Spectrum of Controlling and Observing Complex Networks

TL;DR

This paper investigates how the number of driver nodes affects control energy and observational uncertainty in complex networks, revealing fundamental laws that guide efficient control and observation strategies.

Contribution

It provides a comprehensive analysis of control energy variability based on driver node count, offering new laws for controlling complex networks efficiently.

Findings

Control energy increases sublinearly when all nodes are driven.

Controlling through a single node can be exponentially costly.

Increasing driver nodes exponentially decreases maximum control energy.

Abstract

Observing and controlling complex networks are of paramount interest for understanding complex physical, biological and technological systems. Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. Yet, the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for different directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving to a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.