  In spite of the recent interest and advances in linear controllability of
complex networks, controlling nonlinear network dynamics remains to be an
outstanding problem. We develop an experimentally feasible control framework
for nonlinear dynamical networks that exhibit multistability (multiple
coexisting final states or attractors), which are representative of, e.g., gene
regulatory networks (GRNs). The control objective is to apply parameter
perturbation to drive the system from one attractor to another, assuming that
the former is undesired and the latter is desired. To make our framework
practically useful, we consider RESTRICTED parameter perturbation by imposing
the following two constraints: (a) it must be experimentally realizable and (b)
it is applied only temporarily. We introduce the concept of ATTRACTOR NETWORK,
in which the nodes are the distinct attractors of the system, and there is a
directional link from one attractor to another if the system can be driven from
the former to the latter using restricted control perturbation. Introduction of
the attractor network allows us to formulate a controllability framework for
nonlinear dynamical networks: a network is more controllable if the underlying
attractor network is more strongly connected, which can be quantified. We
demonstrate our control framework using examples from various models of
experimental GRNs. A finding is that, due to nonlinearity, noise can
counter-intuitively facilitate control of the network dynamics.
