Basin of Attraction Determines Hysteresis in Explosive Synchronization

Spontaneous explosive emergent behavior takes place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. A central feature of such explosive transitions is a hysteretic behavior at the transition to synchronization. We unravel the underlying mechanisms and show that the dynamical origin of the hysteresis is a change of basin of attraction of the synchronization state. Our findings hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, reveal how microscopic network parameters such as node degree and frequency affect the global network properties and can be used for network design and control.

Figure 1

Order parameter $r$ as a function of the coupling $\lambda$ for various sizes. The thick lines are theoretical curves obtained by Eq. (5). (a) ($K_1=5$, $K_2=40$), and (b) comparison between with and without noise ($K=10$). $r$ is an arithmetic mean value of [${\mathcal{R}}_{\min }(t)$, ${\mathcal{R}}_{\max }(t)$] over 100 random realizations. Part II without noise while, for parts I and III, there is a random frequency mismatch ${\omega }_j=\omega +{\zeta }_j$ where ${\zeta }_j\in [-0.05,0.05]$ for leaf nodes.

Figure 2

(a) Phase difference $\sin a=\sin ({\varphi }_{K+1}-{\varphi }_1)$ between the hub and the first leaf as a function of $\lambda$ for $\lambda >{\lambda }_c^f$. Both circles and triangles represent the numerical simulation, and the thick lines are theoretical prediction provided by Eq. (3). Network size: $K=5(\circ )$, $K=10(▵)$. (b) Order parameter $r$ on the parameter space of ($K$, $\lambda$) for the forward continuation. The (red) thick line is from fitting the scaling relation provided by Eq. (8), where the parameter $1/B=0.6989$.

Figure 3

Order parameter $r$ on the parameter space ($\delta$, $\lambda$) for the forward continuation ($K=100$). Initial conditions are randomly drawn from an interval [$-\delta$, $\delta$]. The horizontal dashed lines are critical couplings ${\lambda }_c^b$ and ${\lambda }_c^f$ from the theory. (a) without noise effect, (b) with frequency mismatches for leaf nodes, namely, ${\omega }_j=\omega +{\zeta }_j$ where the random value ${\zeta }_j\in [-0.01,0.01]$.

Figure 4

(a) Local order parameter $r_i$ versus coupling strength for two chosen hubs ($K_1=39$ red; $K_2=24$ blue; ${\lambda }_{c,1}^f=4.64$, ${\lambda }_{c,2}^f=3.63$). (b) Critical coupling $⟨{\lambda }_c^f⟩$ vs sizes of SF networks, where $⟨·⟩$ is an ensemble average over 50 network realizations. The dashed line is the theoretical curve predicted by Eq. (9).