# Robustness of Network Controllability with Respect to Node Removals Based on In-Degree and Out-Degree

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## Abstract

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## 1. Introduction

## 2. Network Data

#### 2.1. Directed Synthetic Networks

#### 2.2. Real-World Networks

## 3. Network Controllability

## 4. In-Degree and Out-Degree Node Attacks

## 5. Minimum Fraction of the Number of Driver Nodes under Targeted Node Attacks

#### 5.1. Analytical Approximation

#### 5.1.1. Case: $\alpha =0$

#### 5.1.2. Case: $\alpha =1$

**In-degree:**- In undirected networks, after a fraction p of nodes have been removed based on their degree; specifically, the probability of a node removal is proportional to some power of its degree, see Equation (1); the generating function of the degree distribution, $G\left(x\right)$, transforms into function $\overline{G}\left(x\right)$, which is as follows [28]:$$\overline{G}\left(x\right)=\frac{1}{1-p}\sum _{k=0}^{\infty}{p}_{k}{f}^{{k}^{\alpha}}{(1+\frac{f{G}_{\alpha}^{\prime}\left(f\right)}{<k>}(x-1))}^{k},$$We investigate the extension of prior conclusions to directed networks while removing nodes based on their in-degree. We assume that a node’s in-degree and out-degree are independent and uncorrelated, such that removing a fraction p of nodes based on their in-degree results in the generating function of the in-degree distribution described by Equation (12). Furthermore, the generating function of the out-degree distribution is given by ${\overline{G}}_{out}\left(x\right)={G}_{out}(p+(1-p)x)$ following the equation of random node removals. So, if we remove nodes based on in-degree, function ${\overline{G}}_{in}\left(x\right)$ and function ${\overline{G}}_{in}\left(x\right)$ satisfy$$\begin{array}{cc}\hfill {\overline{G}}_{in}\left(x\right)& =\frac{1}{1-p}\sum _{{k}_{in}=0}^{\infty}{p}_{{k}_{in}}{f}^{{k}_{in}}{(1+\frac{f{G}_{1}^{\prime}\left(f\right)}{<{k}_{in}>}(x-1))}^{{k}_{in}},\hfill \\ \hfill {\overline{G}}_{out}\left(x\right)& ={G}_{out}(p+(1-p)x).\hfill \end{array}$$Then, we can obtain the analytical approximation of the minimum fraction of driver nodes under node removals based on in-degree using Equation (10).
**Out-degree:**- Analogously, if we remove a fraction p of nodes based on their out-degree, we maintain the assumption that the generating function of the out-degree distribution is described by Equation (12). Additionally, the generating function of the in-degree distribution can be expressed as ${\overline{G}}_{in}\left(x\right)={G}_{in}(p+(1-p)x)$. Therefore, we have function ${\overline{G}}_{in}\left(x\right)$ and function ${\overline{G}}_{in}\left(x\right)$ as follows:$$\begin{array}{cc}\hfill {\overline{G}}_{in}\left(x\right)& ={G}_{in}(p+(1-p)x),\hfill \\ \hfill {\overline{G}}_{out}\left(x\right)& =\frac{1}{1-p}\sum _{{k}_{out}=0}^{\infty}{p}_{{k}_{out}}{f}^{{k}_{out}}{(1+\frac{f{G}_{1}^{\prime}\left(f\right)}{<{k}_{out}>}(x-1))}^{{k}_{out}}.\hfill \end{array}$$Furthermore, utilizing Equation (10), we can derive an analytical approximation of the minimum fraction of driver nodes when nodes are removed based on out-degree.

#### 5.1.3. Case: $\alpha =10$

**In-degree:**- In order to estimate the corresponding $\overline{p}$ of a given fraction p under node removals based on in-degree with $\alpha =10$, we adopt the assumption that nodes are removed in descending order of in-degree. Specifically, we first sort the nodes according to their in-degree and then remove nodes starting from the node with the highest in-degree until the targeted fraction p is reached.Next, we calculate the total in-degree of all the removed nodes by utilizing the original in-degree distribution and the targeted removal fraction p. The effective fraction $\overline{p}$ is then obtained by normalizing the total in-degree of all removed nodes with respect to the total in-degree of all nodes in the initial network. This can be calculated as follows:$${\overline{p}}_{in}=\frac{{\sum}_{{k}_{in}={k}_{i{n}_{max}}}^{{k}_{in}={\overline{k}}_{in}}{p}_{{k}_{in}}N{k}_{in}}{N<{k}_{in}>}=\frac{{\sum}_{{k}_{in}={k}_{i{n}_{max}}}^{{k}_{in}={\overline{k}}_{in}}{p}_{{k}_{in}}{k}_{in}}{<{k}_{in}>},\phantom{\rule{6.0pt}{0ex}}$$$$\begin{array}{cc}\hfill {\overline{G}}_{in}\left(x\right)& ={G}_{in}({\overline{p}}_{in}+(1-{\overline{p}}_{in})x),\hfill \\ \hfill {\overline{G}}_{out}\left(x\right)& ={G}_{out}(p+(1-p)x),\hfill \\ \hfill {n}_{D}& =\frac{1}{2}\{{\overline{G}}_{in}\left({\omega}_{2}\right)+{\overline{G}}_{in}(1-{\omega}_{1})-2+{\overline{G}}_{out}\left(\widehat{{\omega}_{2}}\right)+{\overline{G}}_{out}(1-\widehat{{\omega}_{1}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +k(1-\frac{p+{\overline{p}}_{in}}{2})[\widehat{{\omega}_{1}}(1-{\omega}_{2})+{\omega}_{1}(1-\widehat{{\omega}_{2}})]\}(1-\frac{p+{\overline{p}}_{in}}{2})+\frac{p+{\overline{p}}_{in}}{2},\hfill \end{array}\phantom{\rule{6.0pt}{0ex}}$$
**Out-degree:**- Analogously, for targeted node removal based on out-degree with $\alpha =10$, the calculation of fraction ${\overline{p}}_{out}$ follows the same assumption: nodes are removed from the node with the highest out-degree to the node with the lowest out-degree until the removed fraction of nodes reaches p. The effective fraction ${\overline{p}}_{out}$ is the total out-degree of removed nodes normalized by the total out-degree in the original network, which can be calculated by$${\overline{p}}_{out}=\frac{{\sum}_{{k}_{out}={k}_{ou{t}_{max}}}^{{k}_{out}={\overline{k}}_{out}}{p}_{{k}_{out}}N{k}_{out}}{N<{k}_{out}>}=\frac{{\sum}_{{k}_{out}={k}_{ou{t}_{max}}}^{{k}_{out}={\overline{k}}_{out}}{p}_{{k}_{out}}{k}_{out}}{<{k}_{out}>},$$$$\begin{array}{cc}\hfill {\overline{G}}_{in}\left(x\right)& ={G}_{in}(p+(1-p)x),\hfill \\ \hfill {\overline{G}}_{out}\left(x\right)& ={G}_{out}({\overline{p}}_{out}+(1-{\overline{p}}_{out})x),\hfill \\ \hfill {n}_{d}& =\frac{1}{2}\{{\overline{G}}_{in}\left({\omega}_{2}\right)+{\overline{G}}_{in}(1-{\omega}_{1})-2+{\overline{G}}_{out}\left(\widehat{{\omega}_{2}}\right)+{\overline{G}}_{out}(1-\widehat{{\omega}_{1}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +k(1-\frac{p+{\overline{p}}_{out}}{2})[\widehat{{\omega}_{1}}(1-{\omega}_{2})+{\omega}_{1}(1-\widehat{{\omega}_{2}})]\}(1-\frac{p+{\overline{p}}_{out}}{2})+\frac{p+{\overline{p}}_{out}}{2},\hfill \end{array}\phantom{\rule{6.0pt}{0ex}}$$

#### 5.2. Results for Targeted Node Attacks

#### 5.2.1. Case: $\alpha =1$

#### 5.2.2. Case: $\alpha =10$

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ER | Erdös-Rényi networks |

SSNs | Swarm Signal networks |

SFs | Scale-free networks |

RMSE | Root Mean Square Error |

## Appendix A. The Simulation Results Based on Different α Values

**Figure A1.**The minimum fraction of driver nodes ${n}_{D}$ during targeted node removal based on in-degree and out-degree with $\alpha =0$, $\alpha =1$, $\alpha =10$ and $\alpha =100$ for different kinds of networks. The results are the average ${n}_{D}$ calculated by the maximum matching algorithm over 10,000 realizations of real-world networks and 1000 realizations of model networks. The blue, orange, and green dashed lines are the results of simulations with $\alpha =0$, $\alpha =1$ and $\alpha =10$ separately. The pink dotted lines are obtained by the simulation results with $\alpha =100$.

## Appendix B. Comparison with Node Removal Based on Degree with α=1

**Figure A2.**The minimum fraction of driver nodes ${n}_{D}$ during random removal and targeted node removal based on degree, in-degree and out-degree with $\alpha =1$ for three networks. The results are the average ${n}_{D}$ calculated by the maximum matching algorithm over 10,000 realizations of HinerniaGlobal and 1000 realizations of ER(100, 0.04) and SSN(${10}^{4},2$). The blue, red, orange and pink dashed lines are the results of simulations with random removal, target removal based on the total degree with $\alpha =1$, target removal based on in-degree with $\alpha =1$ and target removal based on out-degree with $\alpha =1$ separately.

## Appendix C. Another Real-World Network Results

**Table A1.**The RMSE between the analytical results of random removals and the simulation results under random removals, target removals with $\alpha =1$ and $\alpha =10$, respectively, while removing 10% of the nodes. The analytical method for random removals is from the reference [19].

Network | Random | $\mathit{\alpha}=1$ | $\mathit{\alpha}=10$ | ||||
---|---|---|---|---|---|---|---|

Indegree | Outdegree | Degree | Indegree | Outdegree | Degree | ||

BtNorthAmerica | 0.0097 | 0.0140 | 0.0104 | 0.0121 | 0.0117 | 0.0096 | 0.0101 |

**Table A2.**The RMSE between the analytical results of the proposed analytical methods and the simulation results under different kinds of removals while removing 10% of the nodes. The analytical methods for random removals and targeted node removals based on the total degree are from the reference [19].

Network | Random | $\mathit{\alpha}=1$ | $\mathit{\alpha}=10$ | ||||
---|---|---|---|---|---|---|---|

Indegree | Outdegree | Degree | Indegree | Outdegree | Degree | ||

BtNorthAmerica | 0.0097 | 0.0126 | 0.0175 | 0.0091 | 0.0538 | 0.0612 | 0.0527 |

**Figure A3.**The minimum fraction of driver nodes ${n}_{D}$ during targeted node removal based on in-degree and out-degree with $\alpha =1$ and $\alpha =10$, respectively, for the network BtNorthAmerica.

## Appendix D. Analytical Approximation of Random Node Removals about SFs

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**Figure 1.**The minimum fraction of driver nodes ${n}_{D}$ during targeted node removal based on in-degree with $\alpha =1$ for different kinds of networks.

**Figure 2.**The minimum fraction of driver nodes ${n}_{D}$ during targeted node removal based on out-degree with $\alpha =1$ for different kinds of networks.

**Figure 3.**The minimum fraction of driver nodes ${n}_{D}$ during targeted node removal based on in-degree with $\alpha =10$ for different kinds of networks.

**Figure 4.**The minimum fraction of driver nodes ${n}_{D}$ during targeted node removal based on out-degree with $\alpha =10$ for different kinds of networks.

Name | N | L | $<\mathit{k}>$ |
---|---|---|---|

HinerniaGlobal | 55 | 81 | 2.95 |

Syringa | 74 | 74 | 2.00 |

Interoute | 110 | 146 | 2.65 |

Cogentco | 197 | 243 | 2.47 |

**Table 2.**The RMSE between the analytical results of random removals and the simulation results under random removals, target removals with $\alpha =1$ and $\alpha =10$, respectively, while removing 10% of the nodes. The column labeled “Random” indicates the RMSE under random removals. The columns labeled “$\alpha =1$” and “$\alpha =10$” represent the RMSE under targeted node removals with $\alpha =1$ and $\alpha =10$, respectively. The columns labeled “Indegree”, “Outdegree”, and “Degree” represent the RMSE under targeted node removals based on in-degree, out-degree, and total degree, respectively. The analytical method for random removals is from the reference [19].

Network | Random | $\mathit{\alpha}=1$ | $\mathit{\alpha}=10$ | ||||
---|---|---|---|---|---|---|---|

Indegree | Outdegree | Degree | Indegree | Outdegree | Degree | ||

SF(${10}^{5}$, 3, 5) | 0.0005 | 0.0010 | 0.0010 | 0.0010 | 0.0032 | 0.0032 | 0.0032 |

SF(${10}^{5}$, 3, 10) | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

ER(50, 0.07) | 0.0137 | 0.0164 | 0.0156 | 0.0155 | 0.0190 | 0.0195 | 0.0223 |

ER(100, 0.04) | 0.0079 | 0.0086 | 0.0095 | 0.0094 | 0.0126 | 0.0121 | 0.0156 |

HinerniaGlobal | 0.0039 | 0.0052 | 0.0110 | 0.0084 | 0.0025 | 0.0152 | 0.0152 |

Syringa | 0.0071 | 0.0136 | 0.0217 | 0.0179 | 0.0237 | 0.0263 | 0.0443 |

Interoute | 0.0011 | 0.0008 | 0.0106 | 0.0056 | 0.0064 | 0.0072 | 0.0175 |

Cogentco | 0.0011 | 0.0090 | 0.0053 | 0.0071 | 0.0156 | 0.0091 | 0.0248 |

SSN(${10}^{4}$, 2) | 0.0000 | 0.0103 | 0.0003 | 0.0052 | 0.0143 | 0.0007 | 0.0155 |

SSN(${10}^{4}$, 5) | 0.0000 | 0.0006 | 0.0000 | 0.0003 | 0.0008 | 0.0001 | 0.0008 |

**Table 3.**The RMSE between the analytical results of the proposed analytical methods and the simulation results under different kinds of removals while removing 10% of the nodes. The column labeled “Random” indicates the RMSE under random removals. The columns labeled “$\alpha =1$” and “$\alpha =10$” represent the RMSE under targeted node removals with $\alpha =1$ and $\alpha =10$, respectively. The columns labeled “Indegree”, “Outdegree”, and “Degree” represent the RMSE under targeted node removals based on in-degree, out-degree, and total degree, respectively. The analytical methods for random removals and targeted node removals based on the total degree are from the reference [19].

Network | Random | $\mathit{\alpha}=1$ | $\mathit{\alpha}=10$ | ||||
---|---|---|---|---|---|---|---|

Indegree | Outdegree | Degree | Indegree | Outdegree | Degree | ||

SF(${10}^{5}$, 3, 5) | 0.0005 | 0.0010 | 0.0010 | 0.0546 | 0.0764 | 0.0764 | 0.1573 |

SF(${10}^{5}$, 3, 10) | 0.0001 | 0.0001 | 0.0001 | 0.0555 | 0.0799 | 0.0799 | 0.1600 |

ER(50, 0.07) | 0.0137 | 0.0122 | 0.0113 | 0.0095 | 0.0588 | 0.0595 | 0.0543 |

ER(100, 0.04) | 0.0079 | 0.0058 | 0.0067 | 0.0039 | 0.0189 | 0.0193 | 0.0284 |

HinerniaGlobal | 0.0039 | 0.0089 | 0.0136 | 0.0025 | 0.0281 | 0.0349 | 0.0354 |

Syringa | 0.0071 | 0.0096 | 0.0143 | 0.0061 | 0.0157 | 0.0235 | 0.0142 |

Interoute | 0.0011 | 0.0151 | 0.0242 | 0.0009 | 0.0265 | 0.0454 | 0.0229 |

Cogentco | 0.0011 | 0.0233 | 0.0244 | 0.0050 | 0.0314 | 0.0330 | 0.0322 |

SSN(${10}^{4}$, 2) | 0.0000 | 0.0085 | 0.0002 | 0.0027 | 0.0331 | 0.0006 | 0.0343 |

SSN(${10}^{4}$, 5) | 0.0000 | 0.0094 | 0.0000 | 0.0024 | 0.0167 | 0.0001 | 0.0264 |

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**MDPI and ACS Style**

Wang, F.; Kooij, R.E.
Robustness of Network Controllability with Respect to Node Removals Based on In-Degree and Out-Degree. *Entropy* **2023**, *25*, 656.
https://doi.org/10.3390/e25040656

**AMA Style**

Wang F, Kooij RE.
Robustness of Network Controllability with Respect to Node Removals Based on In-Degree and Out-Degree. *Entropy*. 2023; 25(4):656.
https://doi.org/10.3390/e25040656

**Chicago/Turabian Style**

Wang, Fenghua, and Robert E. Kooij.
2023. "Robustness of Network Controllability with Respect to Node Removals Based on In-Degree and Out-Degree" *Entropy* 25, no. 4: 656.
https://doi.org/10.3390/e25040656