Chen Avin
ABSTRACT
In recent years, protocols that are based on the properties of random walks on graphs have found many applications in communication and information networks, such as wireless networks, peer-to-peer networks, and the Web. For wireless networks, graphs are actually not the correct model of the communication; instead, hyper-graphs better capture the communication over a wireless shared channel. Motivated by this example, we study in this paper random walks on hyper-graphs. First, we formalize the random walk process on hyper-graphs and generalize key notions from random walks on graphs. We then give the novel definition of radio cover time, namely, the expected time of a random walk to be heard (as opposed to visited) by all nodes. We then provide some basic bounds on the radio cover, in particular, we show that while on graphs the radio cover time is O(mn), in hyper-graphs it is O(mnr), where n, m, and r are the number of nodes, the number of edges, and the rank of the hyper-graph, respectively. We conclude the paper with results on specific hyper-graphs that model wireless mesh networks in one and two dimensions and show that in both cases the radio cover time can be significantly faster than the standard cover time. In the two-dimension case, the radio cover time becomes sub-linear for an average degree larger than log2 n.
- }}Aldous, D., and Fill, J. Reversible Markov Chains and Random Walks on Graphs. 1999. Unpublished. http://stat-www.berkeley.edu/users/aldous/RWG/book.html.Google Scholar
- }}Aldous, D. J. Lower bounds for covering times for reversible markov chains and random walks on graphs. J. Theoret. Probab. 2, 1 (1989), 91--100.Google ScholarCross Ref
- }}Aleliunas, R., Karp, R. M., Lipton, R. J., Lovász, L., and Rackoff, C. Random walks, universal traversal sequences, and the complexity of maze problems. In 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979). IEEE, New York, 1979, pp. 218--223. Google ScholarDigital Library
- }}Avin, C., and Brito, C. Efficient and robust query processing in dynamic environments using random walk techniques. In Proc. of the third international symposium on Information processing in sensor networks (2004), pp. 277--286. Google ScholarDigital Library
- }}Avin, C., and Ercal, G. On the cover time and mixing time of random geometric graphs. Theor. Comput. Sci. 380, 1--2 (2007), 2--22. Google ScholarDigital Library
- }}Bar-Yossef, Z., Friedman, R., and Kliot, G. Rawms-: random walk based lightweight membership service for wireless ad hoc network. In MobiHoc '06: Proceedings of the seventh ACM international symposium on Mobile ad hoc networking and computing (New York, NY, USA, 2006), ACM Press, pp. 238--249. Google ScholarDigital Library
- }}Braginsky, D., and Estrin, D. Rumor routing algorthim for sensor networks. In Proc. of the 1st ACM Int. workshop on Wireless sensor networks and applications (2002), ACM Press, pp. 22--31. Google ScholarDigital Library
- }}Broder, A. Generating random spanning trees. Foundations of Computer Science, 1989., 30th Annual Symposium on (1989), 442--447. Google ScholarDigital Library
- }}Broder, A., and Karlin, A. Bounds on the cover time. J. Theoret. Probab. 2 (1989), 101--120.Google ScholarCross Ref
- }}Broder, A. Z., Karlin, A. R., Raghavan, P., and Upfal, E. Trading space for time in undirected s-t connectivity. In STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing (New York, NY, USA, 1989), ACM Press, pp. 543--549. Google ScholarDigital Library
- }}Chandra, A. K., Raghavan, P., Ruzzo, W. L., and Smolensky, R. The electrical resistance of a graph captures its commute and cover times. In Proc. of the 21st annual ACM symposium on Theory of computing (1989), pp. 574--586. Google ScholarDigital Library
- }}Deb, S., Médard, M., and Choute, C. Algebraic gossip: a network coding approach to optimal multiple rumor mongering. IEEE/ACM Trans. Netw. 14, SI (2006), 2486--2507. Google ScholarDigital Library
- }}Doyle, P. G., and Snell, J. L. Random Walks and Electric Networks, vol. 22. The Mathematical Association of America, 1984.Google Scholar
- }}Friedman, R., Kliot, G., and Avin, C. Probabilistic quorum systems in wireless ad hoc networks. In DCN-08. IEEE International Conference on Dependable Systems and Networks (June 2008), pp. 277--286.Google Scholar
- }}Gkantsidis, C., Mihail, M., and Saberi, A. Random walks in peer-to-peer networks. In in Proc. 23 Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM). to appear (2004).Google Scholar
- }}Kempe, D., Dobra, A., and Gehrke, J. Gossip-based computation of aggregate information. In Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science (2003), pp. 482--491. Google ScholarDigital Library
- }}Lun, D., Ho, T., Ratnakar, N., Médard, M., and Koetter, R. Network coding in wireless networks. Cooperation in Wireless Networks: Principles and Applications (2006), 127--161.Google Scholar
- }}Matthews, P. Covering problems for Brownian motion on spheres. Ann. Probab. 16, 1 (1988), 189--199.Google ScholarCross Ref
- }}Sadagopan, N., Krishnamachari, B., and Helmy, A. Active query forwarding in sensor networks (acquire). Journal of Ad Hoc Networks 3, 1 (January 2005), 91--113.Google ScholarCross Ref
- }}Servetto, S. D., and Barrenechea, G. Constrained random walks on random graphs: Routing algorithms for large scale wireless sensor networks. In Proc. of the first ACM Int. workshop on Wireless sensor networks and applications (2002), ACM Press, pp. 12--21. Google ScholarDigital Library
- }}Stirzaker, D. Stochastic Processes and Models. Oxford University Press, 2005.Google Scholar
- }}Synge, J. L. The fundamental theorem of electrical networks. Quarterly of Applied Math., 9 (1951), 113--127.Google ScholarCross Ref
- }}Zhou, D., Huang, J., and Schölkopf, B. Learning with hypergraphs: Clustering, classification, and embedding. In Advances in Neural Information Processing Systems 19, B. Schölkopf, J. Platt, and T. Hoffman, Eds. MIT Press, Cambridge, MA, 2007, pp. 1601--1608.Google Scholar
- }}Zuckerman, D. A technique for lower bounding the cover time. In Proc. of the twenty-second annual ACM symposium on Theory of computing (1990), ACM Press, pp. 254--259. Google ScholarDigital Library
Index Terms
- Radio cover time in hyper-graphs
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