  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 goal of the network designer is to assist the nodes
reach consensus by changing the weights of a limited number of links in the
network. Meanwhile, an adversary strategically disconnects a set of links to
prevent the nodes from converging. We formulate two problems to describe this
competition where the order in which the players act is reversed in the two
problems. We utilize Pontryagin's Maximum Principle (MP) to tackle both
problems and derive the optimal strategies. Although the canonical equations
provided by the MP are intractable, we provide an alternative characterization
for the optimal strategies that highlights a connection with potential theory.
Finally, we provide a sufficient condition for the existence of a saddle-point
equilibrium (SPE) for this zero-sum game.
