Bak-Sneppen model: Local equilibrium and critical value

The Bak-Sneppen (BS) model is a very simple model that exhibits all the richness of self-organized criticality theory. At the thermodynamic limit, the BS model converges to a situation where all particles have a fitness that is uniformly distributed between a critical value $p_c$ and 1. The $p_c$ value is unknown, as are the variables that influence and determine this value. Here we study the BS model in the case in which the lowest fitness particle interacts with an arbitrary even number of $m$ nearest neighbors. We show that $p_c$ verifies a simple local equilibrium relation. Based on this relation, we can determine bounds for $p_c$ of the BS model and exact results for some BS-like models. Finally, we show how transformations of the original BS model can be done without altering the model's complex dynamics.

Figure 1

(a) Fitness of the $N=3000$ particles evolved by the Bak-Sneppen model with $m=4$. (b) Zoom-in of the particles near the lowest fitness particle (2694) at time $t$. The number of neighbors of the lowest particle with fitness below $p_c$ is $S_t=2$. (c) The same particles shown in panel (b) at time $t+1$. In this case, the lowest particle is the number 2695 and $S_{t+1}=1$. The lowest fitness particle at time $t$ is shown with a blue circle, and the $m$ neighbors of the lowest site are shown with red cross circles.

Figure 2

Scheme of the three models studied: random neighbors (left), Bak-Sneppen (middle), and compact neighbors (right). Crosses represent active particles and points represent inactive ones.

Figure 3

Probability density of the fitness of neighbors of $\tilde{k}$ when $\tilde{k}<p_c\left(m\right)$ and when considering (a) a system of $N=m+1$ particles and (b) a system of $N=4000$ ($\sim$ infinity) particles. $h^{\mathrm{closed}}\left(x\right)$ is calculated from Eq. (8). $h\left(x\right)$ is estimated by simulations.

Figure 4

Critical value of the BS model as a function of $m$. Upper and lower bounds are also represented.

Figure 5

Bak–Sneppen model. The update distribution is shown on the left and the equilibrium fitness distribution on the right. Two cases are shown: the (a) uniform (0,1) update distribution and (c) a two-modes update distribution example, $f\left(x\right)$. The dashed line in (c) corresponds to $p_c^{\mathrm{nu}}$, which verifies ${\int }_{-\infty }^{p_c^{\mathrm{nu}}}f\left(x\right)dx=p_c$.

Figure 6

(a) Mean fitness of models 1 and 2 as a function of the number of interacting neighbors. Mean fitness of (b) model 3 and (c) model 4 as a function of $p$ (probability of fitness equal to zero immediately after the update).

Figure 7

Simulations from the BS model for $m=2$. Upper panels: Representation of the BS model where circles represent inactive particles and crosses represent active ones. Lower panels: Proportion of times the lowest fitness particle is the one in the $i\mathrm{th}$ position going clockwise when there are (a) three, (b) four, or (c) five active particles.