Onset of synchronization in complex networks of noisy oscillators

We study networks of noisy phase oscillators whose nodes are characterized by random degrees counting the number of their connections. Both these degrees and the natural frequencies of the oscillators are distributed according to a given probability density. Replacing the randomly connected network by an all-to-all coupled network with weighted edges allows us to formulate the dynamics of a single oscillator coupled to the mean field and to derive the corresponding Fokker-Planck equation. From the latter we calculate the critical coupling strength for the onset of synchronization as a function of the noise intensity, the frequency distribution, and the first two moments of the degree distribution. Our approach is applied to a dense small-world network model, for which we calculate the degree distribution. Numerical simulations prove the validity of the replacement. We also test the applicability to more sparsely connected networks and formulate homogeneity and absence of correlations in the degree distribution as limiting factors of our approach.

Figure 1

The effect of our approximation on a ring network of eight symmetrically coupled oscillators. On the left-hand side the original complex network is shown and on the right-hand side the approximate network is shown, where the thickness of the edges is chosen approximately proportional to the coupling strength. For reasons of clarity, self-coupling is not visualized.

Figure 2

Visualization of a dense small-world network model. If the system size $N$ is duplicated, the average number of connections is duplicated as well. The figure shows one marked node in the network.

Figure 3

Degree distribution $P\left(k\right)$ of dense small-world networks consisting of 400 nodes depicted for different shortcut probabilities $p$ and fractions of regular connections $\alpha$ (inset). For the sake of clarity solid lines are chosen.

Figure 4

Dependency of the topology function on the system size $N$: Solid lines are given by the theory. The thick line corresponds to the thermodynamic limit and slight lines are calculated according to Eqs. (43) and (44) for smaller systems. Markers show results of numerical calculations for indicated system sizes. The parameters are: (a) $\alpha =0.052$ and (b) $p=0.052$.

Figure 5

Numerical determination of critical coupling strength by plotting the order parameter $s$ times $N^{0.25}$ for different system sizes in dependence on the coupling strength. According to Eq. (47) the intersection of the curves indicates the critical coupling strength. Parameter values here are $\alpha =0.05$, $p=0.1$, $D=0.05$, and $\sigma =0.02$.

Figure 6

The dependencies of the critical coupling strength ${\varkappa }_c$ on the shortcut probability $p$ and on the parameter $\alpha$ for the local connections. The remaining parameters are $D=0.05$ and $\sigma =0.2$ together with (a) $\alpha =0.05$ and (b) $p=0.01$.

Figure 7

The dependencies of the critical coupling strength ${\varkappa }_c$ on the noise intensity $D$ and on the diversity parameter $\sigma$. The remaining parameters are $\alpha =0.05$ and $p=0.01$ together with (a) $\sigma =0.2$ and (b) $D=0.05$.

Figure 8

Smallest $p$ (random network) and $\alpha$ (regular network) values for which a synchronization transition with the standard mean-field exponents can be observed, shown as functions of the denseness parameter $q$. We denote these values by $p_{\mathrm{mf}}$ and ${\alpha }_{\mathrm{mf}}$. Dashed lines depict the scaling behaviors.

Figure 9

Numerically calculated average path lengths $L$ and clustering coefficients $C$ as a function of shortcut probability $p$ for a network of 501 nodes. The radius of coupled next neighbors [compare Eq. (40)] equals (a) $K=2(\alpha =0.008)$ or (b) $K=50(\alpha =0.2)$, respectively. Dashed lines indicate the area in which our model generates a small-world network.

Figure 10

The diversity functional for different frequency distributions as a function of noise intensity $D$, standard deviation $\sigma$, or scale parameter $\gamma$. The parameters are: (a) $\sigma =\gamma =1$ and (b) $D=0.4$.