### Abstract

We consider the asymptotic behavior of the following model: balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = n^{p} for p > 1. In this case, one of the two bins eventually obtains a monopoly, in the sense that it obtains all balls thrown past some point. This model is motivated by the phenomenon of positive feedback, where the "rich get richer." We derive a simple asymptotic expression for the probability that bin 1 obtains a monopoly when bin 1 starts with x balls and bin 2 starts with y balls for the case f(n) = n^{p}. We then demonstrate the effectiveness of this approximation with some examples and demonstrate how it generalizes to a wide class of functions f.

Original language | English (US) |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 1 R |

State | Published - Apr 13 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Electronic Journal of Combinatorics*,

*11*(1 R).

**A scaling result for explosive processes.** / Mitzenmacher, M.; Oliveira, R.; Spencer, J.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 11, no. 1 R.

}

TY - JOUR

T1 - A scaling result for explosive processes

AU - Mitzenmacher, M.

AU - Oliveira, R.

AU - Spencer, J.

PY - 2004/4/13

Y1 - 2004/4/13

N2 - We consider the asymptotic behavior of the following model: balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = np for p > 1. In this case, one of the two bins eventually obtains a monopoly, in the sense that it obtains all balls thrown past some point. This model is motivated by the phenomenon of positive feedback, where the "rich get richer." We derive a simple asymptotic expression for the probability that bin 1 obtains a monopoly when bin 1 starts with x balls and bin 2 starts with y balls for the case f(n) = np. We then demonstrate the effectiveness of this approximation with some examples and demonstrate how it generalizes to a wide class of functions f.

AB - We consider the asymptotic behavior of the following model: balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = np for p > 1. In this case, one of the two bins eventually obtains a monopoly, in the sense that it obtains all balls thrown past some point. This model is motivated by the phenomenon of positive feedback, where the "rich get richer." We derive a simple asymptotic expression for the probability that bin 1 obtains a monopoly when bin 1 starts with x balls and bin 2 starts with y balls for the case f(n) = np. We then demonstrate the effectiveness of this approximation with some examples and demonstrate how it generalizes to a wide class of functions f.

UR - http://www.scopus.com/inward/record.url?scp=3042524545&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042524545&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3042524545

VL - 11

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1 R

ER -