A scaling result for explosive processes
The electronic journal of combinatorics, Tome 11 (2004) no. 1
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 $f$.
@article{10_37236_1784,
author = {M. Mitzenmacher and R. Oliveira and J. Spencer},
title = {A scaling result for explosive processes},
journal = {The electronic journal of combinatorics},
year = {2004},
volume = {11},
number = {1},
doi = {10.37236/1784},
zbl = {1054.60076},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1784/}
}
M. Mitzenmacher; R. Oliveira; J. Spencer. A scaling result for explosive processes. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1784
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