Optimizing target nodes selection for the control energy of directed complex networks

TL;DR

This paper introduces an iterative optimization method to select target nodes in directed complex networks, significantly reducing control energy by minimizing path distances from driver nodes, with broad applicability.

Contribution

It presents a novel Stiefel manifold-based algorithm for target node selection that outperforms heuristic strategies in reducing control energy.

Findings

Control energy is minimized when target nodes are close to driver nodes.

The proposed algorithm outperforms heuristics by several orders of magnitude.

Effective in various network topologies including real-world networks.

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.