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Irrelevance of linear controllability to nonlinear dynamical networks

Junjie Jiang,Y. Lai

Published 2019 in Nature Communications

ABSTRACT

There has been tremendous development in 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 the irrelevance of linear controllability to these systems. The recent claim of successful application of linear controllability to Caenorhabditis elegans connectome is examined and discussed. Linear controllability theories have stimulated research on control of complex networks. Here the authors investigate the concordance between linear and nonlinear approaches in ranking the importance of nodes in nonlinear networks, and conclude that linear controllability may not be applicable.

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