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The Convergence-Guaranteed Random Walk and Its Applications in Peer-to-Peer Networks
May 2008 (vol. 57 no. 5)
pp. 619-633
Network structure construction and global state maintenance are expensive in large-scale, dynamic peer-to-peer (p2p) networks. With inherent topology independence and low state maintenance overhead, random walks have been widely used in such network environments. However, the current uses are limited to unguided or heuristic random walks with no guarantee on their converged node visitation probability distribution. Such a convergence guarantee is essential for strong analytical properties and high performance of many p2p applications. In this paper, we investigate an approach for random walks to converge to application-desired node visitation probability distributions while only requiring information about direct neighbors of each peer. Our approach is guided by the Metropolis-Hastings algorithm that is typically used in Monte Carlo Markov Chain sampling. We examine the effectiveness and practical issues of our approach using three application studies: random membership subset management, search, and load balancing. Both search and load balancing desire random walks with biased node visitation distributions to achieve application-specific analytical features. Our theoretical analysis, simulations, and Internet experiments demonstrate the advantage of our random walks compared with alternative topology-independent index-free approaches.

[1] L. Adamic, B. Huberman, R. Lukose, and A. Puniyani, “Search in Power Law Networks,” Physical Rev., vol. 64, pp. 46135-46143, 2001.
[2] M. Adler, E. Halperin, R.M. Karp, and V. Vazirani, “A Stochastic Process on the Hypercube with Applications to Peer-to-Peer Networks,” Proc. 35th ACM Symp. Theory of Computing, pp. 575-584, June 2003.
[3] Y. Azar, A. Broder, A. Karlin, N. Linial, and S. Phillips, “Biased Random Walks,” Proc. 24th ACM Symp. Theory of Computing, pp. 1-9, 1992.
[4] A. Barabási, Linked: How Everything Is Connected to Everything Else and What It Means. Plume, 2003.
[5] A. Barabási and R. Albert, “Emergence of Scaling in Random Networks,” Science, vol. 286, pp. 509-512, 1999.
[6] A.R. Bharambe, M. Agrawal, and S. Seshan, “Mercury: Supporting Scalable Multi-Attribute Range Queries,” Proc. ACM SIGCOMM '04, pp. 353-366, Aug. 2004.
[7] B. Bollobás, Random Graphs. Academic Press, 1985.
[8] B. Bollobás and O. Riordan, “The Diameter of a Scale-Free Random Graph,” Combinatorica, vol. 24, no. 1, pp. 5-34, 2004.
[9] B. Bollobás, O. Riordan, J. Spencer, and G. Tusnady, “The Degree Sequence of a Scale-Free Random Graph Process,” Random Structures and Algorithms, vol. 18, no. 3, pp. 279-290, 2001.
[10] Y. Chawathe, S. Ratnasamy, L. Breslau, N. Lanham, and S. Shenker, “Making Gnutella-Like P2P Systems Scalable,” Proc. ACM SIGCOMM '03, Aug. 2003.
[11] E. Cohen and S. Shenker, “Replication Strategies in Unstructured Peer-to-Peer Networks,” Proc. ACM SIGCOMM '02, Aug. 2002.
[12] B.F. Cooper, “Quickly Routing Searches without Having to Move Content,” Proc. Fourth Int'l Workshop Peer-to-Peer Systems, Feb. 2005.
[13] P. Diaconis and D. Stroock, “Geometric Bounds for Eigenvalues of Markov Chains,” Annals of Applied Probability, vol. 1, pp. 36-61, 1991.
[14] W. Doeblin, “Exposé de la Théorie des Chaînes Simples Constantes de Markov á un Nombre Fini d'États,” Mathématique de l'Union Interbalkanique, vol. 2, pp. 77-105, 1938.
[15] P.T. Eugster, R. Guerraoui, S.B. Handurukande, P. Kouznetsov, and A.-M. Kermarrec, “Lightweight Probabilistic Broadcast,” ACM Trans. Computer Systems, vol. 21, no. 4, pp. 341-374, Nov. 2003.
[16] A.J. Ganesh, A. Kermarrec, and L. Massoulié, “SCAMP: Peer-to-Peer Lightweight Membership Service for Large-Scale Group Communication,” Proc. Third Int'l Workshop Networked Group Comm., pp. 44-55, Nov. 2001.
[17] C. Gkantsidis, M. Mihail, and A. Saberi, “Conductance and Congestion in Power Law Graphs,” Proc. ACM SIGMETRICS '03, pp. 148-159, June 2003.
[18] C. Gkantsidis, M. Mihail, and A. Saberi, “Random Walks in Peer-to-Peer Networks,” Proc. IEEE INFOCOM '04, pp. 120-130, Mar. 2004.
[19] C. Gkantsidis, M. Mihail, and A. Saberi, “Hybrid Search Schemes for Unstructured Peer-to-Peer Networks,” Proc. IEEE INFOCOM '05, pp. 1526-1537, Mar. 2005.
[20] C. Gkantsidis, M. Mihail, and E. Zegura, “The Markov Chain Simulation Method for Generating Connected Power Law Random Graphs,” Proc. Fifth Workshop Algorithm Eng. and Experiments, 2003.
[21] W.K. Hastings, “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, vol. 57, pp. 97-109, 1970.
[22] M. Jelasity, R. Guerraoui, A. Kermarrec, and M.V. Steen, “The Peer Sampling Service: Experimental Evaluation of Unstructured Gossip-Based Implementations,” Proc. Fifth ACM/IFIP/Usenix Int'l Middleware Conf., 2004.
[23] D. Karger and M. Ruhl, “Simple Efficient Load Balancing Algorithms for Peer-to-Peer Systems,” Proc. 16th ACM Symp. Parallelism in Algorithms and Architectures, pp. 36-43, June 2004.
[24] V. King and J. Saia, “Choosing a Random Peer,” Proc. 23rd ACM Symp. Principles of Distributed Computing, pp. 125-130, 2004.
[25] D. Kostić, A. Rodriguez, J. Albrecht, A. Bhirud, and A. Vahdat, “Using Random Subsets to Build Scalable Network Services,” Proc. Fourth Usenix Symp. Internet Technologies and Systems, Mar. 2003.
[26] C. Law and K. Siu, “Distributed Construction of Random Expander Networks,” Proc. IEEE INFOCOM '03, pp. 2133-2143, Mar. 2003.
[27] D. Loguinov, A. Kumar, V. Rai, and S. Ganesh, “Graph-Theoretic Analysis of Structured Peer-to-Peer Systems: Routing Distances and Fault Resilience,” Proc. ACM SIGCOMM '03, pp. 395-406, Aug. 2003.
[28] Q. Lv, P. Cao, E. Cohen, K. Li, and S. Shenker, “Search and Replication in Unstructured Peer-to-Peer Networks,” Proc. ACM Int'l Conf. Supercomputing, pp. 84-95, June 2002.
[29] Q. Lv, S. Ratnasamy, and S. Shenker, “Can Heterogeneity Make Gnutella Scalable?” Proc. First Int'l Workshop Peer-to-Peer Systems, 2002.
[30] G.S. Manku, “Balanced Binary Trees for ID Management and Load Balance in Distributed Hash Tables,” Proc. 23rd ACM Symp. Principles of Distributed Computing, pp. 197-205, July 2004.
[31] G.S. Manku, M. Naor, and U. Wieder, “Know Thy Neighbor's Neighbor: The Power of Lookahead in Randomized P2P Networks,” Proc. 36th ACM Symp. Theory of Computing, 2004.
[32] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chemical Physics, vol. 21, pp. 1087-1092, 1953.
[33] Planetlab, http:/, 2008.
[34] A. Rao, K. Lakshminarayanan, S. Surana, R. Karp, and I. Stoica, “Load Balancing in Structured P2P Systems,” Proc. Second Int'l Workshop Peer-to-Peer Systems, Feb. 2003.
[35] S. Ratnasamy, P. Francis, M. Handley, R. Karp, and S. Shenker, “A Scalable Content-Addressable Network,” Proc. ACM SIGCOMM '01, pp. 161-172, Aug. 2001.
[36] S. Rhea and J. Kubiatowicz, “Probabilistic Location and Routing,” Proc. IEEE INFOCOM '02, pp. 1248-1257, 2002.
[37] A. Rowstron and P. Druschel, “Pastry: Scalable, Distributed Object Location and Routing for Large-Scale Peer-to-Peer Systems,” Proc. IFIP/ACM Int'l Conf. Distributed Systems Platforms, pp. 329-350, Nov. 2001.
[38] K. Shen, “Structure Management for Scalable Overlay Service Construction,” Proc. First Usenix/ACM Symp. Networked Systems Design and Implementation, pp. 281-294, Mar. 2004.
[39] A. Sinclair, “Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow,” Combinatorics, Probability and Computing, vol. 1, pp. 351-370, 1992.
[40] A. Sinclair and M. Jerrum, “Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains,” Information and Computation, vol. 82, pp. 93-133, 1989.
[41] K. Sripanidkulchai, “The Popularity of Gnutella Queries and Its Implications on Scalability,” Proc. O'Reilly Peer-to-Peer and Web Services Conf., 2001.
[42] A.O. Stauffer and V.C. Barbosa, “Probabilistic Heuristics for Disseminating Information in Networks,” IEEE/ACM Trans. Networking, vol. 15, no. 2, pp. 425-435, Apr. 2007.
[43] I. Stoica, R. Morris, D. Karger, M.F. Kaashoek, and H. Balakrishnan, “Chord: A Scalable Peer-to-Peer Lookup Service for Internet Applications,” Proc. ACM SIGCOMM '01, pp. 149-160, Aug. 2001.
[44] D. Tsoumakos and N. Roussopoulos, “Adaptive Probabilistic Search for Peer-to-Peer Networks,” Proc. Third IEEE Int'l Conf. P2P Computing, 2003.
[45] X. Wang, Y. Zhang, X. Li, and D. Loguinov, “On Zone-Balancing of Peer-to-Peer Networks: Analysis of Random Node Join,” Proc. ACM SIGMETRICS '04, June 2004.
[46] B. Yang and H. Garcia-Molina, “Improving Search in Peer-to-Peer Networks,” Proc. 22nd Int'l Conf. Distributed Computing Systems, pp. 5-14, July 2002.
[47] M. Zhong, K. Shen, and J. Seiferas, “Dynamic Load Balancing in Unstructured Peer-to-Peer Networks: Finding Hotspots, Eliminating Them,” unpublished manuscript, http://www.cs.rochester. edu/u/zhong/papers hotspots.pdf, May 2007.

Index Terms:
Distributed networks, Distributed applications, Distributed systems
Ming Zhong, Kai Shen, Joel Seiferas, "The Convergence-Guaranteed Random Walk and Its Applications in Peer-to-Peer Networks," IEEE Transactions on Computers, vol. 57, no. 5, pp. 619-633, May 2008, doi:10.1109/TC.2007.70837
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