Bak-Sneppen-type models and rank-driven processes

The Bak-Sneppen model is a well-known stochastic model of evolution that exhibits self-organized criticality; only a few analytical results have been established for it so far. We report a surprising connection between Bak-Sneppen-type models and more tractable Markov processes that evolve without any reference to an underlying topology. Specifically, we show that in the case of a large number of species, the long time behavior of the fitness profile in the Bak-Sneppen model can be replicated by a model with a purely rank-based update rule whose asymptotics can be studied rigorously.

Figure 1

Plot of $f_N^{\mathrm{aBS}}\left(k\right)$, $k\in \{2,...,150\}$, for $N=250$, 1000, and 20 000.

Figure 2

Plot of $f_{250}^{\mathit{BS}}(k,l)$, $k,l,\in \{2,...,50\}$.

Figure 3

Plot of $g_N^{\mathit{BS}}\left(k\right)$, $k\in \{2,...,150\}$, for $N=250$, 1000, and 20 000.

Figure 4

Size distribution $n\left(l\right)$ of $s$ avalanches in RDP (dashed), aBS model (left) and BS model (right) for $N=2000$.