Explosive synchronization with partial degree-frequency correlation

Networks of Kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of their corresponding vertices exhibit so-called explosive synchronization behavior, which is now under intensive investigation. Here we study and discuss explosive synchronization in a situation that has not yet been considered, namely when only a part, typically a small part, of the vertices is subjected to a degree-frequency correlation. Our results show that in order to have explosive synchronization, it suffices to have degree-frequency correlations only for the hubs, the vertices with the highest degrees. Moreover, we show that a partial degree-frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree-frequency correlation would not allow it. We perform a mean-field analysis and our conclusions were corroborated by exhaustive numerical experiments for synthetic networks and also for the undirected and unweighed version of a typical benchmark biological network, namely the neural network of the worm Caenorhabditis elegans. The latter is an explicit example where partial degree-frequency correlation leads to explosive synchronization with hysteresis, in contrast with the fully correlated case, for which no explosive synchronization is observed.

Figure 1

Synchronization diagrams $r\left(\lambda \right)$ for a Barabasi-Albert network with $k_{\min }=3$ and $\langle k\rangle =6$, with $N=1000$ vertices, and $k_{*}=10$, corresponding approximately to only 10% of vertices with degree-frequency correlation. The left panel corresponds to a situation where the frequencies of the uncorrelated vertices were drawn from the null average homogeneous probability distribution (29) with ${\sigma }_0=1$, while the right one corresponds to the Gaussian (31) with $\Omega$ given by (9) and ${\sigma }_0=1/2$. In both cases, the mean-field analysis predicts ${\lambda }_c$ (the vertical line) but cannot discern between the explosive synchronization (first-order phase transition) of the left panel from the continuous second-order phase transition of the right panel. Our numerical analysis shows that, typically, greater values of $\Omega$ tend to favor second-order phase transitions instead of ES. The simulations corresponding to the $\langle \omega \rangle =0$ case (left panel) were plagued by large statistical fluctuations for small $\lambda$ which origin is unclear. For small $\lambda$, forward and backward continuations do not coincide and large deviations are observed for different runs. Both panel depicts only one run to put in evidence these discrepancies.

Figure 2

The graphics show the synchronization diagrams $r\left(\lambda \right)$ for networks built with the mechanism proposed in Ref. [27]. Panels (a), (b), and (c) show, respectively, the forward and backward continuations for full degree-frequency correlation and $\alpha =0.2$, $\alpha =0.1$, and $\alpha =0$. Panels (d), (e), and (f) are the diagrams when only $10\%$ of the vertices with largest degree have degree-frequency correlation. See the text for further details.

Figure 3

The graphic shows the area $A$ between the forward and backward continuations of the synchronization diagrams $r\left(\lambda \right)$ as a function of the fraction $f$ of vertices for which degree-frequency correlation holds for networks built with the mechanism proposed in Ref. [27]. Different curves show the behavior of $A$ for different values of the parameter $\alpha$ that measures the heterogeneity of the degree distribution. Each point corresponds to an average over 10 different sets of networks and natural frequencies and the error bars represent the corresponding standard deviations.

Figure 4

The graphics show the synchronization diagrams $r\left(\lambda \right)$ for the neural network of the worm C. elegans. The diagram on the left was calculated assuming ${\omega }_i=k_i$ for every vertex $i$ of the network. The inset shows the degree distribution of the network. The diagram at right corresponds to a partial degree-frequency correlation: the correlation holds only for the 20 vertices with largest degrees. The natural frequencies for the remaining vertices were drawn from two different distributions, either from a power-law distribution $g\left(\omega \right)\propto {\omega }^{-\gamma }$, with $\gamma =2.5$ and ${\omega }_0=3$ according to (41), or from a normal $\mathcal{N}(10,4)$ distribution with $\langle \omega \rangle =10$ and ${\sigma }_0=4$, see the insets. For the case of a power law, explosive synchronization holds also for other values of $\gamma$ (not shown).

Figure 5

The figure shows the evolution of the effective frequencies ${\Omega }_i$ of the oscillators on the forward continuation for the C. elegans neural network, for the case with $\gamma =2.5$ of Fig. 4. The red (dashed) lines correspond to the vertices for which degree-frequency correlation holds, whereas black (solid) lines represent the remaining ones.

Figure 6

The figure shows the natural frequencies ${\omega }_i$ of the oscillators as a function of their degrees $k_i$ for the cases analyzed in Fig. 4. Three different cases are depicted, when there is partial correlation, with the noncorrelated vertices having a natural frequency draw from either a power law or a normal distribution as well as when there is a quenched disorder, ${\omega }_i=k_i+{\zeta }_i$, where ${\zeta }_i$ is a random variable drawn from the uniform distribution $\mathcal{U}(-\epsilon ,\epsilon )$, with $\epsilon =12$. The inset shows the region for small degrees and frequencies.