Energy cost for controlling complex networks

TL;DR

This paper investigates the energy required to control complex networks, deriving bounds on the minimum control energy and providing analytical insights to facilitate cost-effective network control strategies.

Contribution

It presents the first analytical derivation of the upper bound scaling behavior of the minimum control energy in complex networks, applicable to various network types and input configurations.

Findings

Analytical expression for the upper bound of control energy scaling.

Validation of theoretical results through numerical simulations.

Framework applicable to networks with any number of input nodes.

Abstract

The controllability of complex networks has received much attention recently, which tells whether we can steer a system from an initial state to any final state within finite time with admissible external inputs. In order to accomplish the control in practice at the minimum cost, we must study how much control energy is needed to reach the desired final state. At a given control distance between the initial and final states, existing results present the scaling behavior of lower bounds of the minimum energy in terms of the control time analytically. However, to reach an arbitrary final state at a given control distance, the minimum energy is actually dominated by the upper bound, whose analytic expression still remains elusive. Here we theoretically show the scaling behavior of the upper bound of the minimum energy in terms of the time required to achieve control. Apart from validating the analytical results with numerical simulations, our findings are feasible to the scenario with any number of nodes that receive inputs directly and any types of networks. Moreover, more precise analytical results for the lower bound of the minimum energy are derived in the proposed framework. Our results pave the way to implement realistic control over various complex networks with the minimum control cost.