The Critical Neighbourhood Range for Asymptotic Overlay Connectivity in Dense Ad Hoc Networks
We define, for an overlay built on top of an ad hoc network, a simple criterion for neighbourhood: two overlay nodes are neighbours if and only if there exists a path between them of at most R hops, and R is called the (overlay) neighbourhood range. A small R may result in a disconnected overlay, while an unnecessarily large R would generate extra control traffic. We are interested in the minimum R ensuring overlay connectivity, the so-called critical R.
We derive a necessary and sufficient condition on R to achieve asymptotic connectivity of the overlay almost surely, i.e. connectivity with probability 1 when the number of overlay nodes tends to infinity, under the hypothesis that the underlying ad hoc network is itself asymptotically almost surely connected.
This condition, though asymptotic, sheds some light on the relation linking the critical R to the number of nodes n, the normalized radio transmission range r and the overlay density D (i.e., the proportion of overlay nodes). This condition can be considered as an approximation when the number of nodes is large enough. Since r is considered as a function of n, we are able to study the impact of topology control mechanisms, by showing how the shape of this function impacts the critical R.
Keywordsad hoc networks overlay connectivity topology control
- Bollobas, B. (1985). Random Graphs. Academic Press, London, England.Google Scholar
- Calomme, S. and Leduc, G. (2004). Performance study of an overlay approach to active routing in ad hoc networks. In Proc. of the Third annual Mediterranean Ad Hoc Networking Workshop (Med-Hoc-Net’04), Bodrum, Turkey.Google Scholar
- Calomme, S. and Leduc, G. (2006). The critical neighbourhood range for asymptotic overlay connectivity in ad hoc networks. To appear in Ad Hoc and Sensor Wireless Networks.Google Scholar
- Erdos, P. and Rényi, A. (1960). On the evolution of random graphs. Hungarian Academy of Science, 5:17–61.Google Scholar
- Goel, A., Rai, S., and Krishnamachari, B. (2004). Sharp thresholds for monotone properties in random geometric graphs. In Proc. of the thirty-sixth annual ACM symposium on Theory of computing (STOC’04), pages 580–586. Chicago, IL.Google Scholar
- Gupta, P. and Kumar, P. (1999). Critical power for asymptotic connectivity in wireless networks. In McEneaney, W.M., Yin, G., and Zhang, Q., editors, Stochastic Analysis, Control, Optimization and Applications, A Volume in Honor of W. H. Fleming. Birkhäuser, Boston.Google Scholar
- Krishnamachari, B., Wicker, S., and Bejar, R. (2001). Phase transition phenomena in wireless ad-hoc networks. In Proc. IEEE Global Conference on Telecommunications (Globecom’01), Symposium on Ad-Hoc Wireless Networks, San Antonio, TX.Google Scholar
- Mohapatra, P., Gui, C., and Li, J. (2004). Group communications in mobile ad hoc networks. IEEE Computer, 37(2):52–59.Google Scholar
- Narayanaswamy, S., Kawadia, V., Sreenivas, R.S., and Kumar, P. R. (2002). Power control in ad-hoc networks: Theory, architecture, algorithm and implementation of the compow protocol. In Proc. of European Wireless 2002. Next Generation Wireless Networks: Technologies, Protocols, Services and Applications, pages 156–162, Florence, Italy.Google Scholar
- Penrose, M. (2003). Random Geometric Graphs. Oxford University Press, England.Google Scholar
- Santi, P. and Blough, D. M. (2003). The critical transmitting range for connectivity in sparse wireless ad hoc networks. IEEE Trans. Mobile Comput., pages 25–39.Google Scholar
- Tang, D. and Baker, M. (2000). Analysis of a local-area wireless network. In Proc. of the Sixth Annual International Conference on Mobile Computing and Networking (MOBICOM’00), Boston, MA.Google Scholar
- Wang, K. H. and Li, B. (2002). Efficient and guaranteed service coverage in partitionable mobile ad-hoc networks. In Proc. Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM’02), New York, NY.Google Scholar