Yang-Yu Liu, Northeastern University "Taming Complexity-The mathematics of network control"

Abstract: The ultimate proof of our understanding of natural or technological systems is reflected in our ability to control them. Although control theory offers mathematical tools for steering engineered and natural systems towards a desired state, a framework to control complex self-organized systems is lacking. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system’s entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network’s degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but that dense and homogeneous networks can be controlled using a few driver nodes. Counterintuitively, we find that in both model and real systems the driver nodes tend to avoid the high-degree nodes. We also introduce the concept of control centrality to quantify the ability of a single node to control the whole network. We map the control centrality into a combinatorial optimization problem and calculate the distribution of control centrality for several real networks. We rigorously prove that in hierarchical networks the control centrality of a node is uniquely determined by its level in the hierarchy, helping us design an efficient attack strategy against the controllability of malicious networks.