The DOI matches its Crossref record; 11 of the 11 references it cites were found in a registry or an index.
On Spreading a Rumor
Abstract
Suppose that one of n people knows a rumor. At the first stage, he passes the rumor to someone chosen at random; at each stage, each person already informed (“knower”) communicates the rumor to a person chosen at random and independently of all other past and present choices. Denote by $S_n $ the random number of stages before everybody is informed. How large is $S_n $ typically? Frieze and Grimmet, who introduced this problem, proved that, in probability, $S_n /( \log _2 n + \log n ) \to 1$. In this paper we show that, in fact, $S_n = \log _2 n + \log n + O( 1 )$ in probability. Our proof demonstrates that the number $I( t )$ of persons informed after t stages obeys very closely, with high probability, a deterministic equation $I( {t + 1} ) = n - ( {n - I( t )} )\exp ( - I( t )/n )$, $t\geqq 0$. A case when each knower passes the rumor to several members at every stage is also discussed.
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Cites
An introduction to probability theory and its applications
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Erdős, P., & Rényi, A. (2022) · Publicationes Mathematicae Debrecen · doi:10.5486/pmd.1959.6.3-4.12 · Cited by 5,337
An Introduction to Probability Theory and Its Applications, Vol. 2
Feller, W., & Teichmann, T. (1967) · Physics Today · doi:10.1063/1.3034322 · Cited by 1,280
DALEY, D. J., & KENDALL, D. G. (1965) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/1.1.42 · Cited by 556
Bingham, N. H., & Billingsley, P. (1980) · Journal of the Royal Statistical Society. Series A (General) · doi:10.2307/2982142 · Cited by 455
The Mathematical Theory of Infectious Diseases and Its Applications
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The shortest-path problem for graphs with random arc-lengths
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Stochastic Models for Social Processes.
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On a simple formula for random mappings and its applications
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Cited by
Gossip-based computation of aggregate information
Kempe, D., Dobra, A., & Gehrke, J. (2004) · 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings. · doi:10.1109/sfcs.2003.1238221 · Cited by 1,480
A survey of gossiping and broadcasting in communication networks
Hedetniemi, S. M., Hedetniemi, S. T., & Liestman, A. L. (1988) · Networks · doi:10.1002/net.3230180406 · Cited by 1,243
Jelasity, M., Voulgaris, S., Guerraoui, R., Kermarrec, A.-M., & van Steen, M. (2007) · ACM Transactions on Computer Systems · doi:10.1145/1275517.1275520 · Cited by 494
Metadata
- Authors
- Pittel, B.
- Venue
- SIAM Journal on Applied Mathematics
- Published
- 1987-02
- Volume
- 47
- Issue
- 1
- Pages
- 213-223
- ISSN
- 0036-1399 (ISSN-L)
- 1095-712X
- OpenAlex
- W1984977694
- Cited by
- 386
- References
- 11
- FWCI
- 1.85
- Type
- journal article
- Language
- en
- Open access
- closed
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