Explosive synchronization in weighted complex networks

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. Given a set of phase oscillators networking with a generic wiring of connections and displaying a generic frequency distribution, we show how combining dynamical local information on frequency mismatches and global information on the graph topology suggests a judicious and yet practical weighting procedure which is able to induce and enhance explosive, irreversible, transitions to synchronization. We report extensive numerical and analytical evidence of the validity and scalability of such a procedure for different initial frequency distributions, for both homogeneous and heterogeneous networks, as well as for both linear and nonlinear weighting functions. We furthermore report on the possibility of parametrically controlling the width and extent of the hysteretic region of coexistence of the unsynchronized and synchronized states.

Figure 1

(a) Synchronization transitions for $N=500$ ER networks, $\langle k\rangle =30$, for unweighted case ($\alpha =0$) (blue squares), and linearly weighted cases ($\alpha =1$) with several frequency distributions within the range $[0,1]$: uniform, Gaussian, Gaussian-derived, Rayleigh, and semi-Gaussian. Solid and dashed lines refer to the forward and backward simulations, respectively. (b) Node strengths $s_i$ (see text for definition) vs natural frequencies ${\omega }_i$, for the unweighted (dark blue squares) and weighted (light blue circles) networks reported in panel (a). Solid line is proportional to the analytical prediction ${(\omega -\frac{a}{2})}^2+\frac{1}{4a}$ in the thermodynamical limit of our model, with $a=1$ the width of the uniform frequency distribution (see text for more details).

Figure 2

Critical scaled couplings ${\sigma }_c/\langle k\rangle$ at the onset of synchronization (forward simulation) and desynchronization (backward simulation) using a linear weighting procedure ${\Omega }_{ij}$ ($\alpha =1$) as a function of (a) $\langle k\rangle$ for several ER network sizes $N$, and of (b) $N$ in all-to-all coupled networks. In panel (a) the vertical dashed line marks the passage from a smooth to an explosive phase transition. In both panels (a) and (b) the upper and lower branches correspond to forward and backward simulations, respectively. Each dot accounts for an average of at least 20 independent runs of uniform frequency distributions. Horizontal dashed lines in panel (b) are close to the analytical values defining the range of the hysteresis in the thermodynamical limit for the Kuramoto model (see explanation in text). Frequencies are uniformly distributed in the range $[0,1]$.

Figure 3

Synchronization transitions for ER networks, $N=500$, uniformly distributed frequencies in the $[0,1]$ range, and nonlinear weighting functions ${\Omega }_{ij}^{\alpha }$. Both plots consider several $\alpha$ values, from sublinear to superlinear weighting [see legend in panel (b)]. (a) ER networks, $\langle k\rangle =30$, (b) regular random networks, $k=30$. In all cases, forward and backward simulations correspond respectively to solid and dashed lines.

Figure 4

Explosive synchronization vs degree of heterogeneity. Synchronization transitions as a function of the coupling strength for linearly weighted networks ($\alpha =1$) with the same average connectivity $\langle k\rangle =30$ and different networks: a regular random (RR) network with homogeneous degree (blue circles), and networks constructed using the model referenced in the text, $p=1$ (ER, red squares), $p=0.2$ (green triangles), and $p=0$ (SF, black diamonds). In all cases, forward and backward simulations correspond respectively to solid and dashed lines. Inset shows log-log plot of the four corresponding degree distributions.

Figure 5

(a) Synchronization transitions for SF networks using different schemes of coupling weighting. In black triangles, the link between nodes $i$ and $j$ is weighted using ${\Omega }_{ij}^{\alpha }$ with $\alpha =1$ (as in Fig. 4), while the rest of cases refer to the weighting function ${\tilde{\Omega }}_{ij}$ of Eq. (3), with the values of the $\beta$ parameter given in the legend. In all cases, forward and backward simulations correspond, respectively, to continuous and dashed lines. (b) Area of the hysteretic region vs $\beta$. Each point is an average of 10 different simulations, each one starting from a different realization of the frequency distributions. In all cases, $\langle k\rangle =30$, $N=1000$, and natural frequencies are uniformly distributed in the interval $[0,1]$.

Figure 6

(a) $I$ as a function of $\mu =\sigma R$ (solid curve) as given by Eq. (9). The dashed lines are those straight lines whose intersection with $I$ marks the backward (${\sigma }_1^c$) and forward (${\sigma }_2^c$) critical points of ES for an all-to-all connected network and for a uniform frequency distribution. (b) The corresponding synchronization order parameter $R$ as a function of the coupling strength. Solid (dashed) curves correspond to the stable (unstable) solution. Dotted vertical lines mark the region of hysteresis defined by ${\sigma }_1^c$ and ${\sigma }_2^c$ in panel (a).