Structural properties of the synchronized cluster on complex networks

We investigate how the largest synchronized connected component (LSCC) is formed and evolves to achieve a global synchronization on complex networks using Kuramoto model. In this study we use two different networks, Erdösi-Rényi network and Barabási-Albert network. From the finite-size scaling analysis, we find that the scaling exponents for the percolation order parameter and mean cluster size on both networks agree with the mean-field percolation theory, $\beta =\gamma =1$. We also find that the finite-size scaling exponent, $\overline{\nu }$, also agrees with the mean-field percolation result, $\overline{\nu }=3$. Moreover, we also show that the cluster size distributions are identical with the mean-field percolation distribution on both networks. Combining with the analysis for the merging clusters, we directly show that the LSCC on both networks evolves by merging clusters of various sizes.

Figure 1

Plot of $J_{\text{max}}\left(N\right)$ in (a) ER networks and (b) BA networks. The intercept with $J_{\text{max}}$-axis corresponds to $J_c$. Plot of $S_{\text{max}}\left(J_{\text{max}},N\right)$ against $N$ on (c) ER networks and (d) BA networks. The solid lines represent the relation (8). (e) and (f) show $P_{\infty }\left(J_{\text{max}},N\right)$ on ER and BA networks, respectively. The solid lines stand for Eq. (9).

Figure 2

Scaling plot of $P_{\infty }$ on (a) ER networks and (b) BA networks. Scaling plot of $S$ on (c) ER networks (d) BA networks.

Figure 3

Plot of $n_s$ at (a) $J\simeq J_{\text{max}}\left(N\right)$ $\left(\approx J_c\right)$ and (b) $J>J_c$ when $N=10000$. The solid lines represent (a) $\tau =2.7$ and (b) $\tau =3.7$.

Figure 4

(a) Plot of $P\left(m\right)$ for $J=0.06$. The lines represent the relation $P\left(m\right)\sim m^{-\delta }$ with $\delta =2.8$ for ER network (solid line) and $\delta =2.9$ for SF network (dashed line). (b) Plot of $P\left(m\right)$ for $N_{\text{LSCC}}\simeq 0.86N$. Solid lines denote $\delta =3.6$.