Abstract
Relay synchronization in multi-layer networks implies inter-layer synchronization between two indirectly connected layers through a relay layer. In this work, we study the relay synchronization in a three-layer multiplex network by introducing degree-based weighting mechanisms. The mechanism of within-layer connectivity may be hubs-repelling or hubs-attracting whenever low-degree or high-degree nodes receive strong influence. We adjust the remote layers to hubs-attracting coupling, whereas the relay layer may be unweighted, hubs-repelling, or hubs-attracting network. We establish that relay synchronization is improved when the relay layer is hubs-repelling compared to the other cases. We determine analytically necessary stability conditions of relay synchronization state using the master stability function approach. Finally, we explore the relation between synchronization and the topological property of the relay layer. We find that a higher clustering coefficient hinders synchronizability, and vice versa. We also look into the intra-layer synchronization in the proposed weighted triplex network and establish that intra-layer synchronization occurs in a wider range when relay layer is hubs-attracting.
Highlights
We have explicitly addressed the emergence of relay synchronization in weighted triplex networks of chaotic non-linear oscillators
The current study evidences an enhancement in the relay synchronization if the middle layer exhibits hubs-repelling interactions compared to hubs-attracting and original unweighted couplings
We have supported numerical findings by analytical treatment of the synchronized state stability conducted via master stability function
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The lack of works on the synchronization in weighted multi-layer networks encouraged us to unveil the effect of weighted networks upon the relay synchronization in a multiplex network In this regard, we would like to highlight the very recent works by Estrada et al [52,53], in which the authors have introduced two opposite approaches to explore the degree-biased mechanisms on the network dynamics based on the generalization of the graph Laplacian operator. The low-degree units produce more potent effects in the second case, illustrated by the hubs-repelling Laplacian [53] These findings have motivated us to dig into the comparison between the weighted directed networks generated by different degree-based weighting mechanisms to achieve synchronization in a relay multiplex network. Apart from this, we examine the emergence of intra-layer synchrony for all three configurations
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