### 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 nodes. 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 (US) |
---|---|

Article number | 28037 |

Journal | Scientific Reports |

Volume | 6 |

DOIs | |

State | Published - Jun 21 2016 |

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### ASJC Scopus subject areas

- General

### Cite this

*Scientific Reports*,

*6*, [28037]. https://doi.org/10.1038/srep28037

**Control efficacy of complex networks.** / Gao, Xin Dong; Wang, Wen Xu; Lai, Ying-Cheng.

Research output: Contribution to journal › Article

*Scientific Reports*, vol. 6, 28037. https://doi.org/10.1038/srep28037

}

TY - JOUR

T1 - Control efficacy of complex networks

AU - Gao, Xin Dong

AU - Wang, Wen Xu

AU - Lai, Ying-Cheng

PY - 2016/6/21

Y1 - 2016/6/21

N2 - 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 nodes. 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.

AB - 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 nodes. 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.

UR - http://www.scopus.com/inward/record.url?scp=84976287441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976287441&partnerID=8YFLogxK

U2 - 10.1038/srep28037

DO - 10.1038/srep28037

M3 - Article

AN - SCOPUS:84976287441

VL - 6

JO - Scientific Reports

JF - Scientific Reports

SN - 2045-2322

M1 - 28037

ER -