Why is the critical state in the Mean Field Bak-Sneppen Model a global attractor?

Question

Simulations of the Bak-Sneppen Model of Species Evolution (introduced in http://link.aps.org/doi/10.1103/PhysRevLett.71.4083) show that it exhibits Self-Organized Criticality where after a transient only mutation through barriers of height $B_i < B_C$ where $B_C = 0.67 \pm 0.01$ occur spontaneously.

As far as I can tell two studies attempt to show the SOC property of the Bak-Sneppen Model in the Mean Field approximation:

1. Flyvbjerg et al. (1993, http://link.aps.org/doi/10.1103/PhysRevLett.71.4087) formulate the evolution equation for the probability distribution of the barrier values $P(B, t)$ but the argument for the existence of a global attractor is given very tersely as:

Our mean field dynamics is an approximation to the master equation for the Markov process of the random neighbor model, both having one unique attractive fixed point. (Page 4088, last paragraph left column)

1. In de Boer et al. (1994, http://link.aps.org/doi/10.1103/PhysRevLett.73.906) the Markov Process hinted at in the previous publication is studied by deriving the Master equation $P_{n,\lambda}(t)$ where $n$ is the number of sites with a barrier value less than $\lambda$. However as $\lambda \rightarrow \frac{1}{2}$, $P_{n,\lambda} \rightarrow 0$, which I interpret as meaning that there exists a $B_C \ge \frac{1}{2}$. But I do not see how this proves that the critical state is an attractor.

Therefore what is the argument for the critical state being a global attractor in the Mean Field Approximation?

Answer 1

The argument they are using is the uniqueness of the stationary distribution of a Markov chain. The master-equation description produces a Markov chain approximation of the dynamics, and then if you find a stationary solution, it must be the unique global attractor.

If you have a Markov process which takes independent steps, you can think of it as a random walk of some kind (at least on a countable state space). If this walk has a nonzero probability for any two initial configurations to collide at some future time, meaning they reach the same point at the same time, then the equilibrium distribution is unique. This condition on Markov chains is called ergodicity.

The proof is a simple coupling argument: consider a doubled walk with two independent walkers which take independent steps according to the Markov process. When they reach the same point, however, they stick together and move together forever. It is easy to see that under the condition above, the two walks will eventually stick together with probability 1.

This proves unique-global-attractorness by letting one walk start in the stationary distribution, and the other to start in an arbitrary distribution. By choosing a long enough time so that they couple with probability $(1-\epsilon)$, the arbitrary distribution must have evolved to be within $(1-\epsilon)$ of the stationary distribution in $L_1$ norm.