  A major difficulty in studying the Bak-Sneppen model is in effectively
comparing it with well-understood models. This stems from the use of two
geometries: complete graph geometry to locate the global fitness minimizer, and
graph geometry to replace the species in the neighborhood of the minimizer. We
present a variant in which only the graph geometry is used. This allows to
obtain the stationary distribution through random walk dynamics. We use this to
show that for constant-degree graphs, the stationary fitness distribution
converges to an IID law as the number of vertices tends to infinity. We also
discuss exponential ergodicity through coupling, and avalanches for the model.
