  We study the spread of a novel state in a network, in the presence of an
exogenous control. The considered controlled evolutionary dynamics is a
non-homogeneous Markov process that describes the evolution of the states of
all nodes in the network. Through a rigorous analysis, we estimate the
performance of the system by establishing upper and lower bounds on the
expected time needed for the novel state to replace the original one. Such
bounds are expressed in terms of the support and intensity of the control
policy (specifically, the set of nodes that can be controlled and its energy)
and of the network topology and establish fundamental limitations on the
system's performance. Leveraging these results, we are able to classify network
structures depending on the possibility to control the system using simple
open-loop control policies. Finally, we propose a feedback control policy that,
using little knowledge of the network topology and of the system's evolution at
a macroscopic level, allows for a substantial speed up of the spreading process
with respect to simple open-loop control policies. All these theoretical
results are presented together with explanatory examples, for which Monte Carlo
simulations corroborate our analytical findings.
