Heterogeneity in Oscillator Networks: Are Smaller Worlds Easier to Synchronize?

Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.

Figure 1

Synchronizability of SF networks of size $N=2^{10}$. (a),(b) The average network distance $\overline{\mathcal{D}}$ and the eigenvalue ratio ${\lambda }_N/{\lambda }_2$ for the semirandom model with $k_0=5$. The inset of (a) shows the mean $\overline{k}$ and the standard deviation $s$ of the connectivity distribution. The maximum ${\ell }_{\max }$ of the normalized load on nodes is plotted in the inset of (b). The horizontal lines in (a) and (b) indicate the values of $\overline{\mathcal{D}}$ and ${\lambda }_N/{\lambda }_2$ computed for the $\gamma =\infty$ case. (c),(d) The same quantities for the growing model with aging of nodes for $n_0=5$ and $m_a=5$. Note that $\overline{k}$ is not shown because it is trivially $\overline{k}=m_a$. The solid curves in (b) and (d) are the lower bounds given in Eq. (2). The upper bounds of Eq. (2) are above the limits, but follow the same trend. All quantities are averaged over 100 realizations.

Figure 2

(a) The average network distance $\overline{\mathcal{D}}$, the clustering coefficient $C$, and (b) the ratio ${\lambda }_N/{\lambda }_2$ for the two-layer model. The inset in (b) shows the maximum value ${\ell }_{\max }$ of the normalized load on nodes. The solid curves in (b) are the bounds given in Eq. (2). All estimates are the results of averaging over 100 realizations, and $\overline{\mathcal{D}}$ and $C$ are normalized by the respective values for $n_c=N$: $\overline{\mathcal{D}}\approx 4.6$ and $C\approx 0.52$. Other parameters are $N=2^{10}$, $\kappa =4$, and $m=2^9$.