Controllability Metrics, Limitations and Algorithms for Complex Networks

TL;DR

This paper introduces new metrics, bounds, and algorithms to analyze and improve the controllability of complex networks, highlighting limitations and properties based on network structure and control node placement.

Contribution

It proposes a controllability difficulty metric, derives bounds relating control energy and control nodes, and develops an open-loop control strategy with performance guarantees.

Findings

Control energy can grow exponentially with network size if control nodes are fixed.

Certain networks can be controlled with constant energy if control nodes are a fixed fraction of the network.

Clustered networks may be easier to control due to their structure.

Abstract

This paper studies the problem of controlling complex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to steer a network to a target state. For this problem (i) we propose a metric to quantify the difficulty of the control problem as a function of the required control energy, (ii) we derive bounds based on the system dynamics (network topology and weights) to characterize the tradeoff between the control energy and the number of control nodes, and (iii) we propose an open-loop control strategy with performance guarantees. In our strategy we select control nodes by relying on network partitioning, and we design the control input by leveraging optimal and distributed control techniques. Our findings show several control limitations and properties. For instance, for Schur stable and symmetric networks: (i) if the number of control nodes is constant, then the control energy increases exponentially with the number of network nodes, (ii) if the number of control nodes is a fixed fraction of the network nodes, then certain networks can be controlled with constant energy independently of the network dimension, and (iii) clustered networks may be easier to control because, for sufficiently many control nodes, the control energy depends only on the controllability properties of the clusters and on their coupling strength. We validate our results with examples from power networks, social networks, and epidemics spreading.