Irrelevance of linear controllability to nonlinear dynamical networks

TL;DR

This paper investigates the relevance of linear controllability to nonlinear networks, revealing that importance rankings differ significantly and linear controllability is largely irrelevant for certain nonlinear systems like biological networks.

Contribution

It demonstrates that linear controllability does not accurately reflect node importance in nonlinear networks, challenging previous assumptions and applications.

Findings

Nodal importance in nonlinear networks correlates with large-degree nodes.

Linear controllability importance is tilted towards small-degree nodes.

Linear controllability is largely irrelevant for biological nonlinear networks.

Abstract

There has been tremendous development of linear controllability of complex networks. Real-world systems are fundamentally nonlinear. Is linear controllability relevant to nonlinear dynamical networks? We identify a common trait underlying both types of control: the nodal "importance." For nonlinear and linear control, the importance is determined, respectively, by physical/biological considerations and the probability for a node to be in the minimum driver set. We study empirical mutualistic networks and a gene regulatory network, for which the nonlinear nodal importance can be quantified by the ability of individual nodes to restore the system from the aftermath of a tipping-point transition. We find that the nodal importance ranking for nonlinear and linear control exhibits opposite trends: for the former large-degree nodes are more important but for the latter, the importance scale is tilted towards the small-degree nodes, suggesting strongly irrelevance of linear controllability to these systems. The recent claim of successful application of linear controllability to C. elegans connectome is examined and discussed.