Analysis of the synchronizability of two-layer chain networks with two inter-layer edges

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The structure and dynamics of multi-layer networks are the frontiers in the current network science research. The inter-layer connections in a multi-layer network directly affect the dynamics, propagation, diffusion, and synchronization of the entire multi-layer network. However, there is rarely systematic theory or method for the general structure of multi-layer networks. People can only start with a simple structure to explore how the inter-layer connections affect the overall dynamic behaviors of multiple layers. The work of this paper aims at the two-layer networks where each layer is a chain network. Through graph theory and numerical calculation, this problem is explored to find the optimization method of the inter-layer connections. First of all, according to the master stability function method, this paper proves that when the synchronous region is unbounded, a multi-layer network with the same network structure at each layer has a certain inter-layer coupling strength threshold, so that the entire network reaches the maximum synchronizability, which is the synchronizability of its single-layer network. Secondly, the simulations show that when there are only two edges between layers, the optimal positions are respectively located at $1/4$ and $3/4$ of each chain, while the worst positions are the end node of the chain and its neighbor node. Furthermore, we discuss in detail about the optimal connection modes of two-layer chain networks when the number of single-layer nodes is $4N+x~(x=0,~1,~2,~3)$ using the master stability function method and “network shortest distance” structure index. In the case that the single-layer nodes number is $4N$, there are $4$ optimal connection modes; in the other three cases, there is only one optimal connection mode respectively. Finally, we propose the expression of the optimal inter-layer coupling strength, and then verify the correctness of the expression by calculating the optimal inter-layer coupling strength values of the two-layer chain network with different sizes.