### Rank Driven Dynamics

Peter Veerman

Abstract.
We investigate a class of models related to the Bak-Sneppen (BS)
model, initially proposed to study evolution. The BS model is extremely
simple and yet captures some forms of =E2=80=9Ccomplex behavior=E2=80=9D su=
ch as punctuated
equilibrium that is often observed in physical and biological systems.
In the BS model, random numbers in [0,1] (interpreted as fitnesses of
agents) distributed according to some cumulative distribution function
R: [0, 1] -> [0, 1] are placed at the vertices of a graph G. At every time-step
the lowest number and its immediate neighbors are replaced by new random
numbers. We approximate this dynamics by making the assumption that the
numbers to be replaced are independently distributed. We then use Order
Statistics to define a dynamical system on the cumulative distribution
functions R of the collection of numbers.
For this simplified model we can find the limiting marginal distribution as
a function of the initial conditions. Agreement with experimental results
of the BS model is excellent.
We analyze two main cases: The exogenous case where the new fitnesses are
taken from an a priori fixed distribution, and the endogenous case where
the new fitnesses are taken from the current distribution as it evolves.