Complex Systems Group &GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.
Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain.
Sci Rep. 2017 Apr 4;7:45475. doi: 10.1038/srep45475.
Inter-layer synchronization is a dynamical process occurring in multi-layer networks composed of identical nodes. This process emerges when all layers are synchronized, while nodes in each layer do not necessarily evolve in unison. So far, the study of such inter-layer synchronization has been restricted to the case in which all layers have an identical connectivity structure. When layers are not identical, the inter-layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few links, an approximate treatment is still feasible, and allows one to gather information on whether and how the system may wander around an inter-layer synchronous configuration. We report the details of an approximate analytical treatment for a two-layer multiplex, which results in the introduction of an extra inertial term accounting for structural differences. Numerical validation of the predictions highlights the usefulness of our approach, especially for small or moderate topological differences in the intra-layer coupling. Moreover, we identify a non-trivial relationship connecting the betweenness centrality of the missing links and the intra-layer coupling strength. Finally, by the use of multiplexed layers of electronic circuits, we study the inter-layer synchronization as a function of the removed links.
层间同步是由相同节点组成的多层网络中发生的动态过程。当所有层都同步时，就会出现这种过程，而每个层中的节点不一定会协同演化。到目前为止，这种层间同步的研究仅限于所有层都具有相同连接结构的情况。当层不相同时，层间同步状态不再是系统的稳定解。然而，当层只有少数几个链路不同时，近似处理仍然是可行的，并且可以收集有关系统是否以及如何在层间同步配置周围徘徊的信息。我们报告了两层多路复用的近似分析处理的细节，这导致引入了一个额外的惯性项来解释结构差异。对预测的数值验证突出了我们方法的有用性，特别是对于层内耦合中的小或中等拓扑差异。此外，我们确定了一个非平凡的关系，将缺失链路的中间中心度与层内耦合强度联系起来。最后，通过使用电子电路的多路复用层，我们研究了作为去除链路函数的层间同步。
