Physica A 391 (2012) 4420–4425 Contents lists available at SciVerse ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Robustness analysis of network controllability Cun-Lai Pu a,b,c,∗ , Wen-Jiang Pei b , Andrew Michaelson d a School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China b School of Information Science and Engineering, Southeast University, Nanjing 210096, People’s Republic of China c Center for Complex Network Research, Department of Physics, Northeastern University, Boston, MA 02115, USA d Department of Bioengineering, Northeastern University, Boston, MA 02115, USA article info Article history: Received 1 September 2011 Received in revised form 7 April 2012 Available online 24 April 2012 Keywords: Network controllability Robustness Cascade failure abstract Structural controllability, which is an interesting property of complex networks, attracts many researchers from various fields. The maximum matching algorithm was recently applied to explore the minimum number of driver nodes, where control signals are injected, for controlling the whole network. Here we study the controllability of directed Erdös–Rényi and scale-free networks under attacks and cascading failures. Results show that degree-based attacks are more efficient than random attacks on network structural controllability. Cascade failures also do great harm to network controllability even if they are triggered by a local node failure. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Complex networks, composed of interacting individual nodes abstracted from natural or technological systems, have received great attention from scientific communities in past decades [1–5]. Advances have focused on network topological characteristics and network dynamics [6–15]. However, the ultimate goal for studying complex networks is not only to explore underlying principles of complex systems, but also to learn how to control them more efficiently [16–30]. It is difficult to control a complex system due to the unknown architecture of the system and the complex dynamics rules that govern the time-dependent interactions between the components. Recent advances in structures of complex networks stimulate the research in efficient control of large complex systems. Yang et al. [18] found in the contact process dynamics spreading can be maximized when the contact probability is chosen to be inversely proportional to the node degree. Zhang et al. [19] obtained the transmission efficiency of scale-free networks can be dramatically enhanced by kicking out the edges linking to nodes with large betweenness. Wang and Chen [20] demonstrate that, in scale-free networks it is more efficient to choose the large-degree nodes rather than the small-degree nodes as controllers in order to achieve a desired stabilization of the network. Li et al. [21] developed a virtual control strategy in which the pinned nodes virtually control other dynamical nodes with links between them, and found that after the pinned nodes were stabilized, the whole network is virtually broken into parts. It is more important to study the control of a system in the absence of key nodes and links. Motter [22] obtained that a selective further removal of nodes, right after the initial attack or failure, can prevent the cascading failures from propagating through the entire network. Motter et al. [23] showed that in metabolic networks of single-cell organisms, some mutants that are unable to grow due to perturbations caused by genetic or epigenetic defects can be turned into viable organisms through additional gene deletions. Sahasrabudhe and Motter [24] found that the consequence of the extinction ∗ Corresponding author at: School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China. E-mail addresses: pucunlai@gmail.com, pucunlai@yahoo.cn (C.-L. Pu). 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.04.019