Synchronization transition in scale-free networks: Clusters of synchrony

We study the synchronization transition in scale-free networks that display power-law asymptotic behaviors in their degree distributions. The critical coupling strength and the order-parameter critical exponent derived by the mean-field approach depend on the degree exponent $\lambda$, which implies a close connection between structural organization and the emergence of dynamical order in complex systems. We also derive the finite-size scaling behavior of the order parameter, finding that the giant cluster of synchronized nodes is formed in different ways between scale-free networks with $2<\lambda <3$ and those with $\lambda >3$.

Figure 1

Order parameter $r={\lim }_{t\to \infty }r\left(t\right)$ versus the coupling strength $J$ for $N=1600$, $⟨k⟩=4$, and $\lambda =6.4$, 3.6, and 2.4. Inset: Time evolution of $r\left(t\right)$ for $\lambda =3.6$ with $J=0.0$, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.8, and 3.2 from bottom to top.

Figure 2

(Color online) Phase diagram and typical dynamical states of oscillators on SF networks. The phase boundary in Eq. (9) is drawn with $p_d\left(k\right)=k^{-\lambda }∕\zeta \left(\lambda \right)$ for $k⩾1$ and $\zeta \left(\lambda \right)$ the Riemann-zeta function. The two network configurations represent dynamical states of coupled oscillators that have evolved with $J=1.0$ (desynchronized) and $J=4.0$ (synchronized), respectively, on SF networks with $\lambda =3.6$. In both, synchronized nodes, those having their phases ${\varphi }_i$ so close to the average phase $\theta$ that $\mid {\varphi }_i-\theta \mid <0.5$, are drawn as filled boxes inside the circle while the others are as empty boxes at the circumference. Edges connecting synchronized nodes are drawn as thick lines.

Figure 3

Data collapse of scaled order parameter for $⟨k⟩=4$, (a) $\lambda =6.4$, (b) $\lambda =3.6$, and (c) $\lambda =2.4$ according to Eqs. (15, 16). In (a) and (b), insets show the crossing of scaled order parameters at $J_c$, which is found to be (a) 1.85(5) and (b) 1.15(5), larger than 1.22 and 0.84, respectively, evaluated from Eq. (9). In the inset of (c), asymptotic behavior of the scaling function is explicitly shown. The solid line has slope $1∕(3-\lambda )=5∕3$.