  We study the critical properties of the Bak-Sneppen coevolution model on
scale-free networks by Monte Carlo method. We report the distribution of the
avalanche size and fractal activity through the branching process. We observe
that the critical fitness $f_c (N)$ depends on the number of the node such as
$f_c (N) \sim 1/ \log (N)$ for both the scale-free network and the directed
scale-free network. Near the critical fitness many physical quantities show
power-law behaviors. The probability distribution $P(s)$ of the avalanche size
at the critical fitness shows a power-law like $P(s) \sim s^{-\tau}$ with
$\tau=1.80(3)$ regardless of the scale-free network and the directed scale free
network. The probability distribution $P_f (t)$ of the first return time also
shows a power-law such as $P_f (t) \sim t^{-\tau_f}$. The probability
distribution of the first return time has two scaling regimes. The critical
exponents $\tau_f$ are equivalent for both the scale-free network and the
directed scale-free network. We obtain the critical exponents as $\tau_{f1}
=2.7(1)$ at $t < t_c$ and $\tau_{f2} = 1.72(3)$ at $ t >t_c$ where the
crossover time $t_c \sim 100$. The Bak-Sneppen model on the scale-free network
and directed scale-free network shows a unique universality class. The critical
exponents are different from the mean-field results. The directionality of the
network does not change the universality on the network.
