# 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 |

## References

- Çolak, S.; Lima, A.; González, M.C. Understanding congested travel in urban areas. Nat. Commun.
**2016**, 7, 10793. [Google Scholar] [CrossRef][Green Version] - Brittain, M.; Wei, P. Autonomous air traffic controller: A deep multi-agent reinforcement learning approach. arXiv
**2019**, arXiv:1905.01303. [Google Scholar] - Rehman, A. Machine learning based air traffic control strategy. Int. J. Mach. Learn. Cybern.
**2021**, 12, 2151–2161. [Google Scholar] [CrossRef] - Ren, G.; Lu, C.; Zhu, J.; Liu, X. Analyzing the topological characteristic and key nodes of Chinese air sector network. Int. J. Mod. Phys. B
**2019**, 33, 1950100. [Google Scholar] [CrossRef] - Isufaj, R.; Omeri, M.; Piera, M.A.; Saez Valls, J.; Verdonk Gallego, C.E. From Single Aircraft to Communities: A Neutral Interpretation of Air Traffic Complexity Dynamics. Aerospace
**2022**, 9, 613. [Google Scholar] [CrossRef] - Angeloudis, P.; Fisk, D. Large subway systems as complex networks. Phys. A Stat. Mech. Its Appl.
**2006**, 367, 553–558. [Google Scholar] [CrossRef] - Porta, S.; Crucitti, P.; Latora, V. The network analysis of urban streets: A primal approach. Environ. Plan. B Plan. Des.
**2006**, 33, 705–725. [Google Scholar] [CrossRef][Green Version] - Sharifi, A. Resilient urban forms: A review of literature on streets and street networks. Build. Environ.
**2019**, 147, 171–187. [Google Scholar] [CrossRef] - Sen, P.; Dasgupta, S.; Chatterjee, A.; Sreeram, P.; Mukherjee, G.; Manna, S. Small-world properties of the Indian railway network. Phys. Rev. E
**2003**, 67, 036106. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bombelli, A.; Santos, B.F.; Tavasszy, L. Analysis of the air cargo transport network using a complex network theory perspective. Transp. Res. Part E Logist. Transp. Rev.
**2020**, 138, 101959. [Google Scholar] [CrossRef] - Lordan, O.; Sallan, J.M.; Simo, P. Study of the topology and robustness of airline route networks from the complex network approach: A survey and research agenda. J. Transp. Geogr.
**2014**, 37, 112–120. [Google Scholar] [CrossRef] - Zanin, M.; Lillo, F. Modelling the air transport with complex networks: A short review. Eur. Phys. J. Spec. Top.
**2013**, 215, 5–21. [Google Scholar] [CrossRef][Green Version] - Amaral, L.A.N.; Scala, A.; Barthelemy, M.; Stanley, H.E. Classes of small-world networks. Proc. Natl. Acad. Sci. USA
**2000**, 97, 11149–11152. [Google Scholar] [CrossRef][Green Version] - Kaiquan, C.; Jun, Z.; Wenbo, D.; Xianbin, C. Analysis of the Chinese air route network as a complex network. Chin. Phys. B
**2012**, 21, 028903. [Google Scholar] - Hossain, M.; Alam, S.; Rees, T.; Abbass, H. Australian airport network robustness analysis: A complex network approach. In Proceedings of the 36th Australasian Transport Research Forum (ATRF), Brisbane, Australia, 2–4 October 2013. [Google Scholar]
- Albert, R.; Jeong, H.; Barabási, A.-L. Error and attack tolerance of complex networks. Nature
**2000**, 406, 378–382. [Google Scholar] [CrossRef][Green Version] - Janssen, M.A.; Schoon, M.L.; Ke, W.; Börner, K. Scholarly networks on resilience, vulnerability and adaptation within the human dimensions of global environmental change. Glob. Environ. Chang.
**2006**, 16, 240–252. [Google Scholar] [CrossRef][Green Version] - Qian, B.; Zhang, N. Topology and Robustness of Weighted Air Transport Networks in Multi-Airport Region. Sustainability
**2022**, 14, 6832. [Google Scholar] [CrossRef] - Başpınar, B.; Gopalakrishnan, K.; Koyuncu, E.; Balakrishnan, H. An empirical study of the resilience of the US and European air transportation networks. J. Air Transp. Manag.
**2023**, 106, 102303. [Google Scholar] [CrossRef] - Chicco, G.; Mancarella, P. Matrix modelling of small-scale trigeneration systems and application to operational optimization. Energy
**2009**, 34, 261–273. [Google Scholar] [CrossRef] - Sergeeva, M.; Delahaye, D.; Mancel, C.; Vidosavljevic, A. Dynamic airspace configuration by genetic algorithm. J. Traffic Transp. Eng. (Engl. Ed.)
**2017**, 4, 300–314. [Google Scholar] [CrossRef] - Zhang, X.; Miller-Hooks, E.; Denny, K. Assessing the role of network topology in transportation network resilience. J. Transp. Geogr.
**2015**, 46, 35–45. [Google Scholar] [CrossRef][Green Version] - Bin, J.; Claramunt, C. Topological analysis of urban street networks. Environ. Plan. B Plan. Des.
**2004**, 31, 151–162. [Google Scholar] - Barrat, A.; Barthelemy, M.; Pastor-Satorras, R.; Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 3747–3752. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mahyar, H.; Hasheminezhad, R.; Ghalebi, E.; Nazemian, A.; Grosu, R.; Movaghar, A.; Rabiee, H.R. Compressive sensing of high betweenness centrality nodes in networks. Phys. A Stat. Mech. Its Appl.
**2018**, 497, 166–184. [Google Scholar] [CrossRef] - Said, A.; Abbasi, R.A.; Maqbool, O.; Daud, A.; Aljohani, N.R. CC-GA: A clustering coefficient based genetic algorithm for detecting communities in social networks. Appl. Soft Comput.
**2018**, 63, 59–70. [Google Scholar] [CrossRef] - Berahmand, K.; Bouyer, A.; Samadi, N. A new centrality measure based on the negative and positive effects of clustering coefficient for identifying influential spreaders in complex networks. Chaos Solitons Fractals
**2018**, 110, 41–54. [Google Scholar] [CrossRef] - Roy, S.; Xue, M.; Sridhar, B. Vulnerability metrics for the airspace system. In Proceedings of the 2017 FAA/Eurocontrol Air Traffic Management Research and Development Seminar, Seattle, WA, USA, 29 June 2017. [Google Scholar]
- Hébert-Dufresne, L.; Allard, A.; Young, J.-G.; Dubé, L.J. Global efficiency of local immunization on complex networks. Sci. Rep.
**2013**, 3, 1–8. [Google Scholar] [CrossRef] [PubMed][Green Version] - Veremyev, A.; Prokopyev, O.A.; Boginski, V.; Pasiliao, E.L. Finding maximum subgraphs with relatively large vertex connectivity. Eur. J. Oper. Res.
**2014**, 239, 349–362. [Google Scholar] [CrossRef] - Saleh, M.; Esa, Y.; Mohamed, A. Applications of Complex Network Analysis in Electric Power Systems. Energies
**2018**, 11, 1381. [Google Scholar] [CrossRef][Green Version] - Du, W.B.; Liang, B.Y.; Hong, C.; Lordan, O. Analysis of the Chinese provincial air transportation network. Phys. A Stat. Mech. Its Appl.
**2016**, 465, 579–586. [Google Scholar] [CrossRef][Green Version] - Wang, I.L.; Johnson, E.L.; Sokol, J.S. A multiple pairs shortest path algorithm. Transp. Sci.
**2005**, 39, 465–476. [Google Scholar] - Zaidi, F. Small world networks and clustered small world networks with random connectivity. Soc. Netw. Anal. Min.
**2013**, 3, 51–63. [Google Scholar] [CrossRef][Green Version] - Yazdani, A.; Otoo, R.A.; Jeffrey, P. Resilience enhancing expansion strategies for water distribution systems: A network theory approach. Environ. Model. Softw.
**2011**, 26, 1574–1582. [Google Scholar] [CrossRef] - Nie, T.; Guo, Z.; Zhao, K.; Lu, Z.-M. New attack strategies for complex networks. Phys. A Stat. Mech. Its Appl.
**2015**, 424, 248–253. [Google Scholar] [CrossRef] - Cohen, R.; Havlin, S. Complex Networks: Structure, Robustness and Function; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Zanin, M.; Sun, X.; Wandelt, S. Studying the topology of transportation systems through complex networks: Handle with care. J. Adv. Transp.
**2018**, 2018, 3156137. [Google Scholar] [CrossRef][Green Version] - Xia, Y.; Hill, D.J. Attack vulnerability of complex communication networks. IEEE Trans. Circuits Syst. II Express Briefs
**2008**, 55, 65–69. [Google Scholar] [CrossRef] - Perea, F.; Puerto, J. Revisiting a game theoretic framework for the robust railway network design against intentional attacks. Eur. J. Oper. Res.
**2013**, 226, 286–292. [Google Scholar] [CrossRef][Green Version] - Cook, A.; Blom, H.A.; Lillo, F.; Mantegna, R.N.; Micciche, S.; Rivas, D.; Zanin, M. Applying complexity science to air traffic management. J. Air Transp. Manag.
**2015**, 42, 149–158. [Google Scholar] [CrossRef][Green Version] - Muñiz, A.S.G.; Raya, A.M.; Carvajal, C.R. Key sectors: A new proposal from network theory. Reg. Stud.
**2008**, 42, 1013–1030. [Google Scholar] [CrossRef][Green Version] - Boccaletti, S.; Buldú, J.; Criado, R.; Flores, J.; Latora, V.; Pello, J.; Romance, M. Multiscale vulnerability of complex networks. Chaos Interdiscip. J. Nonlinear Sci.
**2007**, 17, 043110. [Google Scholar] [CrossRef][Green Version] - Ash, J.; Newth, D. Optimizing complex networks for resilience against cascading failure. Phys. A Stat. Mech. Its Appl.
**2007**, 380, 673–683. [Google Scholar] [CrossRef] - Dobson, I.; Carreras, B.A.; Lynch, V.E.; Newman, D.E. Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization. Chaos Interdiscip. J. Nonlinear Sci.
**2007**, 17, 026103. [Google Scholar] [CrossRef] [PubMed] - Bellingeri, M.; Bevacqua, D.; Scotognella, F.; Cassi, D. The heterogeneity in link weights may decrease the robustness of real-world complex weighted networks. Sci. Rep.
**2019**, 9, 10692. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bellingeri, M.; Bevacqua, D.; Scotognella, F.; Alfieri, R.; Cassi, D. A comparative analysis of link removal strategies in real complex weighted networks. Sci. Rep.
**2020**, 10, 3911. [Google Scholar] [CrossRef][Green Version] - Wandelt, S.; Lin, W.; Sun, X.; Zanin, M. From Random Failures to Targeted Attacks in Network Dismantling. Reliab. Eng. Syst. Saf.
**2021**, 218, 108146. [Google Scholar] [CrossRef]

**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