A unified framework for synchronization optimization in directed multiplex networks

摘要

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.