On Quantitatively Measuring Controllability of Complex Networks

📝 Abstract

This letter deals with the controllability issue of complex networks. An index is chosen to quantitatively measure the extent of controllability of given network. The effect of this index is analyzed based on empirical studies on various classes of network topologies, such as random network, small-world network, and scale-free network.

💡 Analysis

This letter deals with the controllability issue of complex networks. An index is chosen to quantitatively measure the extent of controllability of given network. The effect of this index is analyzed based on empirical studies on various classes of network topologies, such as random network, small-world network, and scale-free network.

📄 Content

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On Quantitatively Measuring Controllability of Complex Networks

CAI, Ning1, 2 1College of Electrical Engineering, Northwest University for Nationalities, Lanzhou, China 2School of Automation, Hangzhou Dianzi University, Hangzhou, China E-mail: caining91@tsinghua.org.cn

Abstract: This letter deals with the controllability issue of complex networks. An index is chosen to quantitatively measure the extent of controllability of given network. The effect of this index is observed based on empirical studies on various classes of network topologies, such as random network, small-world network, and scale-free network.

Key Words: Controllability; Complex Network; Quantitative Measure

1. Introduction The controllability of a dynamic system reflects the ability of influencing from external input information to the motion of the overall system. Observability and controllability are dual alternatives. Integrated with stability, they form the theoretical foundation for most of the systems analysis and synthesis problems. Therefore, controllability has been one of the most important concepts in modern control theory.
   Since about a decade ago, the controllability problems of dynamic networked large-scale systems have intrigued many scholars from both the control [1-7] and the physics [8-13] community, and will surely continue to attract the attention from more and more disciplines.
   ~ 2 ~ Tanner [1] earlier studied the controllability of systems with single leader and conjectured that excessive connectivity might even be detrimental to controllability, while giving a definition of graph controllability based on a partition of the associated Laplacian matrix. Paying attention to the relationship between the extent of symmetry of graph and controllability, Rahmani and Mesbahi [2] further extended the results in [1]. Cai et al. addressed the controllability problems of a class of high-order systems, proposing a scheme of controllability improvement [3-4]. Liu et al. [5] concerned the controllability of discrete-time systems with switching graph topologies. Ji et al. [6-7] dealt with the interactive protocols, endeavoring to integrate the influence of three facets upon controllability, which are the protocol, the vertex dynamics and the network topology, respectively.
   Liu et al. [8] addressed the structural controllability of complex networks. They selected an index denoted by ND to quantitatively measure the extent of controllability of complex networks, namely the least amount of independent input signals required. Along the route of [8], there have emerged plenty of researches from the physics community, e.g. [9-13]. Particularly, Yan et al. [9] concentrated upon the problem of minimal energy cost for maneuvering the nodes. Yuan et al. concerned the exact controllability [10] of undirected networks with identical edge weights and discovered certain consistency with [8].
   The concept of controllability for dynamic systems was originally raised by Kalman, with a set of algebraic criteria to check whether or not a given system is exactly controllable. It has formed the foundation of the controllability theory. However, there are two intrinsic problems limiting the study of the controllability of complex networks from the viewpoint of exact controllability. The first problem is that almost any arbitrary system is controllable in the sense of exact controllability. This fact reduces the significance of being controllable. As the second problem, it is rather difficult to translate those algebraic criteria into straightforward conditions for the topologies of networks. In comparison with exact controllability, the concept of structural controllability possesses some advantages for coping with the controllability problems of networked systems. First, it bears no ambiguity like almost exact controllability. Being structurally controllable or not is essentially distinct for any system. Second, it is possible to acquire concise and straightforward criteria about the topologies of networks to check whether or not they are structurally controllable. Nevertheless, the essence of structural controllability is a minimal requirement for the availability of ~ 3 ~ input information across the network, which is only a necessary prerequisite for controlling, whereas it cannot facilitate to evaluate the efficiency of control. Therefore, structural controllability is also restrictive.
   In the current letter, a third angle on controllability is addressed, other than the exact controllability and structural controllability. We shall endeavor to explore the possible way to measure the extent of controllability of any given network, quantitatively. It is motivated by a wish to overcome some of the limitations of exact controllability and structural controllability, and to extend the methodology for controllability analysis of networks, from qualitative

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