# Study on Characteristics and Invulnerability of Airspace Sector Network Using Complex Network Theory

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## Abstract

**:**

## 1. Introduction

- This paper expands the research horizon of aviation networks by incorporating complex network theory into ATC sectors, offering a complementary viewpoint to existing studies on aviation networks and route networks.
- The invulnerability of various sectors under different attack strategies is evaluated, and critical sectors are identified by considering the network’s global efficiency and the relative size of its connected component.
- This research provides a perspective on alleviating air traffic congestion and improving airspace efficiency by improving the sector structure in air traffic control systems. Additionally, this research method also provides a reference for analysis in other complex network engineering projects.

## 2. Methods of Modeling and Topological Property Measurement

#### 2.1. Modeling of Airspace Sector Network

#### 2.2. Modeling of Airspace Sector Network

- The degree of a node, denoted as ${k}_{i}$, reflects the node’s significance in the network and is defined as the number of edges connected to it. In the context of the airspace sector network, the degree ${k}_{i}$ of sector $i$ represents the number of sectors that are geographically adjacent and have a direct air traffic connection with sector $i$ [23]. This study expands the research scope of aviation networks by applying complex network theory to ATC sectors, providing a complementary perspective to previous studies on aviation and route networks.
- Intensity, as expressed in Equation (1):$${S}_{i}={{\displaystyle \sum}}_{j\in V\left(i\right)}{W}_{ij}$$
- Average path length, as expressed in Equation (2):$$\text{}\begin{array}{c}l=\frac{1}{N\left(N-1\right)}{{\displaystyle \sum}}_{i\ne j}\text{}{l}_{ij}\end{array}$$
- Betweenness centrality, as expressed in Equation (3):$$\text{}{B}_{i}={{\displaystyle \sum}}_{j,k\in F,j\ne k}\text{}\frac{{n}_{jk}\left(i\right)}{{n}_{jk}}$$
- Clustering coefficient, as expressed in Equation (4) for local clustering coefficient of node $i$ and Equation (5) for network average clustering coefficient:$$\begin{array}{c}{C}_{i}=\frac{2{e}_{i}}{{k}_{i\text{}}\left({k}_{i}-1\right)}\end{array}$$$$\begin{array}{c}C=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}\text{}{C}_{i}\end{array}$$

#### 2.3. Modeling of Airspace Sector Network

- The global efficiency of the network is a measure of its connectivity, which is represented as the average of the inverse of the distances between sectors [29]. It is expressed in Equation (6).$$\begin{array}{c}E=\frac{1}{M\left(M-1\right)}{\displaystyle \sum}_{i\ne j}\frac{1}{{l}_{ij}}\end{array}$$
- The relative size of the connected component is a metric that measures the proportion of nodes that remain connected after a set of nodes or edges have been removed from the network. This is expressed as a ratio of the number of nodes in the remaining connected component to the total number of nodes in the original network, as shown in Equation (7).$$\begin{array}{c}G=\frac{{g}^{\prime}}{g}\end{array}$$

## 3. Analysis of Sector Network Characteristics

#### 3.1. Construction of Airspace Sector Network

#### 3.2. Topological Properties of Sector Network

#### 3.3. Analysis of Network Structural Characteristics

## 4. Invulnerability of Sector Network

#### 4.1. Attack Strategies

#### 4.2. Results of Invulnerability Assessment

#### 4.3. Impact of Attacks on Critical Sectors

## 5. Conclusions

- The study reveals that the airspace sector network in North China exhibits a compact and well-established structure, as indicated by its small average path length and clustering coefficient. However, the connections between adjacent sectors are scattered, posing challenges for efficient air traffic control coordination and collaboration. The sector degree distribution is relatively even, and the workload of controllers in different sectors varies significantly, as evidenced by the power-law distribution of intensity and betweenness. Therefore, optimizing sector management to balance controller workload is crucial for efficient air traffic control.
- We evaluated the invulnerability of the airspace sector network by analyzing two key measurement indices: network global efficiency and the relative size of connected components. The results indicate that the sector network is more resilient to random attacks but less so to intentional attacks. In the case of intentional attacks, degree and betweenness are crucial indices that can help identify critical sectors and potential critical sectors. Since betweenness has a higher impact than degree in intentional attacks, it was chosen as the primary critical index to analyze the effect of critical sectors and identify potential critical sectors. Critical sectors play a pivotal role in the overall invulnerability of the sector network. If critical sectors are targeted, it could significantly increase the flight traffic of potential critical sectors, leading to traffic congestion and further damage to other sectors.
- This study utilized complex network theory to investigate the airspace network, specifically analyzing the ATC sector network in North China using historical air traffic data. The study obtained a comprehensive understanding of the topological characteristics and invulnerability of the sector network from the perspective of air traffic control. This provides insights for alleviating air traffic congestion and lays the foundation for future planning of sector selection in airspace.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Sector Code | Degree Priority | Betweenness Priority | Random Attack |
---|---|---|---|

0 | 0.4969 | 0.4969 | 0.4969 |

1 | 0.4193 | 0.4193 | 0.4510 |

2 | 0.3728 | 0.3102 | 0.4210 |

3 | 0.3159 | 0.2700 | 0.3851 |

4 | 0.2724 | 0.2200 | 0.3219 |

5 | 0.238 | 0.1900 | 0.2529 |

6 | 0.1958 | 0.1900 | 0.2529 |

7 | 0.1958 | 0.1500 | 0.2309 |

8 | 0.1481 | 0.0900 | 0.2031 |

9 | 0.0960 | 0.0740 | 0.1673 |

10 | 0.0640 | 0.0630 | 0.1376 |

11 | 0.0542 | 0.0390 | 0.0873 |

12 | 0.0336 | 0.0336 | 0.0626 |

13 | 0.0277 | 0.0336 | 0.0626 |

14 | 0.0178 | 0.0200 | 0.0507 |

15 | 0.0119 | 0.013 | 0.0375 |

16 | 0.0119 | 0.0079 | 0.0257 |

17 | 0.0079 | 0.0079 | 0.0138 |

18 | 0.0040 | 0.0040 | 0.0079 |

19 | 0.0040 | 0.0040 | 0.0040 |

20 | 0.0040 | 0 | 0.0040 |

21 | 0 | 0 | 0 |

22 | 0 | 0 | 0 |

23 | 0 | 0 | 0 |

**Table A2.**The relative size of connected components of sector network under different attack strategies.

Sector Coding | Degree Priority | Betweenness Priority | Random Attack |
---|---|---|---|

0 | 1 | 1 | 1 |

1 | 0.96 | 0.96 | 0.96 |

2 | 0.91 | 0.87 | 0.91 |

3 | 0.87 | 0.87 | 0.87 |

4 | 0.83 | 0.739 | 0.83 |

5 | 0.739 | 0.739 | 0.78 |

6 | 0.695 | 0.695 | 0.73 |

7 | 0.695 | 0.65 | 0.695 |

8 | 0.65 | 0.43 | 0.65 |

9 | 0.43 | 0.43 | 0.61 |

10 | 0.3 | 0.26 | 0.56 |

11 | 0.174 | 0.26 | 0.3 |

12 | 0.174 | 0.174 | 0.22 |

13 | 0.174 | 0.174 | 0.22 |

14 | 0.174 | 0.13 | 0.22 |

15 | 0.13 | 0.087 | 0.22 |

16 | 0.087 | 0.087 | 0.17 |

17 | 0.087 | 0.087 | 0.13 |

18 | 0.087 | 0.04348 | 0.087 |

19 | 0.087 | 0.04348 | 0.087 |

20 | 0.087 | 0.04348 | 0.087 |

21 | 0.04348 | 0.04348 | 0.04348 |

22 | 0.04348 | 0.04348 | 0.04348 |

23 | 0.04348 | 0.04348 | 0.04348 |

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**Figure 1.**(

**a**) Structure of North-China-Controlled Airspace Sector; (

**b**) Constructed Network of North-China-Controlled Airspace Sector.

**Figure 4.**Correlation analysis of the degree, intensity, agglomeration coefficient, intermediate number, and other topological property indices of the North China airspace sector network. (

**a**) Degree–average nearest neighbor degree (${R}^{2}=0.47$); (

**b**) intensity–average nearest neighbor intensity; (

**c**) degree–intensity; (

**d**) degree–betweenness (${R}^{2}=0.81$); (

**e**) degree–clustering coefficient (${R}^{2}=0.33$); (

**f**) intensity–clustering coefficient; (

**g**) intensity–betweenness; (

**h**) betweenness–clustering coefficient (${R}^{2}=0.35$).

**Figure 5.**(

**a**) Trends of network global efficiency under different attack strategies; (

**b**) trends of relative size of connected components under different attack strategies.

No. | Sector | Degree | Intensity | Betweenness | Clustering Coefficient |
---|---|---|---|---|---|

1 | Hohhot01 | 4 | 283 | 34.08 | 0.33 |

2 | Hohhot02 | 3 | 261 | 6.07 | 0.33 |

3 | Taiyuan01 | 3 | 447 | 20.15 | 0.33 |

4 | Taiyuan02 | 5 | 475 | 58.50 | 0.25 |

5 | Taiyuan03 | 3 | 256 | 25.75 | 0.33 |

6 | Taiyuan04 | 7 | 272 | 53.43 | 0.57 |

7 | Beijing01 | 8 | 823 | 156.03 | 0.4 |

8 | Beijing02 | 5 | 1923 | 51.81 | 0.55 |

9 | Beijing03 | 6 | 626 | 74.43 | 0.43 |

10 | Beijing04 | 7 | 708 | 103.47 | 0.33 |

11 | Beijing05 | 4 | 343 | 22.93 | 0.5 |

12 | Beijing06 | 4 | 406 | 13.59 | 0.5 |

13 | Beijing07 | 3 | 378 | 3.74 | 0.67 |

14 | Beijing08 | 3 | 924 | 6.33 | 0.33 |

15 | Beijing09 | 5 | 884 | 32.16 | 0.4 |

16 | Beijing10 | 4 | 636 | 24.66 | 0.5 |

17 | Beijing11 | 6 | 623 | 51.92 | 0.43 |

18 | Beijing12 | 3 | 456 | 2.72 | 0.67 |

19 | Beijing13 | 4 | 544 | 20.99 | 0.5 |

20 | Beijing14 | 3 | 578 | 2.16 | 0.67 |

21 | Beijing15 | 4 | 1203 | 12.96 | 0.5 |

22 | Beijing16 | 4 | 1656 | 12.42 | 0.5 |

23 | Beijing17 | 4 | 1320 | 8.67 | 0.67 |

Type of Intentional Attack | Description |
---|---|

Degree priority | Sort the network nodes by degree from high to low, and sequentially remove the nodes and associated edges from the network. |

Betweenness priority | Sort the network nodes by betweenness from high to low, and sequentially remove the nodes and associated edges from the network. |

Critical Sector | Network Efficiency | Potential Critical Sector | Betweenness | Betweenness Change Rate (%) |
---|---|---|---|---|

Beijing01 | 0.4454 | Hohhot01 | 54.42 | 159.7 |

Beijing04 | 0.435 | Beijing02 | 81.67 | 157.62 |

Beijing03 | 0.4437 | Beijing05 | 46.78 | 204.01 |

Taiyuan02 | 0.4193 | Taiyuan01 | 220.62 | 1094.89 |

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**MDPI and ACS Style**

Liang, H.; Zhang, S.; Kong, J.
Study on Characteristics and Invulnerability of Airspace Sector Network Using Complex Network Theory. *Aerospace* **2023**, *10*, 225.
https://doi.org/10.3390/aerospace10030225

**AMA Style**

Liang H, Zhang S, Kong J.
Study on Characteristics and Invulnerability of Airspace Sector Network Using Complex Network Theory. *Aerospace*. 2023; 10(3):225.
https://doi.org/10.3390/aerospace10030225

**Chicago/Turabian Style**

Liang, Haijun, Shiyu Zhang, and Jianguo Kong.
2023. "Study on Characteristics and Invulnerability of Airspace Sector Network Using Complex Network Theory" *Aerospace* 10, no. 3: 225.
https://doi.org/10.3390/aerospace10030225