Perfect synchronization in complex networks with higher-order interactions

Achieving perfect synchronization in a complex network, specially in the presence of higher-order interactions (HOIs) at a targeted point in the parameter space, is an interesting, yet challenging task. Here we present a theoretical framework to achieve the same under the paradigm of the Sakaguchi-Kuramoto (SK) model. We analytically derive a frequency set to achieve perfect synchrony at some desired point in a complex network of SK oscillators with higher-order interactions. Considering the SK model with HOIs on top of the scale-free, random, and small world networks, we perform extensive numerical simulations to verify the proposed theory. Numerical simulations show that the analytically derived frequency set not only provides stable perfect synchronization in the network at a desired point but also proves to be very effective in achieving a high level of synchronization around it compared to the other choices of frequency sets. The stability and the robustness of the perfect synchronization state of the system are determined using the low-dimensional reduction of the network and by introducing a Gaussian noise around the derived frequency set, respectively.

Figure 1

Perfect synchronization for different networks. Numerically computed order parameter $r$ as a function of pairwise coupling strength $K_1$ shown for phase lags $\alpha =0.1$ (blue) and 0.5 (magenta). The triadic coupling $K_2$ is set to 0.5. The green dots indicate perfect synchronization states at targeted coupling $K_1=0.1$. (a) The ER network achieves $r=1$ at $K_1=0.1$ in both forward and backward simulations. (b) and (c) The SF network with $\gamma =2.5$ and 3.2 attains $r=1$ at $K_1=0.1$ in both forward and backward simulations. (d) The SW network achieves perfect synchronization at $K_1=0.1$ in forward and backward simulations only for $\alpha =0.1$. For the other lag, the backward path only attains $r=1$.

Figure 2

Synchronization diagram for fixed $K_2=\alpha =0.5$ with natural frequencies drawn from the normal, uniform, and derived (perfect) frequency will ensure perfect synchronization at $K_1=0.1$ in different networks of size $N=1000$: (a) $r$ vs $K_1$ for the ER network; (b) and (c) $r$ vs $K_1$ for the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) $r$ vs $K_1$ for the SW network. For the green, red, and blue curves in panels (a)–(d), the natural frequencies are drawn from the normal, uniform, and derived (perfect) frequency distributions, respectively.

Figure 3

Synchronization of the SK model with HOIs for different networks. Synchronization diagrams describe the order parameter $r$ as a function of triadic coupling $K_2$ for fixed $K_1=0.1$. The blue and magenta curves present the transitions for $\alpha =0.1$ and 0.5, respectively. The green dots denote the perfect synchronization states. (a) The ER network achieves $r=1$ at $K_2=0.5$ for considered lags in both paths. (b) and (c) The SF networks with $\gamma =2.5$ and 3.2 attain perfect synchronization at $K_2=0.5$ for considered $\alpha$ values in both paths. (d) The SW network attains $r=1$ at $K_2=0.5$ only during forward transition (except $\alpha =0.1$).

Figure 4

Synchronization diagram for fixed $K_1=0.1$ and $\alpha =0.5$ with natural frequencies drawn from the normal, uniform, and derived (perfect) frequency distributions for achieving perfect synchronization at $K_2=0.5$ in different networks of size $N=1000$: (a) $r$ vs $K_2$ for the ER network; (b) and (c) $r$ vs $K_2$ for the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) $r$ vs $K_2$ for the SW network. The color convention is same as the one used in Fig. 2.

Figure 5

Synchronization diagram for fixed $K_1=0.1$ and $K_2=0.5$ with natural frequencies drawn from the normal, uniform, and derived (perfect) frequency distributions for achieving perfect synchronization at $\alpha =0.3$ in different networks of size $N=1000$: (a) $r$ vs $\alpha$ for the ER network; (b) and (c) $r$ vs $\alpha$ for the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) $r$ vs $\alpha$ for the SW network. The color convention is same as the one used in Fig. 2.

Figure 6

Distribution of the frequency for perfect synchronization using the four different networks, namely, (a) and (b) ER (blue); (c) and (d) SF with $\gamma =2.5$ (red); (e) and (f) SF with $\gamma =3.2$ (purple); and (g) and (h) SW (brown), of network size $N=1000$. In the first row the frequency is calculated for the phase lag $\alpha =0.1$ and in the second row it is calculated for the phase lag $\alpha =0.7$. We take the pairwise coupling strength $K_1^{\left(p\right)}=0.1$ and the higher-order coupling strength $K_2^{\left(p\right)}=0.5$.

Figure 7

Distribution of mean of average frequencies over 50 realizations of networks is shown as a function of network size for (a) the ER network; (b) and (c) the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) the SW network. The error bars represents the standard deviation over 50 realizations of networks and the dots on the blue curve represents the mean of the same. The insets represent the distribution of the mean of average frequencies of network sizes $N=1000$ and 3000.

Figure 8

Synchronization profile for the derived frequency set for $\alpha =0.2, \beta =0.1, K_1=0.1$, and $K_2=0.5$ for different networks of size $N=1000$: (a) $r$ vs $K_1$ for the ER network; (b) and (c) $r$ vs $K_1$ for the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) $r$ vs $K_1$ for the SW network. The green dots show the perfect synchronization states at the targeted coupling $K_1=0.1$ for all the networks.

Figure 9

Synchronization profile for the derived frequency set for $\alpha =0.2, \beta =0.1, K_1=0.1$, and $K_2=0.5$ for different networks of size $N=1000$: (a) $r$ vs $K_2$ for the ER network, (b) and (c) $r$ vs $K_2$ for the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) $r$ vs $K_2$ for the SW network. All considered networks attain $r=1$ (green dot) at the targeted point $K_2=0.5$ for both forward (red) and backward (blue) continuation.

Figure 10

Variation of $g$ obtained in Eq. (14) as a function of $\chi$ under derived frequency for the ER network. (a) Coupling $K_2$ is set to 0.5 and the values of $K_1$ are taken as 0.06 (cyan), 0.08 (green), 0.092 (magenta), 0.1 (blue), and 0.12 (red). (b) Coupling $K_1$ is fixed to 0.1 and the value of $K_2$ varies as 0.01 (cyan), 0.15 (green), 0.28 (magenta), 0.5 (blue), and 0.8 (red). (c) Derived (perfect) frequency (blue) and normally (red), and uniformly (green) distributed frequencies are plotted, showing that only the derived frequency reaches the stable synchronization state.

Figure 11

Synchronization error $\left(\rho \right)$ under deviated frequency in different networks of size $N=1000$: (a) $\rho$ vs $\sigma$ in the ER network; (b) and (c) $\rho$ vs $\sigma$ in the SF networks with $\gamma =2.5$ and 3.2, respectively; and (d) $\rho$ vs $\sigma$ in the SW network. The data represented by the blue stars are obtained from numerical simulation, and the red lines represent the analytically obtained function $\rho \sim {\sigma }^2$.

Figure 12

Order parameter $r$ is plotted as a function of pairwise coupling $K_1$ for $K_2=0.5$ and $\alpha =0.3$ and 0.7. The green dots denote the perfect synchronization states. (a) The ER network attains $r=1$ at targeted coupling $K_1=0.1$ in both forward and backward paths. (b) The SF network with $\gamma =2.5$ attains $r=1$ at $K_1=0.1$ except in the forward path for $\alpha =0.7$. (c) The SF with $\gamma =3.2$ achieves perfect synchronization at $K_1=0.1$ for considered lag values in both paths. (d) The SW network achieves perfect synchronization at $K_1=0.1$ in the backward paths only.

Figure 13

Order parameter $r$ is plotted as a function of higher-order coupling $K_2$ for $K_1=0.1$ and $\alpha =0.3$ and 0.7. The green dots denote the perfect synchronization states. (a) The ER network shows perfect synchronization at predicted coupling $K_2=0.5$ in both forward and backward directions. (b) The SF network with $\gamma =2.5$ attains $r=1$ at $K_2=0.5$ except in the forward path for lag value 0.7. (c) The SF network with $\gamma =3.2$ achieves perfect synchronization at $K_2=0.5$ for considered lag values in both paths. (d) The SW network achieves $r=1$ at $K_2=0.5$ in the forward path only for $\alpha =0.3$.

Figure 14

Distribution of the derived frequency using the four different networks, namely, (a) and (b) ER (blue); (c) and (d) SF with $\gamma =2.5$ (red); (e) and (f) SF with $\gamma =3.2$ (purple); and (g) and (h) SW (brown), taking the average across 100 realizations of network size 5000 in each case. In the first row the frequency is calculated for phase lag $\alpha =0.1$, and in the second row it is calculated for phase lag $\alpha =0.7$. The pairwise coupling strength is set to $K_1^{\left(p\right)}=0.1$ and the higher-order coupling strength is set to $K_2^{\left(p\right)}=0.5$.