Minimum energy control for complex networks

TL;DR

This paper investigates the minimal energy required to control complex networks, revealing how eigenvalues influence control energy and proposing algorithms and heuristics for optimal driver node selection.

Contribution

It introduces new insights into control energy dependence on eigenvalues and develops algorithms for driver node selection in complex networks.

Findings

Control energy decreases as eigenvalues approach the imaginary axis.

Algorithms for driver node selection are effective in networks with purely imaginary eigenvalues.

Controlling nodes with high outdegree to indegree ratio reduces overall control cost.

Abstract

The aim of this paper is to shed light on the problem of controlling a complex network with minimal control energy. We show first that the control energy depends on the time constant of the modes of the network, and that the closer the eigenvalues are to the imaginary axis of the complex plane, the less energy is required for complete controllability. In the limit case of networks having all purely imaginary eigenvalues (e.g. networks of coupled harmonic oscillators), several constructive algorithms for minimum control energy driver node selection are developed. A general heuristic principle valid for any directed network is also proposed: the overall cost of controlling a network is reduced when the controls are concentrated on the nodes with highest ratio of weighted outdegree vs indegree.