  We study the interaction between a network designer and an adversary over a
dynamical network. The network consists of nodes performing continuous-time
distributed averaging. The adversary strategically disconnects a set of links
to prevent the nodes from reaching consensus. Meanwhile, the network designer
assists the nodes in reaching consensus by changing the weights of a limited
number of links in the network. We formulate two Stackelberg games to describe
this competition where the order in which the players act is reversed in the
two problems. Although the canonical equations provided by the Pontryagin's
maximum principle seem to be intractable, we provide an alternative
characterization for the optimal strategies that makes connection to potential
theory. Finally, we provide a sufficient condition for the existence of a
saddle-point equilibrium for the underlying zero-sum game.
