  With synchronization being one of nature's most ubiquitous collective
behaviors, the field of network synchronization has experienced tremendous
growth, leading to significant theoretical developments. However, most of these
previous studies consider uniform connection weights and undirected networks
with positive coupling. In the present article, we incorporate the asymmetry in
a two-layer multiplex network by assigning the ratio of the adjacent nodes'
degrees as the weights to the intralayer edges. Despite the presence of
degree-biased weighting mechanism and attractive-repulsive coupling strengths,
we are able to find the necessary conditions for intralayer synchronization and
interlayer antisynchronization and test whether these two macroscopic states
can withstand demultiplexing in a network. During the occurrence of these two
states, we analytically calculate the oscillator's amplitude. In addition to
deriving the local stability conditions for interlayer antisynchronization via
the master stability function approach, we also construct a suitable Lyapunov
function to determine a sufficient condition for global stability. We provide
numerical evidence to show the necessity of negative interlayer coupling
strength for the occurrence of antisynchronization, and such repulsive
interlayer coupling coefficients can not destroy intralayer synchronization.
