We report on extensive numerical simulations on the Bak-Sneppen model
in high dimensions. We uncover a very rich behavior as a function of d
imensionality. For d > 2 the avalanche cluster becomes fractal and for
d greater than or equal to 4 the process becomes transient. Finally,
the exponents reach their mean field values for d = d(c) = 8, which is
then the upper critical dimension of the Bak-Sneppen model.