Abstract
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.
Highlights
Inter-layer synchronization is a dynamical process occurring in multi-layer networks composed of identical nodes
The object of our study is a multiplex of two layers, formed by N identical m dimensional dynamical systems, whose states are xi, yi, ∈ m for and encoded by represented by i = 1, 2, ..., N the elements of the mN × 1 column vectors X = [x1, x2, ..., xN]T and Y = [y1, y2, ..., yN]T with
The chaotic dynamics of the Rössler-like circuits is well approximated by an electronic model, where the nodes’ state variables x, y, z are translated into three different voltages ν1, ν2 and ν3 that evolve in time as follows: Figure 6
Summary
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. The study of such inter-layer synchronization has been restricted to the case in which all layers have an identical connectivity structure. In the majority of these studies, the multi-layer structure of connections supports a global synchronous state in which all the nodes in all the layers behave coherently. We have provided analytical, numerical and experimental evidence of inter-layer synchronization[21], based on a version of the MSF for reducing the system dimensionality, and under the assumption that different layers are structurally identical.
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