Synchronization ability of two-layer chain network: Directed vs undirected

This study systematically investigates the synchronization ability of two-layer networks with identical chain structures, with particular focus on the influence of inter-layer edges. For undirected chain networks, we identify the optimal placement of two directed inter-layer edges (co-directional or reverse-directional) that maximize synchronization ability and analyze how synchronization ability varies with inter-layer coupling strength for directed and undirected edges at these optimal positions. Extending to two-layer directed chain networks, we analyze how the number of inter-layer edges (whether directed or undirected) impacts synchronization ability, contrast the coupling-strength-dependent synchronization ability between networks configured with two undirected vs two directed inter-layer edges, and examine the effect of edge positional changes with a fixed number of inter-layer edges. The findings offer theoretical guidance for optimizing synchronization through strategic structural design in multi-layer chain networks.