Individual dynamics induces symmetry in network controllability

TL;DR

This paper reveals a global symmetry in network controllability influenced by the diversity and density of dynamic units, showing optimal control when different types are equally represented.

Contribution

It introduces a theoretical framework demonstrating the invariance of controllability under exchange of dynamic unit densities, regardless of network topology.

Findings

Controllability is invariant under swapping densities of different dynamic units.

Maximum controllability occurs when different dynamic types are equally dense.

Minimum controllability is linked to uniform presence or absence of self-loops.

Abstract

Controlling complex networked systems to a desired state is a key research goal in contemporary science. Despite recent advances in studying the impact of network topology on controllability, a comprehensive understanding of the synergistic effect of network topology and individual dynamics on controllability is still lacking. Here we offer a theoretical study with particular interest in the diversity of dynamic units characterized by different types of individual dynamics. Interestingly, we find a global symmetry accounting for the invariance of controllability with respect to exchanging the densities of any two different types of dynamic units, irrespective of the network topology. The highest controllability arises at the global symmetry point, at which different types of dynamic units are of the same density. The lowest controllability occurs when all self-loops are either completely absent or present with identical weights. These findings further improve our understanding of network controllability and have implications for devising the optimal control of complex networked systems in a wide range of fields.