  We analyze the ordering efficiency and the structure of disordered
configurations for the zero-temperature Glauber model on Watts-Strogatz
networks obtained by rewiring 2D regular square lattices. In the small-world
regime, the dynamics fails to reach the ordered state in the thermodynamic
limit. Due to interplay of the perturbed regular topology and the energy
neutral stochastic state transitions, the stationary state consists of two
intertwined domains, manifested as multi-clustered states on the original
lattice. Moreover, for intermediate rewiring probabilities, one finds an
additional source of disorder due to the low connectivity degree, which gives
rise to small isolated droplets of spins. We also examine the ordering process
in paradigmatic two-layer networks with heterogeneous rewiring probabilities.
Comparing the cases of a multiplex network and the corresponding network with
random inter-layer connectivity, we demonstrate that the character of the final
state qualitatively depends on the type of inter-layer connections.
