Determination of the critical coupling of explosive synchronization transitions in scale-free networks by mean-field approximations

An explosive synchronization can be observed in scale-free networks when Kuramoto oscillators have natural frequencies equal to their number of connections. The present paper reports on mean-field approximations to determine the critical coupling of such explosive synchronization. It has been verified that the equation obtained for the critical coupling has an inverse dependence on the network average degree. This expression differs from those whose frequency distributions are unimodal and even. In this case, the critical coupling depends on the ratio between the first and second statistical moments of the degree distribution. Numerical simulations were also conducted to verify our analytical results.

Figure 1

Distribution of the phases $\varphi$ as a function of the degree $k$ for a BA network with $N=3\times {10}^3$ nodes and average degree $〈k〉=6$. The value of $r$ considered in each theoretical curve (solid line) is calculated using the discrete form of Eq. (5), i.e., $r{e}^{i\psi \left(t\right)}={\sum }_i{k}_i{e}^{i{\theta }_i\left(t\right)}/{\sum }_i{k}_i$, at the given coupling $\lambda$.

Figure 2

Distribution of the phases $\varphi$ as a function of the degree $k$ for a ER network with $N=3\times {10}^3$ nodes and average degree $〈k〉=6$. The value of $r$ considered in each theoretical curve (solid line) is calculated using the discrete form of Eq. (5), i.e., $r{e}^{i\psi \left(t\right)}={\sum }_i{k}_i{e}^{i{\theta }_i\left(t\right)}/{\sum }_i{k}_i$, at the given coupling $\lambda$.

Figure 3

Synchronization diagram for ER networks with forward continuation of the coupling strength $\lambda$ with steps of $\delta \lambda =0.02$. The networks have $N={10}^3,2\times {10}^3,\text{and}3\times {10}^3$ and the same average degree (a) $〈k〉=6$ and (b) $〈k〉=8$. Each point is an average over 30 networks. The error bars have the size of the points.

Figure 4

Synchronization diagram for BA networks with forward continuation of the coupling strength $\lambda$ with steps of $\delta \lambda =0.02$. The networks have $N={10}^3,2\times {10}^3,\text{and}3\times {10}^3$ and the same average degree (a) $〈k〉=6$ and (b) $〈k〉=8$. Each point is an average over 30 networks. The error bars have the size of the points.

Figure 5

Coherence diagrams for configurational models with degree distribution $P\left(k\right)\sim k^{-\gamma }$: (a) $\gamma =2.4$ (b) $\gamma =2.6$ (c) $\gamma =2.8$, and (d) $\gamma =3.0$, with forward continuation of the coupling strength $\lambda$ with steps of $\delta \lambda =0.02$.

Figure 6

Scaling plot of $r$ for ER networks with (a) $〈k〉=6$ and (b) $〈k〉=8$.