  The problem of achieving consensus in a network of connected systems arises
in many science and engineering applications. In contrast to previous works, we
focus on the system reactivity, i.e., the initial amplification of the norm of
the system states. We identify a class of networks that we call minimally
reactive, which are such that the indegree and the outdegree of each node of
the network are the same. We propose several optimization procedures in which
minimum perturbations (links or link weights) are imposed on a given network
topology to make it minimally reactive. A new concept of structural reactivity
is introduced which measures how much a given network is far from becoming
minimally reactive by link perturbations. The structural reactivity of directed
random graphs is studied.
