We investigate Bak-Sneppen coevolution models on scale-free networks with various degree exponents $\gamma$ including random networks. For $\gamma >3$, the critical fitness value $f_c$ approaches a nonzero finite value in the limit $N\to \infty$, whereas $f_c$ approaches zero as $2<\gamma ⩽3$. These results are explained by showing analytically $f_c\left(N\right)\simeq A∕{⟨{(k+1)}^2⟩}_N$ on the networks with size $N$. The avalanche size distribution $P\left(s\right)$ shows the normal power-law behavior for $\gamma >3$. In contrast, $P\left(s\right)$ for $2<\gamma ⩽3$ has two power-law regimes. One is a short regime for small $s$ with a large exponent ${\tau }_1$ and the other is a long regime for large $s$ with a small exponent ${\tau }_2({\tau }_1>{\tau }_2)$. The origin of the two power regimes is explained by the dynamics on an artificially made star-linked network.

• Received 22 November 2004

DOI:https://doi.org/10.1103/PhysRevE.71.057102