### Abstract

Controlling complex networks has become a forefront research area in network science and engineering.
Recent efforts have led to theoretical frameworks of controllability to fully control a network through
steering a minimum set of driver nodes. However, in realistic situations not every node is accessible or can
be externally driven, raising the fundamental issue of control efficacy: if driving signals are applied to an
arbitrary subset of nodes, how many other nodes can be controlled? We develop a framework to determine
the control efficacy for undirected networks of arbitrary topology. Mathematically, based on non-singular
transformation, we prove a theorem to determine rigorously the control efficacy of the network and to
identify the nodes that can be controlled for any given driver node. Physically, we develop the picture of
diffusion that views the control process as a signal diffused from input signals to the set of controllable
nodes. The combination of mathematical theory and physical reasoning allows us not only to determine the
control efficacy for model complex networks and a large number of empirical networks, but also to uncover
phenomena in network control, e.g., hub nodes in general possess lower control centrality than an average
node in undirected networks.

Original language | English |
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Article number | 28037 |

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Scientific Reports |

Volume | 6 |

DOIs | |

Publication status | Published - 21 Jun 2016 |

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## Cite this

Gao, X-D., Wang, W-X., & Lai, Y-C. (2016). Control efficacy of complex networks.

*Scientific Reports*,*6*, 1-12. [28037]. https://doi.org/10.1038/srep28037