1. Introduction
Network controllability is a crucial area of research that has been explored in various types of networks, including biological networks [
1], transportation networks [
2], and corruption networks [
3]. The controllability of a network refers to the ability to steer the states of its nodes to any desired state in a finite time by manipulating the input to a subset of its nodes. Nodes whose inputs are imposed are named driver nodes. In linear time-invariant systems, Kalman’s controllability rank condition [
4] is the classic method of assessing controllability. However, the method has limitations such as computation complexity and the lack of information about the system’s interaction matrix and input matrix. To overcome these limitations, the concept of structural controllability was proposed [
5]. Structural controllability is a property of structural linear time-invariant systems with independently free parameters or fixed zero elements in their interaction and input matrices that satisfy the controllability rank condition. Directed networks are structural systems. Liu et al. [
6] developed the algorithm and analytical methods to obtain the minimum number of driver nodes in directed networks with the assumption that the directed network has no self-links and a node’s internal state can only be modified upon interaction with neighboring nodes [
7]. Throughout this paper, we will adhere to this assumption. Besides structural controllability, Yuan et al. introduced an exact controllability paradigm to determine the minimum number of driver nodes for undirected networks with arbitrary weights by using the maximum multiplicity [
8].
In recent years, network structural controllability has gained increasing attention as a tool to measure and enhance network robustness. Robustness is commonly assessed by measuring network performance under various perturbations [
9]. One approach is to randomly remove nodes or links and observe the resulting changes in network performance, while another approach involves targeted attack strategies exploiting specific features of network topology such as betweenness, degree, and closeness. Several studies have investigated the effectiveness of different targeted attack strategies on network controllability. For example, degree-based attacks have been found to be more harmful to network controllability than random attacks [
10], while betweenness-based attacks are more damaging in most real-world networks [
11]. Additionally, attacking bridge links, which results in a disconnected network, has been shown to be an effective way to destroy network controllability [
12]. Another approach to targeted attack strategies involves identifying critical nodes and links whose removal increases the number of driver nodes [
6]. Protecting critical links can make random link attacks less efficient [
13]. Some studies have found that hierarchical attack strategies targeting critical nodes and links first are more efficient than metric-based attack strategies, such as betweenness- or degree-based strategies in interdependent networks [
14]. In addition to assessing the robustness of network controllability under perturbations, some studies have focused on enhancing it. For example, increasing the density of nodes with an in-degree and out-degree equal to one or two has been shown to improve network controllability [
15]. Adding links to low-degree nodes and creating multi-loop structures have also been found to increase the robustness of network controllability [
16]. Furthermore, different redundant design strategies of interdependent networks, such as betweenness-based and degree-based strategies for node backup and high degree first strategy for edge backup, have been investigated to optimize the robustness of network controllability [
17].
In addition to qualitative research, quantitative studies have been carried out to explore the robustness of network controllability under different types of perturbations. Lu et al. [
11] developed numerical approximations of random and targeted node attacks based on the degree on Erdös-Rényi (ER) networks, which fit well when the fraction of nodes is below 20%. Sun et al. [
13] derived closed-form approximations of the minimum number of driver nodes under various types of attacks, including random link attacks, targeted attacks, and random attacks with protection. Chen et al. [
18] developed analytical approximations for the minimum number of driver nodes during random link removal using generating functions. Wang et al. [
19] later conducted analytical methods based on generating functions to approximate the network controllability during random and targeted node removal based on the total degree of different kinds of networks. In addition to analytical methods, machine learning has been employed to predict network controllability robustness. Dhiman et al. [
20] used machine-learning-based approximations to quantify the minimum fraction of driver nodes under random and targeted link attacks, which performed better than the closed-form approximation proposed by Sun et al. [
13]. Meanwhile, by utilizing deep learning techniques, Lou et al. have developed a series of works that employ different convolutional neural network (CNN) frameworks, treating the adjacency matrix as a visual representation, to predict network controllability under random node or link attacks, degree-based targeted node or link attacks, and betweenness-based targeted node and link attacks [
21,
22,
23]. Through the use of these models, they have achieved increasingly precise controllability predictions and demonstrated improved scalability. The quantitative studies provide valuable insights into the robustness of network controllability.
As the analytical approximations for targeted node removals based on node in-degree and out-degree are still lacking, in this paper, we aim to utilize the structural controllability framework for directed networks proposed by Liu et al. [
6] to make the analytical approximation for those two kinds of targeted node removals. We validate our proposed methods by applying them to three types of synthetic networks and four real-world communication networks.
The remainder of the paper is structured as follows. In
Section 2, we introduce the networks used in our study.
Section 3 presents the analytical results of network controllability under the two classes of targeted attacks. Finally, we conclude and discuss the implications of our findings in
Section 4.
3. Network Controllability
Consider a linear, time-invariant networked system of N nodes, where each node’s state is governed by , with being the state vector. The matrix A represents the interactions among the network components, and the matrix B specifies which nodes are under the direct control of the control input vector .
A linear, time-invariant networked system is controllable if it can reach any desired state within a finite time by applying external inputs. The Kalman rank criterion requires that the rank of the controllability matrix
equals
N for the system to be fully controllable. Liu et al. introduced the maximum matching method and the minimum inputs theorem to determine the minimum number of driver nodes required to ensure network structural controllability [
6]. The number of driver nodes,
, can be obtained by mapping a directed network into a bipartite network [
13], obtaining a maximum matching edge set using the maximum matching algorithm [
27], and then calculating
, where
is the number of directed edges in the maximum matching set without sharing the same source or end nodes.
4. In-Degree and Out-Degree Node Attacks
Centrality analysis is an essential research area in studying network robustness [
28]. Nodes with a high degree are known to have a substantial impact on network functioning and are more susceptible to targeted attacks. In this study, our objective is to investigate an analytical approximation of network controllability during targeted node removal based on two types of degrees: in-degree and out-degree.
Assuming that the probability of node attack is proportional to some power of its in-degree and out-degree, we can express the probability of removing node
i based on its in-degree
as
and based on its out-degree
as
. The formula for calculating these probabilities is given as follows:
In the node removal process, after some nodes are removed, we recalculate the removal probabilities for the remaining nodes using Equation (
1). We then select nodes to remove based on the recalculated probabilities until all nodes are removed.
When
, the aforementioned equations become
which indicates that each node has an equal probability of being removed, resulting in a random removal strategy. On the other hand, for
, nodes with higher degrees have a greater likelihood of being removed, while for
, nodes with lower degrees are more likely to be removed.
In this study, we investigate the impact of degree-based node removal strategies on network robustness. To this end, we focus on
, as higher-degree nodes are commonly targeted for attack in real-world scenarios. Specifically, we consider two values of
, namely
and
, to evaluate the impact of removing nodes proportional to their degree and removing high-degree nodes more aggressively, respectively. By using Equation (
1), we obtain the probabilities of the node being removed based on in-degree and out-degree when
as follows:
Analogously, the node removal probabilities based on in-degree or out-degree with
can be calculated by
Our results show that, for
, the removal of high-degree nodes does not lead to a significant reduction in network robustness in the beginning stage. For several networks, there are no significant differences between the results with
and
. Interestingly, we observe that increasing the value of
to 100 does not result in further performance gains, as the performance of attacks with
is similar to that of attacks with
. Additional details on these findings can be found in
Appendix A. Furthermore, we find that when
, the removal strategies based on in-degree or out-degree can be more detrimental to certain networks than node removal based on the total degree. However, for some other networks, the harmful effects of these strategies are comparable. The results are presented in
Appendix B.
6. Conclusions and Discussion
This study introduces analytical methods based on generating functions to determine the minimum fraction of driver nodes required to maintain network controllability in directed networks under node failures based on in-degree and out-degree. We develop separate analytical techniques for two scenarios, namely and . Our proposed analytical methods demonstrate reasonable results to predict the minimum fraction of driver nodes under targeted attacks. Furthermore, our investigation indicates that random node removal may also serve as a reliable predictor of the results of various targeted node removals, particularly when the fraction of removed nodes is minimal (below 10%).
In addition to the findings presented in this paper, we have endeavored to apply our simulations to various other real-world networks. Our analysis reveals that the minimum fraction of driver nodes calculated by the proposed analytical method utilizing generating functions does not coincide with the results obtained using the maximum matching algorithm before node removal. As such, our proposed methods are inadequate for predicting the minimum fraction of driver nodes under node removal for these networks. When targeted node removal is based on in-degree and out-degree with , our approximation method assumes that nodes are removed in descending order of in-degree and out-degree. However, the assumption does not reflect the actual removal process, as we recalculated the removal probabilities to choose nodes at each step. This discrepancy is one of the reasons for the inaccurate results obtained. Moreover, we acknowledge that further improvements are required to enhance the method’s efficacy. Notably, the numerical solution of the predicted outcomes can be challenging to obtain, particularly when attempting to acquire the results for SFs with some other parameters.
The approximation of node removals based on in- or out-degree involves an assumption that the in-degree distribution and out-degree distribution evolve independently. However, the assumption requires further investigation to ensure its validity. To address this issue, an avenue of promising research involves examining the relationship between in-degree and out-degree distributions through the randomization of networks. Such analyses may provide upper and lower bounds for analytical methods, contributing to the improvement of predictions about network controllability under targeted attacks based on in-degree and out-degree.
In the future, we aim to broaden the scope of our findings by including other types of node attacks, specifically localized node attacks, as documented in [
28]. Furthermore, we intend to verify our conclusions on a more comprehensive collection of real-world networks and various types of networks, such as interdependent networks. We also plan to apply additional prediction techniques, such as machine learning methods, to assess network controllability under node removals concerning in-degree and out-degree.