Nodal Dynamics, Not Degree Distributions, Determine the Structural Controllability of Complex Networks

Noah J Cowan,Carl T Bergstrom,James S Freudenberg,Daril A Vilhena,Erick J Chastain,Frank Emmert-Streib

doi:10.1371/journal.pone.0038398

Abstract

Structural controllability has been proposed as an analytical framework for making predictions regarding the control of complex networks across myriad disciplines in the physical and life sciences (Liu et al., Nature:473(7346):167–173, 2011). Although the integration of control theory and network analysis is important, we argue that the application of the structural controllability framework to most if not all real-world networks leads to the conclusion that a single control input, applied to the power dominating set, is all that is needed for structural controllability. This result is consistent with the well-known fact that controllability and its dual observability are generic properties of systems. We argue that more important than issues of structural controllability are the questions of whether a system is almost uncontrollable, whether it is almost unobservable, and whether it possesses almost pole-zero cancellations.

Highlights

Both conclusions hinge on a critical assumption of the model in [10]: the results (implicitly) require that the ‘‘default’’ structures of the dynamical systems at the nodes of the network have infinite time constants
How can we control complex networks of dynamical systems [1,2,3,4,5,6,7,8,9]? Is it sufficient to control a few nodes, or are inputs needed at a large fraction of the nodes in the network? Which nodes need to be controlled? A recent paper [10] suggests that we can address these problems using the concept of structural controllability [11], and in doing so we may be able to forge new connections between control theory and complex networks
Assuming arbitrary linear dynamics, we show here that (1) a single time-dependent input is all that is needed for structural controllability, and (2) that this input should be applied to the power dominating set (PDS) [16] of the network

Summary

Introduction

Both conclusions hinge on a critical assumption of the model in [10]: the results (implicitly) require that the ‘‘default’’ structures of the dynamical systems at the nodes of the network have infinite time constants. The assumption of an infinite time constant implies that a certain parameter in the mathematical model of the system is equal to zero, and that term is off-limits as far as structural controllability is concerned. We show in this paper that all networks with finitedimensional linear dynamics (save a special set of parameters of zero measure) are controllable with a single input.

Results

Conclusion