Optimal Target Control of Complex Networks

A control strategy based on a general cost function is defined for dynamical complex networks that is optimal in terms of the magnitudes of the input, the state, and their time derivatives. Here we show that by controlling a network’s output rather than the state of every node, the required energy to control the network can be reduced substantially. In particular, by only targeting a subset of the nodes in the network, the energy requirements exponentially decay, suggesting that large networks can be controlled by a relatively small number of inputs as long as the target set is appropriately sized. We also show that the solution to the minimum energy control problem well approximates the energy required for optimal control strategies with respect to a large family of cost objectives. Thus the benefits of target control extend beyond the minimum energy control scheme considered in previous work. We validate our conclusions in model and real networks to arrive at an energy scaling law to better design control objectives regardless of system size, energy restrictions, state restrictions, driver node choices and target node choices. Recent years have witnessed increased interest from the scientific community on the control of complex dynamical networks [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Some common types of networks examined throughout the literature are power grids, communication networks, gene regulatory networks, neuronal systems, food webs, and social systems. Recent works [6, 8, 9, 10, 11, 12, 14] have explored our ability to control these systems when control signals are injected into a subset of the network nodes (the ‘driver nodes’). If the network is controllable, the control signals, designed to attain the desired control goal in a predefined time, are not unique. An important metric to characterize these control strategies is the energy that each strategy requires. In general, for controllable networks, it is possible to define the particular control law that minimizes the control energy, i.e., the minimum energy control strategy (MECS). The MECS provides an energetic theoretical limit, as any alternative strategy will necessarily require more energy. Therefore, knowledge of the minimum control energy is crucial in order to understand how expensive it can be to control a given network. Recent work [15, 16] has shown that based on the underlying network structure and on the set of driver nodes, the minimum energy to control a network can span a broad range of orders of magnitude, depending on the specific action required. With respect to the existing literature, here we advance the current understanding of the control of complex networks in two important directions: 1. We consider more general control objectives. While Refs. [15, 16] have investigated the MECS for dynamical networks, here we extend this type of analysis to a much broader family of control objectives. 1 ar X iv :1 60 3. 07 66 0v 2 [ cs . S Y ] 3 1 M ar 2 01 6 2. We extend our optimal control analysis to the more general case of output control and target control. In real applications involving large complex networks, achieving control over all the network nodes is often unfeasible and ultimately unnecessary [17, 14]. Two recent papers [15, 16] have investigated the minimum control energy for large complex networks and found that depending on a number of factors, such as the choice of the driver nodes and the desired final state, the minimum control energy can be extremely large. What appears to be a more reasonable goal is that of affecting a subset of the network nodes, chosen as the target of the control action. This type of control action is typically what is needed in applications in gene regulatory networks, financial networks, and social systems [17, 14]. Our main results are the following: R1. For a broad range of control objectives (involving the magnitude of the control inputs, of the nodes’ states, and of their time-derivatives) the order of magnitude of the required control energy is still the same as that of the MECS. R2. The minimum control energy rapidly (exponentially) decreases as the number of target nodes is reduced, making target control of complex networks feasible.