Universality Class of Bak-Sneppen Model on Scale-Free Network

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.