Advances in Mathematical Physics

Volume 2018, Article ID 2586536, 9 pages

https://doi.org/10.1155/2018/2586536

## Quantitative Controllability Index of Complex Networks

School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao, China

Correspondence should be addressed to Lifu Wang; moc.qq@zkflw

Received 25 June 2018; Revised 23 September 2018; Accepted 2 October 2018; Published 22 October 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 Lifu Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, the controllability issue of complex network is discussed. A new quantitative index using knowledge of control centrality and condition number is constructed to measure the controllability of given networks. For complex networks with different controllable subspace dimensions, their controllability is mainly determined by the control centrality factor. For the complex networks that have the equal controllable subspace dimension, their different controllability is mostly determined by the condition number of subnetworks’ controllability matrix. Then the effect of this index is analyzed based on simulations on various types of network topologies, such as ER random network, WS small-world network, and BA scale-free network. The results show that the presented index could reflect the holistic controllability of complex networks. Such an endeavour could help us better understand the relationship between controllability and network topology.

#### 1. Introduction

In recent years, the study of complex networks has drawn the attention of many scholars from both the science and the engineering communities [1–5]. Studies of theirs impact our understanding and control of a wide range of systems, from the Internet and the power-grid to cellular and ecological networks. While there are many challenges in the process of research due to the diversity of complex networks, one most challenging issue in modern science and engineering networks is the control of complex networks [6, 7]. The controllability of a dynamical system reflects the ability of external input information to influence the motion of the overall system. A complex network is controllable if, imposing appropriate external signals, the system can be driven from any initial state to any final state in finite time [8–10]. Although great effort has been devoted to understanding the interplay between complex networks and dynamical processes taking place on them in various natural and technological systems [11–17], the control of complex dynamical networks remains to be an outstanding problem. The Kalman controllability formed the foundation of the controllability theory by a set of algebraic criteria to check whether or not a given system is controllable. However, the Kalman controllability is qualitative; we cannot know how it is difficult or easy to control, that is, cannot know the size of network controllability for one network system. Therefore, it is necessary to find a method that can quantitatively measure controllability of a given system.

With deeper understanding of the controllability of complex networks [18–26], more scholars have begun to explore the ability to control complex networks through various indicators. For example, Liu et al. [18] addressed the structural controllability of arbitrary complex directed networks, identifying a minimal set of driver nodes that can guide the system to any desired state. They selected an index denoted by to quantitatively measure the extent of controllability of complex networks. Liu et al. [19] propose the concept of node control centrality to quantify the ability of a single node of complex network and calculate the distribution of control centrality for some networks. Jia et al. [20] proposed the concept of node control capability in the further study; that is, the possibility of node becoming a driving node was calculated. Many research works have been published under the theoretical framework of structural controllability [21–26]. However, these quantitative controllability methods only roughly measure the dimensionality of controllable subsystems; they could not better distinguish the controllability of systems with the same number of dimensions of controllable subsystems. Cai Ning [27] studied a way to quantitatively measure the extent of controllability of any given controllable network. This method is based on fully controllable networks; for uncontrollable networks, it could not make a judgement.

In the current paper, we will try to explore the possible ways to quantitatively measure the extent of controllability of any given network. An index will be proposed to assess the controllability of a dynamical network, that is, the network whether or not being easily or difficulty controlled via the input information. While the previous Kalman controllability index is qualitative, only defining whether or not a system is controllable, the one in our paper is quantitative. Particularly, for a not completely controllable system, we can quantitative measure controllability of the controllable subsystem by decomposing the network into a controllable subsystem and an uncontrollable subsystem, so that we can measure how far the network is from being uncontrollable. Such a route may be called quantitative index measure controllability of complex network. Then simulations will be performed to show the effect of controllability index between different network topologies on three distinct types of model networks, namely, the random networks, the small-world networks, and the BA scale-free networks. Through simulation results, it can be found that controllability of given network is related to their topologies, such as the number of nodes and edges and edge density, so it is possible to improve the controllability of the network by adjusting certain parameters, such as the connectivity probability and the number of nodes .

#### 2. Controllability of Complex Networks

Consider a complex system described by a directed weighted network of nodes whose time evolution follows the linear time invariant dynamics [28].where is the state vector of each node at time . denotes the state matrix that depicts the linking strength between the state nodes. The matrix element gives the strength or weight that node can affect node . is the input matrix, indicating the nodes that are controlled by the time-dependent input vector with independent signals imposed by an outside controller. The element represents the control strength from the control node to the state node . A system is controllable if we can drive it from any initial state to any desired final state in finite time; otherwise, it is uncontrollable.

System (1), also denoted as (), is controllable if and only if its controllability matrix has full rank; that is,which are the criteria often called Kalman’s controllability rank condition [8].

The rank of the controllability matrix* C*, denoted by rank(*C*), provides the dimension of the controllable subspace of system (). The system is not completely controllable if . Then the system can be decomposed into a controllable subsystem and an uncontrollable subsystem through a linear transformation [29], which iswhere is the controllable subsystem whose dimension is , and is the uncontrollable subsystem whose dimension is .

When we control node only ( is the* i*th node in the network), reduces to the vector with a single nonzero entry, and we denote with . We can therefore use rank as a natural measure of node ’s ability to control the system: if , then node alone can control the whole system; that is, it can drive the system to travel between any points in the N-dimensional state space in finite time. Any value of less than provides the dimension of the subspace that can control. In particular, if , then node can only control itself.

Kalman controllability theory provides great convenience for checking whether or not a given network is controllable. However, from the point of view of Kalman controllability, there are some problems that restrict the study of the controllability of complex networks to some extent. First, almost any arbitrary system is completely controllable in the sense of Kalman controllability [27]. This fact reduces the significance of Kalman controllability theory. In addition, Kalman controllability of dynamical system is qualitative, which means that, in Kalman's theory, the system could be only defined as either controllable or uncontrollable. However, for many real-world network systems, people need to know more information about the controllability issue such as how much our control force should be to control the whole system or maybe part of the system. In the next section, one index is brought up with to measure the controllability of networks quantitatively.

#### 3. Quantitative Index Measuring Controllability

Before we begin to analyze quantitative controllability, we first introduce the definition of conditional number.

*Definition 1 (see [30]). *The condition number of a square matrix iswhere denotes 2-norm of the square matrix .

The condition number can measure the nonsingularity of a matrix, with range of variation in . When is identity matrix, . On the contrary, when is nearly singular matrix, . That is, the greater the condition number is, the closer the matrix is to being singular.

##### 3.1. Examples of Analysis Controllability

Let us start the analysis on quantitative index measuring controllability by investigating two typical examples.

Consider two simple networks of nodes with different structures. The controlled network has an input connecting to a state node . The network topology is shown in Figure 1. The controllability of node to the whole network is analyzed as in Figure 1.