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Large Connectivity for Dynamic Random Geometric Graphs
June 2009 (vol. 8 no. 6)
pp. 821-835
Josep Díaz, UPC, Barcelona
Dieter Mitsche, Institut für Theoretische Informatik, ETH, Zürich
Xavier Pérez-Giménez, UPC, Barcelona
We provide the first rigorous analytical results for the connectivity of dynamic random geometric graphs—a model for mobile wireless networks in which vertices move in random directions in the unit torus. The model presented here follows the one described in [11]. We provide precise asymptotic results for the expected length of the connectivity and disconnectivity periods of the network. We believe that the formal tools developed in this work could be extended to be used in more concrete settings and in more realistic models, in the same manner as the development of the connectivity threshold for static random geometric graphs has affected a lot of research done on ad hoc networks.

[1] M. Appel and R.P. Russo, “The Connectivity of a Graph on Uniform Points on $[0,1]^d$ ,” Statistics & Probability Letters, vol. 60, no. 4, pp.351-357, 2002.
[2] C. Avin, M. Koucky, and Z. Lotker, “How to Explore a Fast-Changing World (On the Cover Time of Dynamic Graphs),” Proc. 35th Int'l Colloquium Automata, Languages and Programming, 2008.
[3] B. Bollobás, Random Graphs, second ed. Cambridge Univ. Press, 2001.
[4] A. Boukerche and L. Bononi, “Simulation and Modeling of Wireless Mobile and Ad-Hoc Networks,” Mobile Ad-Hoc Networking, S. Basagni, M. Conti, S. Giordano, and I. Stojmenovic, eds., pp.373-410, Wiley Interscience, 2004.
[5] T. Camp, J. Boleng, and V. Davies, “Mobility Models for Ad-Hoc Network Research,” Mobile Ad-Hoc Networking: Research, Trends and Applications, vol. 2, no. 5, pp.483-502, 2002.
[6] J. Díaz, X. Pérez, M. Serna, and N. Wormald, “Walkers on the Cycle and the Grid,” SIAM J. Discrete Math., vol. 22, no. 2, pp.747-775, 2008.
[7] J. Díaz, D. Mitsche, and X. Pérez-Giménez, “On the Probability of the Existence of Fixed-Size Components in Random Geometric Graphs,” Annals of Applied Probability, to appear.
[8] J. Díaz, J. Petit, and M. Serna, “A Guide to Concentration Bounds,” Handbook of Randomized Computing, S. Rajasekaran, J. Reif, and J.Rolim, eds., vol.2, chapter12, pp.457-507, Kluwer, 2001.
[9] A. Goel, S. Rai, S. Suri, and B. Krishnamachari, “Monotone Properties of Random Geometric Graphs Have Sharp Thresholds,” Annals of Applied Probability, vol. 15, no. 4, pp.2535-2552, 2005.
[10] L. Guibas, J. Hershberger, S. Suri, and Li Zhang, “Kinetic Connectivity for Unit Disks,” Discrete and Computational Geometry, vol. 25, pp.591-610, 2001.
[11] R.A. Guerin, “Channel Occupancy Time Distribution in a Cellular Radio System,” IEEE Trans. Vehicular Technology, vol. 36, no. 3, pp.89-99, 1987.
[12] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless Networks,” Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W. McEneaney, G.G. Yin, and Q. Zhang, eds., pp.547-566, Birkhäuser, 1999.
[13] Y. Han, R.J. La, A.M. Makowski, and S. Lee, “Distribution of Path Durations in Mobile Ad-Hoc Networks—Palm's Theorem to the Rescue,” Computer Networks J., special issue on network modeling and simulation, vol.50, no. 12, pp.1887-1900, 2006.
[14] R. Hekmat, Ad-Hoc Networks: Fundamental Properties and Network Topologies. Springer, 2006.
[15] A. Jardosh, E. Belding-Royer, K. Almeroth, and S. Suri, “Towards Realistic Mobility Models for Mobile Ad Hoc Networks,” Proc. ACM MobiCom, 2003.
[16] A. Nain, D. Towsley, B. Liu,k, and Z. Liu, “Properties of Random Direction Models,” Proc. ACM MobiHoc, 2005.
[17] M. Penrose, “The Longest Edge of the Random Minimal Spanning Tree,” Annals of Applied Probability, vol. 7, no. 2, pp.340-361, 1997.
[18] M. Penrose, “On K-Connectivity for a Geometric Random Graph,” Random Structures and Algorithms, vol. 15, pp.145-164, 1999.
[19] M. Penrose, Random Geometric Graphs, Oxford Studies in Probability. Oxford Univ. Press, 2003.
[20] P. Santi and D.M. Blough, “An Evaluation of Connectivity in Mobile Wireless Ad Hoc Networks,” Proc. Int'l Conf. Dependable Systems and Networks, pp.89-98, 2002.

Index Terms:
Mobile communication systems, dynamic random geometric graphs, connectivity period.
Josep Díaz, Dieter Mitsche, Xavier Pérez-Giménez, "Large Connectivity for Dynamic Random Geometric Graphs," IEEE Transactions on Mobile Computing, vol. 8, no. 6, pp. 821-835, June 2009, doi:10.1109/TMC.2009.42
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