Factors that predict better synchronizability on complex networks

While shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network, the suppressing tendency of the heterogeneity of the degree distribution, even for shorter characteristic path length, has also been reported. To see this, we investigate the effects of various factors such as the degree, characteristic path length, heterogeneity, and betweenness centrality on synchronization, and find a consistent trend between the synchronization and the betweenness centrality. The betweenness centrality is thus proposed as a good indicator for synchronizability.

Figure 1

Behavior of the ratio of the maximum eigenvalue ${\lambda }_{\max }$ to the smallest nonvanishing eigenvalue ${\lambda }_{\min }$ with the rewiring probability $p$. As $p$ is raised, the ratio is shown to reduce, yielding better synchronization.

Figure 2

Behavior of the variance ${\sigma }_k^2$ of the degree distribution with the rewiring probability $p$. Inset: characteristic path length $\ell$ vs the rewiring probability $p$. Here and in subsequent figures lines are merely guides to the eye.

Figure 3

Behavior of the maximum value $B^{\max }$ of the betweenness centrality with the rewiring probability $p$. $B^{\max }$ is shown to decrease as $p$ is increased, corresponding to more heterogeneous degree distributions.

Figure 4

The maximum degree $k^{\max }$ versus the rewiring probability $p$ on a WS small-world network. It is shown that $k^{\max }$ increases as the degree distribution becomes more heterogeneous.

Figure 5

Behavior of the difference $\delta$ of the eigenvalue ratio is displayed with the rewiring probability $p$. The data (open circles and squares) have been obtained for the network with the number of nodes $N=2000$ and $r=3$. Due to strong finite-size effects at $p=0.1$, we use the data (filled circle and square) for large system size of $N=5000$. See the text for the explanation of the data symbols.