Explosive synchronization caused by optimizing synchrony of coupled phase oscillators on complex networks

Abstract

We optimize synchrony in networks of heterogeneous phase oscillators. By analyzing the eigenvalues of the Jacobian matrix at full synchronization, two structure indicators related to the critical coupling strength are constructed. When the network structure is fixed, the current scheme of optimization on the frequency picking and allocation converts the synchronization from a second-order continuous to a first-order discontinuous phase transition. When the frequency distribution is deterministic and uniform, the obtained best network structure has a critical coupling strength that approaches the theoretical minimum value. It is found that the requirement for explosive synchronization is consistent with that for optimizing synchrony, which requires that the frequency difference between connected nodes be large, and the distance from the node to the average frequency is proportional to the node degree. The connection between optimization and explosive synchronization deepens our understanding of the latter and provides an effective way of control.

Graphic abstract

Appendix A: Derivation of structural index \(S_1\), \(S_2\)

First of all, we need to get the phase $\varvec{\theta }$ of the nodes in the fully synchronized state [21]. When the coupling strength k is large enough, $\theta _j-\theta _i \approx 0$ , Eq.  (1) can be transformed into

$$\begin{aligned} \bar{\omega }=\omega _{i}+\frac{k}{\bar{d}}\sum _{j=1}^{N} A_{ij} (\theta _j-\theta _i) ,i=1,2...,N, \end{aligned}$$

(A.1)

which can be written in a matrix form:

$$\begin{aligned} \varvec{\varOmega }=\frac{k}{\bar{d}} L \varvec{\theta }, \end{aligned}$$

(A.2)

where $\varOmega _i = \omega _i - \bar{\omega }$, L is the Laplacian matrix of A, the eigenvalues and eigenvectors of L are $\gamma _i,\varvec{u}^i$ respectively. If the network is connected, then the eigenvalues of L satisfy $\gamma _N>= \cdots>=\gamma _2 > \gamma _1 = 0$, and the generalized inverse matrix of L is $L^\dagger =\sum _{i=2}^{N}\gamma _i^{-1} \varvec{u}^i \varvec{u}^{iT}$. According to Eq. (4):

$$\begin{aligned} \varvec{\theta } = \frac{\bar{d}}{k} L^\dagger \varvec{\varOmega }. \end{aligned}$$

(A.3)

Let $\varvec{\phi } = \bar{d} L^\dagger \varvec{\varOmega }$ , then $\varvec{\theta } = \varvec{\phi }/k$.

Let $G_{ij}=A_{ij}\cos (\theta _j-\theta _i), G_{ii}=-\sum _{j=1}^{N}{A_{ij}\cos (\theta _j-\theta _i)}$, then the Jacobian matrix J of Eq. (1) is $J = {k}/{\bar{d}}G$. Because ${k}/{\bar{d}} > 0$, the positive and negative signs of the eigenvalues of matrix J and G are the same, so we can use the eigenvalues of G to judge the stability of the system. Let $\lambda _i,\varvec{v}^i$ be eigenvalues and eigenvectors of matrix G. When the coupling strength $k = k_0$ is large enough, the phase $\varvec{\theta }(k_0) = \varvec{\phi }/k_0$, when the coupling strength $k = k_0 + \varepsilon$($|\varepsilon | \ll 1$), the phase $\varvec{\theta }(k_0 + \varepsilon ) ={\varvec{\phi }}/{(k_0 + \varepsilon )}$, and the phase difference $\varDelta \varvec{\theta }={\varvec{\phi }}/{(k_0 + \varepsilon )}-{\varvec{\phi }}/{k_0}= {-\varepsilon \varvec{\phi }}/{(k_0(k_0+\varepsilon ))} \approx -\varepsilon \varvec{\phi }/k_0^2$. We carry out a perturbation analysis [56,57,58] by writing

$$\begin{aligned}&G(k_0+\varepsilon ) = G^{(0)} + \varepsilon G^{(1)} + \varepsilon ^2 G^{(2)} + \cdots , \end{aligned}$$

(A.4)

$$\begin{aligned}&\lambda _l(k_0+\varepsilon ) = \lambda _l^{(0)} + \varepsilon \lambda _l^{(1)} + \cdots , l=1,2,\cdots ,N, \end{aligned}$$

(A.5)

where $G^{(1)}_{ij}\approx A_{ij}\sin ((\phi _j-\phi _i)/k_0)(\phi _j-\phi _i)/k_0^2, i \ne j$, and $G^{(1)}_{ii}\approx -\sum _{j=1}^{N}{A_{ij}\sin ((\phi _j-\phi _i)/k_0)(\phi _j-\phi _i)/k_0^2}$. When the coupling strength $k = k_0$ is large, $\theta _j \approx \theta _i$, $cos(\theta _j-\theta _i) \approx 1$, so $G^{(0)} = G(k_0) \approx -L$, eigenvalues and eigenvectors of matrix G: $\lambda _l \approx - \gamma _l, \varvec{v}^l \approx \varvec{u}^l$, according to perturbation theory:

$$\begin{aligned}&\lambda _l^{(1)}(k_0)=\varvec{v}^{lT} (k_0) G^{(1)} \varvec{v}^l(k_0) \approx \varvec{u}^{lT} G^{(1)} \varvec{u}^l \nonumber \\&\approx \sum _{i=1}^{N}\sum _{j=1}^{N}(u_i^l(u_j^l-u_i^l)A_{ij} \frac{\phi _j-\phi _i}{k_0^2} \sin (\frac{\phi _j-\phi _i}{k_0})) \nonumber \\&\approx \sum _{i=1}^{N}\sum _{j=1}^{N}(u_i^l(u_j^l-u_i^l)A_{ij}\sum _{s=1}^{\infty }{ \frac{(-1)^{s+1}(\phi _j-\phi _i)^{2s}}{(2s-1)! k_0^{(2s+1)}}}), \end{aligned}$$

(A.6)

where $\sum _{s=1}^{\infty }{ {(-1)^{s+1}(\phi _j-\phi _i)^{2s}}/{((2s-1)! k_0^{(2s+1)})}}$ is the Taylor expansion of ${(\phi _j-\phi _i)} \sin ({(\phi _j-\phi _i)}/{k_0}))/{k_0^2}$ near 0. According to Eq. (A.5): $\lambda _l^{(1)}(k_0)=(\lambda _l(k_0+\varepsilon )-\lambda _l(k_0))/ \varepsilon -\varepsilon \lambda _l^{(2)}-\varepsilon ^2\lambda _l^{(3)}-\cdots$, when $\varepsilon \rightarrow 0$, $\lambda _l^{(1)}(k_0) = \dot{\lambda _l}(k_0)$. Then $\lambda _l(k)$ can be obtained by integrating $\lambda _l^{(1)}(k)$:

$$\begin{aligned} \lambda _l=\int \dot{\lambda _l}\mathrm {d}k=\int \lambda _l^{(1)}\mathrm {d}k = -\gamma _l+\sum _{s=1}^{\infty }{(-1)^{s+1}\frac{a_s^l}{ k^{2s}}}, \end{aligned}$$

(A.7)

where $a_s^l =\sum _{i=1}^{N}\sum _{j=1}^{N}(u_i^l(u_j^l-u_i^l)A_{ij}\frac{(\phi _j-\phi _i)^{2s}}{(2s)!})$,$\gamma _l$ is the eigenvalue of L. With the increment of s, $a_s^l$ will quickly converge to 0, so we only need to calculate of the first two terms.

The critical coupling $k_c$ corresponds to the zero crossing point of $\lambda _l$, so scaling $\lambda _l$ will not affect the $k_c$. For the convenience of comparing $\lambda _l$ of different l, the eigenvalue $\lambda _l$ is normalized:

$$\begin{aligned} \tilde{\lambda _l}=\frac{\lambda _l}{\gamma _l}\approx -1+\sum _{s=1}^{\infty }{(-1)^{s+1} \frac{b_s^l}{k^{2s}}} \approx -1+ \frac{b_1^l}{k^2} - \frac{b_2^l}{k^4}, \end{aligned}$$

(A.8)

where $b_s^l = a_s^l / \gamma _l$. When k decreases from infinity(only a larger value is needed, such as $10k_c$) to critical coupling strength $k_c$, $\mathop {max}\limits _{l}(\tilde{\lambda _l})$ increases from $-1$ to 0. If the change trend of $\tilde{\lambda _l}$ is slower, the coupling strength k is smaller when $\tilde{\lambda _l}$ reaches 0, and the critical coupling strength $k_c$ is smaller. When k is large, $b_1^l$ has far more influence on $\tilde{\lambda _l}$ then $b_2^l$, so only $b_1^l$ is analyzed, then the critical coupling strength $k_c$ can be reduced as long as $\mathop {max}\limits _{l}(b_1^l)$ is reduced.

However, the form of $\mathop {max}\limits _{l}(b_1^l)$ is very complicated. Let us make a further treatment of $\mathop {max}\limits _{l}(b_1^l)$ for simplicity. Let $B_{ij}=A_{ij}(\phi _j-\phi _i)^2$, $C=diag(\sum _{j=1}^{N}B_{1j},\cdots ,\sum _{j=1}^{N}B_{Nj})-B$, then

$$\begin{aligned} b_1^l=\frac{a_1^l}{\gamma _l}=\frac{\varvec{u}^{lT} C \varvec{u}^l}{2\gamma _l}. \end{aligned}$$

(A.9)

Let the mean square of the difference $\phi$ of the connected nodes be set as follows:

$$\begin{aligned} \bar{\delta }=\frac{1}{N\bar{d}}\sum _{i=1}^{N}\sum _{j=1}^{N}{A_{ij}(\phi _j-\phi _i)^2} =\frac{2}{N\bar{d}}{\varvec{\phi }}^T L \varvec{\phi }. \end{aligned}$$

(A.10)

Let $(\phi _j-\phi _i)^2 = \bar{\delta }+\delta _{ij}$, $H_{ij}=A_{ij}\delta _{ij}$, $D=diag(\sum _{j=1}^{N}H_{1j},\cdots , \sum _{j=1}^{N}H_{Nj})-H$, then $C=\bar{\delta }L+D$. From Eq. (A.9), we can get that

$$\begin{aligned} b_1^l=\frac{\varvec{u}^{lT} C \varvec{u}^l}{2\gamma _l}=\frac{\varvec{u}^{lT} \bar{\delta }L\varvec{u}^l+\varvec{u}^{lT} D\varvec{u}^l}{2\gamma _l} =\frac{\bar{\delta }}{2}+\frac{\varvec{u}^{lT} D\varvec{u}^l}{2\gamma _l}. \end{aligned}$$

(A.11)

So $\mathop {max}\limits _{l}(b_1^l)={\bar{\delta }}/{2}+\mathop {max}\limits _{l}({\varvec{u}^{lT} D\varvec{u}^l}/(2\gamma _l))$. Therefore, to reduce the critical coupling strength, only $\bar{\delta }$, $\mathop {max}\limits _{l}({\varvec{u}^{lT} D\varvec{u}^l}/(2\gamma _l))$ can be reduced. $\bar{\delta }$ and $\mathop {max}\limits _{l}({\varvec{u}^{lT} D\varvec{u}^l}/(2\gamma _l))$ are further discussed below.

Consider $\bar{\delta }$ first, let $\varvec{\varOmega }=\sum _{j=1}^{N}{\alpha _j \varvec{u}^j}$, then $\alpha _1=\varvec{\varOmega }^T \varvec{u}^1=\sum _{j=1}^{N}{\varOmega _j/\sqrt{N}}=0$, $\varvec{\phi }=\bar{d}L^\dagger \varvec{\varOmega }=\sum _{j=2}^{N}{\alpha _j\varvec{u}^j/\gamma _j}$, according to Eq. (A.10):

$$\begin{aligned} \bar{\delta } =\frac{2}{N\bar{d}}{\varvec{\phi }}^T L \varvec{\phi }=\frac{2}{N\bar{d}}\sum _{j=2}^{N}\frac{\alpha _j^2}{\gamma _j}. \end{aligned}$$

(A.12)

Because $\gamma _N>= \cdots>=\gamma _2 > \gamma _1 = 0$. To make $\sum _{j=2}^{N}{\alpha _j^2}/{\gamma _j}$ smaller, we only need to make $\sum _{j=2}^{N}{\alpha _j^2 \gamma _j}$ larger, As $\gamma _j$ and $\varvec{u}^j$ are the eigenvalues and eigenvectors of L, respectively, so $\sum _{j=2}^{N}{\alpha _j^2 \gamma _j}=\varvec{\varOmega }^T L \varvec{\varOmega }=0.5\sum _{i=1}^{N}\sum _{j=1}^{N}{A_{ij}(\omega _i-\omega _j)^2}$. Therefore, to make $\bar{\delta }$ smaller, nodes with larger frequency difference should be connected as much as possible. So we get the first structural indicator $S_1$, which is the average value of the frequency difference of the connected nodes:

$$\begin{aligned} S_1=\frac{1}{N\bar{d}}\sum _{i=1}^{N}\sum _{j=1}^{N}{A_{ij}(\omega _i-\omega _j)^2} =\frac{2}{N\bar{d}}\sum _{j=2}^{N}{\alpha _j^2 \gamma _j}.\qquad \end{aligned}$$

(A.13)

When $S_1$ is larger, $\bar{\delta }$ is smaller.

Next, let us consider $\mathop {max}\limits _{l}({\varvec{u}^{lT} D\varvec{u}^l}/(2\gamma _l))$, to make $\mathop {max}\limits _{l}({\varvec{u}^{lT} D\varvec{u}^l}/(2\gamma _l))$ small, every element in D should be as close as possible to 0, that is, the difference value of the connected nodes $\phi$ is approximately equal, let $|\phi _j-\phi _i|\approx \varTheta , A_{ij}=1$. When the coupling strength k is large, the system is completely synchronous state, $\dot{\theta _i}(t) = \bar{\omega }$, and Eq. (1) can be reduced to

$$\begin{aligned} \varOmega _i=\frac{k}{\bar{d}}\sum _{j=1}^{N}{A_{ij}\sin (\phi _j-\phi _i)}= \frac{k}{\bar{d}}d_ir_i\sin (\varPsi _i-\phi _i),\qquad \end{aligned}$$

(A.14)

where $r_i e^{I\varPsi _i}=({1}/{d_i})\sum _{j=1}^{N}{A_{ij}e^{I\phi _j}}, I=\sqrt{-1}$. When $S_1$ is large, the frequency difference of connected nodes is large, and $\phi _j$ of nodes connected with ith node are distributed on the same side of $\phi _i$. Otherwise, if $\phi _{j_1}$ and $\phi _{j_2}$ are distributed on both sides of $\phi _i$ and $S_1$ is larger, $A_{j_1 j_2}=1$ and $|\phi _{j_1}-\phi _{j_2}| \approx 2|\phi _{j_1}-\phi _i| \approx 2\varTheta$ are contradictory to the former condition $|\phi _j-\phi _i|\approx \varTheta , A_{ij}=1$. So $|\sin (\varPsi _i-\phi _i)|=\bar{d}|\varOmega _i|/(k d_i r_i) \approx \sin (\varTheta ) \approx \bar{d}|\varOmega _j|/(k d_j r_j)$, while $r_i$, $r_j$ is approximately equal during synchronization, $\bar{d}|\varOmega _i|/d_i \approx \bar{d}|\varOmega _j|/d_j$, that is, the distance from node frequency to average frequency is proportional to node degree. Let $\mu _i=\bar{d}|\varOmega _i|/d_i$, to make $\mu _i$ approximate, we only need to make $\mathop {max}\limits _{i}(\mu _i)$ as small as possible. So we get the second structural indicator $S_2$:

$$\begin{aligned} S_2=\mathop {max}\limits _{i}(\mu _i)=\mathop {max}\limits _{i}(\bar{d}|\varOmega _i|/d_i).\qquad \end{aligned}$$

(A.15)

When $S_1$ is large and $S_2$ is small, $\mathop {max}\limits _{l}({\varvec{u}^{lT} D\varvec{u}^l}/(2\gamma _l))$ can be small. Therefore, we obtain two structural indices $S_1$ and $S_2$ from Eq. (A.13) and Eq. (A.15). When $S_1$ is larger and $S_2$ is smaller, the critical coupling strength $k_c$ is smaller.