Many complex biological and technological systems can be represented
by multilayer networks where the nodes are coupled via several
independent networks. Despite its significance from both the
theoretical and the application perspectives, synchronization in
multilayer networks and its dependence on the network topology remain
poorly understood. In this talk, we will present a universal
connection graph-based method which opens up the possibility of
explicitly assessing critical multilayer-induced interactions which
can hamper network synchronization. The method reveals striking,
counterintuitive effects caused by multilayer coupling. It
demonstrates that a coupling which is favorable to synchronization in
single-layer networks can reverse its role and destabilize
synchronization when used in a multilayer network. This property is
controlled by the traffic load on a given edge when the replacement of
a lightly loaded edge in one layer with a coupling from another layer
can promote synchronization, but a similar replacement of a highly
loaded edge can break synchronization, forcing a "good" link to go
"bad."  We will also discuss open problems and challenges in studying
cooperative dynamics of multilayer networks.