Abstract:
We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two
types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other
is the interlayer connection between the layers. Here all types of interactions within and between the layers are
allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of
the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer
synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal
and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish
after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for
local stability of the interlayer synchronization state using master stability function approach and a sufficient
condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that
intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections.
Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring
probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.
