  This paper studies the resilience of second-order networked dynamical systems
to strategic attacks. We discuss two widely used control laws, which have
applications in power networks and formation control of autonomous agents. In
the first control law, each agent receives pure velocity feedback from its
neighbor. In the second control law, each agent receives its velocity relative
to its neighbors. The attacker selects a subset of nodes in which to inject a
signal, and its objective is to maximize the $\mathcal{H}_2$ norm of the system
from the attack signal to the output. The defender improves the resilience of
the system by adding self-feedback loops to certain nodes of the network with
the objective of minimizing the system's $\mathcal{H}_2$ norm. Their decisions
comprise a strategic game. Graph-theoretic necessary and sufficient conditions
for the existence of Nash equilibria are presented. In the case of no Nash
equilibrium, a Stackelberg game is discussed, and the optimal solution when the
defender acts as the leader is characterized. For the case of a single attacked
node and a single defense node, it is shown that the optimal location of the
defense node for each of the control laws is determined by a specific network
centrality measure. The extension of the game to the case of multiple attacked
and defense nodes is also addressed.
