn, J., Szemerédi, E.: On the second eigenvalue of random regular graphs. In: Proceedings of the Twenty-first Annual ACM Symposium on Theory of Computing, STOC 1989, pp. 587–598. ACM, New York (1989)CrossRefGoogle Scholar
  17. 17.
    Füredi, Z., Komlós, J.: The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE/ACM Trans. Netw. 14(SI), 2508–2530 (2006)MathSciNetGoogle Scholar
  19. 19.
    Avin, C., Ercal, G.: Bounds on the mixing time and partial cover of ad-hoc and sensor networks. In: Proceedings of the 2nd European Workshop on Wireless Sensor Networks (EWSN 2005), pp. 1–12 (2005)Google Scholar
  20. 20.
    Avin, C., Ercal, G.: On the cover time and mixing time of random geometric graphs. Theor. Comput. Sci. 380(1-2), 2–22 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chung, F.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  22. 22.
    Flaxman, A.D.: Expansion and lack thereof in randomly perturbed graphs. Internet Mathematics 4(2), 131–147 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2011

Authors and Affiliations

  • Gunes Ercal
    • 1
  1. 1.University of KansasUSA

Personalised recommendations