A unified framework for synchronization optimization in directed multiplex networks

Abstract

The multiplex network paradigm has been instrumental in revealing many unexpected phenomena and dynamical regimes in complex interacting systems. Nevertheless, most of the current research focuses on undirected multiplex structures, whereas real-world systems predominantly involve directed interactions. Here, we present an analytical framework for attaining optimal synchronization in directed multiplex networks composed of phase oscillators, considering both frustrated and non-frustrated regimes. A multiplex synchrony alignment function (MSAF) is introduced for this purpose, whose formulation integrates structural properties and dynamical characteristics of the individual directed layers. Using this function, we derive two classes of frequency distributions: one that yields perfect synchronization at a prescribed coupling strength in the presence of phase-lag, and another that optimizes synchronization over a broad range of coupling strengths. Numerical simulations on various directed duplex topologies demonstrate that both frequency sets substantially outperform conventional distributions. We also explore network optimization through a directed link rewiring strategy aimed at minimizing the MSAF, along with a swapping algorithm for optimally assigning fixed frequencies on both layers of a given directed duplex network. Examination of synchrony-optimized directed networks uncovers three notable correlations: a positive relationship between frequency and out-degree, a negative correlation between neighboring frequencies, and an anti-correlation between mirror node frequencies across directed layers.

Figures (12)

• Figure 1: Schematic diagram of a duplex network composed of two directed layers. The dashed lines represent interlayer links between mirror nodes in the two layers.
• Figure 2: Optimal synchronization in a directed multiplex network of non-frustrated phase oscillators for three duplex configurations: SF–SF (a–c), SF–ER (d–f), and ER–ER (g–i). Panels (a), (d), and (g) show the variation of the order parameter $R_{1}$ of the first layer with coupling strength $K$; panels (b), (e), and (h) display the order parameter $R_{2}$ of the second layer; and panels (c), (f), and (i) depict the global order parameter $R$ of the entire duplex network. The red curves with open circles represent the order parameters obtained using the optimal frequency sets derived in Eq. (\ref{['eqn16']}). For comparison, the orange, magenta, and purple curves with open circles correspond to the order parameters obtained using uniform, normal, and Lorentzian distributions of the natural frequencies $\pmb{\omega}^{(l)}$, respectively.
• Figure 3: Optimal synchronization in directed frustrated multiplex networks with $\alpha^{(1)}=0.2$ and $\alpha^{(2)}=0.4$ for three duplex configurations: SF–SF (a), SF–ER (b), and ER–ER (c). In each panel, the red, green, and blue curves with open circles denote the layer-specific order parameters $R_1$ and $R_2$, and the global order parameter $R$, respectively, obtained using the optimal frequency sets from Eq. (\ref{['eqn16']}). The orange, magenta, and purple curves with open circles show the global order parameter for uniform, normal, and Lorentzian frequency distributions, respectively, serving as a benchmark for comparison.
• Figure 4: Perfect synchronization in directed frustrated multiplex networks with $\alpha^{(1)}=0.2$ and $\alpha^{(2)}=0.4$, targeted at $K_{\mathrm{p}}=0.3$, for three duplex configurations: SF–SF (a), SF–ER (b), and ER–ER (c). In each panel, the red, green, and blue curves denote the layer-specific order parameters $R_1$ and $R_2$, and the global order parameter $R$, respectively, obtained using the perfect frequency sets from Eq. (\ref{['eqn14']}). The orange, magenta, and purple curves correspond to the global order parameter $R$ for uniform, normal, and Lorentzian frequency distributions, respectively.
• Figure 5: Perfect synchronization in directed frustrated multiplex networks $(\alpha^{(1)}=0.2,\alpha^{(2)}=0.4)$ at the targeted coupling strength $K_{\mathrm{p}}=0.5$ for three duplex configurations: SF–SF (a), SF–ER (b), and ER–ER (c). Red, green, and blue curves denote $R_1$, $R_2$, and $R$, respectively, obtained using the perfect frequency sets from Eq. (\ref{['eqn14']}).