We consider continuous time Hopfield-like recurrent networks as dynamical
models for gene regulation and neural networks. We are interested in networks
that contain n high-degree nodes preferably connected to a large number of Ns
weakly connected satellites, a property that we call n/Ns-centrality. If the
hub dynamics is slow, we obtain that the large time network dynamics is
completely defined by the hub dynamics. Moreover, such networks are maximally
flexible and switchable, in the sense that they can switch from a globally
attractive rest state to any structurally stable dynamics when the response
time of a special controller hub is changed. In particular, we show that a
decrease of the controller hub response time can lead to a sharp variation in
the network attractor structure: we can obtain a set of new local attractors,
whose number can increase exponentially with N, the total number of nodes of
the nework. These new attractors can be periodic or even chaotic. We provide an
algorithm, which allows us to design networks with the desired switching
properties, or to learn them from time series, by adjusting the interactions
between hubs and satellites. Such switchable networks could be used as models
for context dependent adaptation in functional genetics or as models for
cognitive functions in neuroscience
