  The mathematical framework of multiplex networks has been increasingly
realized as a more suitable framework for modelling real-world complex systems.
In this work, we investigate the optimization of synchronizability in multiplex
networks by evolving only one layer while keeping other layers fixed. Our main
finding is to show the conditions under which the efficiency of convergence to
the most optimal structure is almost as good as the case where both layers are
rewired during an optimization process. In particular, inter-layer coupling
strength responsible for the integration between the layers turns out to be
crucial factor governing the efficiency of optimization even for the cases when
the layer going through the evolution has nodes interacting much weakly than
those in the fixed layer. Additionally, we investigate the dependency of
synchronizability on the rewiring probability which governs the network
structure from a regular lattice to the random networks. The efficiency of the
optimization process preceding evolution driven by the optimization process is
maximum when the fixed layer has regular architecture, whereas the optimized
network is more synchronizable for the fixed layer having the rewiring
probability lying between the small-world transition and the random structure.
