Extremal dynamics on complex networks: Analytic solutions

The Bak-Sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks. By using the rate equation and the random walk approaches, we obtain the analytic solution of the fitness threshold $x_c$ to be $1∕({⟨k⟩}_f+1)$, where ${⟨k⟩}_f=⟨k^2⟩∕⟨k⟩$ $(=⟨k⟩)$ in the quenched (annealed) updating case, where $⟨k^n⟩$ is the $n\text{th}$ moment of the degree distribution. Thus, the threshold is zero (finite) for the degree exponent $\gamma <3$ $(\gamma >3)$ for the quenched case in the thermodynamic limit. The theoretical value $x_c$ fits well to the numerical simulation data in the annealed case only. Avalanche size, defined as the duration of successive mutations below the threshold, exhibits a critical behavior as its distribution follows a power law, $P_a\left(s\right)\sim s^{-3∕2}$.

Figure 1

The data of $f_k$ (엯) and $k{p}_k∕⟨k⟩$ (˟) for the Erdős-Rényi network (a) and for the scale-free network with $\gamma =3.6$ (b) in the quenched cases. To generate the scale-free network, we use the static model [20]. Both networks have the average degree $⟨k⟩=4$ and the system size $N={10}^6$.

Figure 2

(a) and (b) The avalanche size distribution for the regular network with the degree distribution of the $\delta$-function. In the quenched case (a), the avalanche size distribution follows a power law when $\lambda$ is chosen as 0.253 (엯), larger than the theoretical value $1∕(⟨k⟩+1)=0.2$ (◻). In the annealed case (b), the theoretical value $x_c=0.2$ (엯) works well to generate the power-law behavior of $P_a\left(s\right)$. (c) and (d) Same plot for the Erdős-Rényi network [14]. In the quenched case (c), the power-law behavior of $P_a\left(s\right)$ occurs at $x_c=0.207$, larger than the theoretical value, $x_c=1∕6$. In the annealed case (d), the theoretical value $x_c=0.2$ generates the power-law behavior. (e) and (f) Same plot for the scale-free network with $\gamma =3.6$. In the quenched case (e), the power-law behavior of $P_a\left(s\right)$ occurs at $x_c=0.1$, which is larger than the theoretical value $x_c=0.08$. In the annealed case, the theoretical value $x_c=0.2$ yields the power-law behavior of $P_a\left(s\right)$. The mean degree is fixed to be $⟨k⟩=4$ and the system size is $N={10}^6$ in all cases. The straight lines have slope $-3∕2$ in all cases.