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# Randomized Rumour Spreading: The Effect of the Network Topology

Published online by Cambridge University Press:  06 May 2014

## Abstract

We consider the popular and well-studied push model, which is used to spread information in a given network with n vertices. Initially, some vertex owns a rumour and passes it to one of its neighbours, which is chosen randomly. In each of the succeeding rounds, every vertex that knows the rumour informs a random neighbour. It has been shown on various network topologies that this algorithm succeeds in spreading the rumour within O(log n) rounds. However, many studies are quite coarse and involve huge constants that do not allow for a direct comparison between different network topologies. In this paper, we analyse the push model on several important families of graphs, and obtain tight runtime estimates. We first show that, for any almost-regular graph on n vertices with small spectral expansion, rumour spreading completes after log2n + log n+o(log n) rounds with high probability. This is the first result that exhibits a general graph class for which rumour spreading is essentially as fast as on complete graphs. Moreover, for the random graph G(n,p) with p=c log n/n, where c > 1, we determine the runtime of rumour spreading to be log2n + γ (c)log n with high probability, where γ(c) = clog(c/(c−1)). In particular, this shows that the assumption of almost regularity in our first result is necessary. Finally, for a hypercube on n=2d vertices, the runtime is with high probability at least (1+β) ⋅ (log2n + log n), where β > 0. This reveals that the push model on hypercubes is slower than on complete graphs, and thus shows that the assumption of small spectral expansion in our first result is also necessary. In addition, our results combined with the upper bound of O(log n) for the hypercube (see ) imply that the push model is faster on hypercubes than on a random graph G(n, clog n/n), where c is sufficiently close to 1.

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Combinatorics, Probability and Computing , March 2015 , pp. 457 - 479

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## References

Alon, N. and Chung, F. R. K. (1988) Explicit construction of linear sized tolerant networks. Discrete Math. 72 1519.CrossRefGoogle Scholar
Alon, N. and Spencer, J. (2008) The Probabilistic Method, third edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
Boyd, S., Ghosh, A., Prabhakar, B. and Shah, D. (2006) Randomized gossip algorithms. IEEE Trans. Inform. Theory 52 25082530.CrossRefGoogle Scholar
Broder, A. Z., Frieze, A. M., Suen, S. and Upfal, E. (1998) Optimal construction of edge-disjoint paths in random graphs. SIAM J. Comput. 28 541573.CrossRefGoogle Scholar
Chierichetti, F., Lattanzi, S. and Panconesi, A. (2010) Almost tight bounds for rumour spreading with conductance. In 42nd Annual ACM Symposium on Theory of Computing: STOC'10, pp. 399–408.Google Scholar
Cooper, C. and Frieze, A. M. (2007) The cover time of sparse random graphs. Random Struct. Alg. 30 116.CrossRefGoogle Scholar
Doerr, B., Friedrich, T. and Sauerwald, T. (2008) Quasirandom rumor spreading. In 19th Annual ACM-SIAM Symposium on Discrete Algorithms: SODA'08, pp. 773–781. arXiv.1012.5351Google Scholar
Dubhashi, D. and Panconesi, A. (2009) Concentration of Measure for the Analysis of Randomized Algorithms, Cambridge University Press.CrossRefGoogle Scholar
Elsässer, R. and Sauerwald, T. (2009) On the runtime and robustness of randomized broadcasting. Theoret. Comput. Sci. 410 34143427.CrossRefGoogle Scholar
Elsässer, R. and Sauerwald, T. (2009) Cover time and broadcast time. In 26th International Symposium on Theoretical Aspects of Computer Science: STACS'09, pp. 373–384.Google Scholar
Feige, U., Peleg, D., Raghavan, P. and Upfal, E. (1990) Randomized broadcast in networks. Random Struct. Alg. 1 447460.CrossRefGoogle Scholar
Fountoulakis, N. and Panagiotou, K. (2010) Rumor spreading on random regular graphs and expanders. In 14th International Workshop on Randomization and Computation: RANDOM'10, pp. 560–573.Google Scholar
Fountoulakis, N., Huber, A. and Panagiotou, K. (2010) Reliable broadcasting in random networks and the effect of density. In 29th IEEE Conference on Computer Communications: INFOCOM'10, pp. 2552–2560.Google Scholar
Friedrich, T., Gairing, M. and Sauerwald, T. (2012) Quasirandom load balancing. SIAM J. Comput. 41 747771.CrossRefGoogle Scholar
Frieze, A. and Grimmett, G. (1985) The shortest-path problem for graphs with random arc-lengths. Discrete Appl. Math. 10 5777.CrossRefGoogle Scholar
Füredi, Z. and Kómlos, J. (1981) The eigenvalues of random symmetric matrices. Combinatorica 3 233241.CrossRefGoogle Scholar
Giakkoupis, G. (2011) Tight upper bounds for rumor spreading in graphs of a given conductance. In 28th International Symposium on Theoretical Aspects of Computer Science: STACS'11, pp. 57–68.Google Scholar
Giakkoupis, G. and Sauerwald, T. (2012) Rumor spreading and vertex expansion. In 23rd Annual ACM-SIAM Symposium on Discrete Algorithms: SODA'12,, pp. 1623–1641.Google Scholar
Hoory, S., Linial, N. and Wigderson, A. (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. 43 439561.CrossRefGoogle Scholar
Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
Karp, R., Schindelhauer, C., Shenker, S. and Vöcking, B. (2000) Randomized rumor spreading. In 41st Annual IEEE Symposium on Foundations of Computer Science: FOCS'00, pp. 565–574.Google Scholar
Krivelevich, M. and Sudakov, B. (2006) Pseudo-random graphs. In More Sets, Graphs and Numbers, Vol. 15 of Bolyai Society Mathematical Studies, Springer, pp. 199262.CrossRefGoogle Scholar
McDiarmid, C. (1989) On the method of bounded differences. In Surveys in Combinatorics, Vol. 141 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 148188.Google Scholar
Mitzenmacher, M. and Upfal, E. (2005) Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press.CrossRefGoogle Scholar
Mosk-Aoyama, D. and Shah, D. (2008) Fast distributed algorithms for computing separable functions. IEEE Trans. Inform. Theory 54 29973007.CrossRefGoogle Scholar
Vu, V. H. (2007) Spectral norm of random matrices. Combinatorica 27 721736.CrossRefGoogle Scholar
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