The Bak-Sneppen model is a discrete-time Markov chain in which the site with the minimum value and its neighboring sites receive new independently sampled values. Bak and Sneppen (1993) introduced it as a simple model of biological evolution. The values are conventionally called fitnesses.
In the standard ring version, there are 
sites with fitnesses in the unit
interval, so the state space is ![[0,1]^N](/images/equations/Bak-SneppenModel/Inline2.svg)
. At each step, find a site 
of minimum fitness and replace
the values at sites 
,
, and 
by three independent
samples from the uniform distribution on
![[0,1]](/images/equations/Bak-SneppenModel/Inline7.svg)
. Site indices are interpreted modulo
. Ties may be resolved by taking the
first minimum in index order. They have probability
zero under independent continuous initialization.
In the illustrated eight-site ring, site 8 has the minimum fitness, so sites 7, 8, and 1 are updated. The three displayed replacement values illustrate one transition.
For each finite 
, the ring process has a unique stationary distribution (Alberts et al.
2017). For four sites, its joint probability
density function is
 |
(1)
|
where
![q(x)=3/2(1-(1-x)^3)/([(1-x)^3+2]^2)](/images/equations/Bak-SneppenModel/NumberedEquation2.svg) |
(2)
|
for 
(Schlemm 2012, Alberts et al. 2017). Since 
, the one-site marginal
distribution has probability density
function
 |
(3)
|
Thus 
alone is not the one-site probability
density function.
The open-chain version used by Trott (2004, sec. 1.5.6) refreshes only two sites when the minimum occurs at an endpoint. It therefore has a different boundary rule from the ring process. Discrete-valued variants also require separate analysis (Meester and Znamenski 2002).
