Controllability Metrics and Algorithms for Complex Networks

Abstract—This paper studies the problem of controlling com-plex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to steer the network to a target state. For this problem (i) we propose a metric to quantify the difficulty of the control problem as a function of the required control energy, (ii) we derive bounds based on the system dynamics (network topology and weights) to characterize the tradeoff between the control energy and the number of control nodes, and (iii) we propose a distributed strategy with performance guarantees for the control of complex networks. In our strategy we select control nodes by relying on network partitioning, and we design the control input by leveraging optimal and distributed control techniques. Our findings show for instance that (i) if the number of control nodes is constant, then the control energy increases exponentially with the number of the network nodes, (ii) if the number of control nodes is a fixed fraction of the network nodes, then certain networks can be controlled with constant energy independently of the network dimension, and (iii) clustered networks may be easier to control because, for sufficiently many control nodes, the control energy depends only on the controllability properties of the clusters and on their coupling strength. We validate our results with examples from power networks, social networks, and epidemics spreading. I