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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.

In recent years, the study of complex networks has drawn the attention of many scholars from both the science and the engineering communities [

With deeper understanding of the controllability of complex networks [

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

Consider a complex system described by a directed weighted network of

System (

The rank of the controllability matrix

When we control node

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 [

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

The condition number of a square matrix is

The condition number can measure the nonsingularity of a matrix, with range of variation in

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

Consider two simple networks of

There are two simple networks of

For Figure

Similarly, for Figure

In this subsection, a more precise index of evaluating the controllability is presented. More than just saying “controllable” or “uncontrollable” to a network system, it can quantify the controllability of the given network. Such a route may be called quantitative controllability index.

The quantitative index measuring controllability of single node

By the above definition, we can calculate the controllability of each node for the given network. Then we can get the average quantitative controllability index of the whole network:

Quantitative controllability indexes

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Figure | 6.34 | 4 | 4 | 4 | 2.71 | 2 | 2 | 3.5774 |

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Figure | 6.29 | 3 | 2 | 3.71 | 3.71 | 4.38 | 2 | 3.5837 |

The change of number of nodes does influence the

The controllability index

The controllability of complex networks is related to their network topology. Therefore, it is significant to find the relationship between quantitative index measuring controllability and the structure parameters of complex network. In this section, we will conduct a series of simulation analyses for three different typical complex network models to observe the regularity with which the network controllability changes with the variations of parameters.

Liu et al. [

Trend of controllability and drive nodes of ER network with connectivity probability

Consider the ER network with twenty nodes,

We can see from Figure

Then we fix the connection probability

Trend of controllability of ER network with the number of nodes

For comparison, we apply our controllability index

For WS small-world network model, it could been seen from Figure

Trend of controllability and drive nodes of WS small-world network with

Set

Trend of controllability of WS small-world network with

Then we set

Trend of controllability of WS small-world network with the number of nodes

The algorithm of the BA scale-free model [

For the controllability of BA scale-free network model, firstly, let the start nodes number

Trend of controllability of BA scale-free network with the number of new edges added to each node

In Figure

When setting

Trend of controllability of BA scale-free network with the number of nodes

In this paper, we mainly discussed the problem of quantitative measurement controllability for arbitrary given network. A quantitative index is presented based on the control centrality and conditional number of the controllability matrix of controllable network (or subnetwork) system. And the index can quantitatively measure the controllability of each node in a given network. The effect of this controllability index is observed and discussed mainly by a series of experiments on various types of networks, namely, E–R networks, WS small-world networks, and BA scale-free networks. For ER networks, the controllability of networks increases with the increase of connection probability p; however, when p reached a certain value, the controllability will decrease with the increase of connection probability. Besides, the controllability of the network is negatively correlated with the number of nodes when the connectivity probability is fixed, so the controllability of the large-scale network is poor. For WS small-world network model, the controllability of the network is negatively correlated to both K and the number of nodes. For the BA scale-free network, due to the different number of new edges added when one node is trying to connect to the previous network, the controllability of the network presents different trends, and specific reasons of this phenomenon need to be further studied. These findings can help us better understand the relationship between controllability and network topology.

Much work still remains to be done concerning the controllability issue of complex networks. There still exist abundant potential future extensions:

The simulation data used to support the findings of this study are included within the article. If other data or MATLAB programs used to support the findings of this study are needed, you can obtain them from the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by Natural Science Foundation of Hebei Province (Grants nos. F2016501023 and F2017501041), Fundamental Research Funds for the Central Universities (Grant no. N172304030), and National Natural Science Foundation of China (Grant no. 61402088).