Abstract
The energy needed in controlling a complex network is a problem of practical importance. Recent works have focused on the reduction of control energy either via strategic placement of driver nodes, or by decreasing the cardinality of nodes to be controlled. However, optimizing control energy with respect to target nodes selection has yet been considered. In this work, we propose an iterative method based on Stiefel manifold optimization of selectable target node matrix to reduce control energy. We derive the matrix derivative gradient needed for the search algorithm in a general way, and search for target nodes which result in reduced control energy, assuming that driver nodes placement is fixed. Our findings reveal that the control energy is optimal when the path distances from driver nodes to target nodes are minimized. We corroborate our algorithm with extensive simulations on elementary network topologies, random and scale-free networks, as well as various real networks. The simulation results show that the control energy found using our algorithm outperforms heuristic selection strategies for choosing target nodes by a few orders of magnitude. Our work may be applicable to opinion networks, where one is interested in identifying the optimal group of individuals that the driver nodes can influence.
Highlights
The energy needed in controlling a complex network is a problem of practical importance
Using (7) and (8) and applying it to the trace-constraint-based projected gradient method (TPGM) algorithm, we solve for the energy-optimal target node matrix and obtain the optimal solution, C∗
Based on experimentation with elementary topologies, we find that numerical contribution characteristics similar to row b usually correspond to an optimal target node
Summary
The energy needed in controlling a complex network is a problem of practical importance. The motivation to study and understand these complex systems can be traced to our desire to obtain control over them[6] In this case, control refers to exerting influence on the networked system via external control signals to steer the state vector of the networked system from its arbitrary initial, to a predefined goal state vector in finite time [t0, tf ]6. Sun and Motter explored the numeric success rate of network controllability when numerically computing the controllability Gramian matrix when using energy optimal control s ignal[9,10] to steer the network[11] They found that for a complex network with more than a handful of nodes, using the minimum driver node set is computationally insufficient as the computation of the controllability Gramian will become ill-conditioned or nearly singular. Attaching additional control signals onto a networked system beyond the minimum driver node set is one way to achieve network control with reduced energy cost
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