  Extremal dynamics is the mechanism that drives the Bak-Sneppen model into a
(self-organized) critical state, marked by a singular stationary probability
density $p(x)$. With the aim of understanding this phenomenon, we study the BS
model and several variants via mean-field theory and simulation. In all cases,
we find that $p(x)$ is singular at one or more points, as a consequence of
extremal dynamics. Furthermore we show that the extremal barrier $x_i$ always
belongs to the `prohibited' interval, in which $p(x)=0$. Our simulations
indicate that the Bak-Sneppen universality class is robust with regard to
changes in the updating rule: we find the same value for the exponent $\pi$ for
all variants. Mean-field theory, which furnishes an exact description for the
model on a complete graph, reproduces the character of the probability
distribution found in simulations. For the modified processes mean-field theory
takes the form of a functional equation for $p(x)$.
