  Some aspects of nonlocal dynamics on directed and undirected networks for an
initial value problem whose Jacobian matrix is a variable-order fractional
power of a Laplacian matrix are discussed here. This is a new extension to
non-stationary behavior of a class of non-local phenomena on complex networks
for which both directed and undirected graphs are considered. Under appropriate
assumptions, the existence, uniqueness, and uniform asymptotic stability of the
solutions of the underlying initial value problem are proved. Some examples
giving a sample of the behavior of the dynamics are also included.
