# Modeling and Feature Analysis of Air Traffic Complexity Propagation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Propagation Models

#### 2.1. Definition of Air Traffic Complexity Propagation Behavior

#### 2.2. Modeling of Air Traffic Complexity Propagation

## 3. Influence Factors of the Propagation Model

#### 3.1. Relative Velocity and Duration

#### 3.2. Clustering Degree

#### 3.2.1. Aircraft Group Division

- Preprocessing of trajectory data

- 2.
- Clustering algorithm

- 3.
- Determination of clustering result.

#### 3.2.2. Computation of Clustering Degree

#### 3.3. Nodes and Network Evolution Features

## 4. Propagation Model Solving and Propagation Ability Division

#### 4.1. Solving the Differential Equation of the Propagation Model

#### 4.2. Optimization of Differential Equation Parameters

#### 4.3. Division of Propagation Ability

## 5. Case Validation

#### 5.1. Description of Real Data

#### 5.2. Calculated Results of Propagation Model Parameters

^{2}of the propagation model is 0.917, indicating that the solved parameters can better describe the propagation process of air traffic complexity.

#### 5.3. Results of Propagation Ability Division

#### 5.4. Analytical Results of Influence Factors

- Aircraft with high propagation capability (Q1)

- 2.
- Aircraft with medium propagation capability (Q2)

- 3.
- Aircraft with low propagation capability (Q3)

## 6. Conclusions

^{2}of multiple linear regression still has room for improvement, and more influencing factors need to be further studied in the future. This model provides a feasible way to explore the complexity evolution mechanism and control method.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Schmidt, D.K. On modeling ATC workload and sector capacity. J. Aircr.
**1976**, 13, 531–537. [Google Scholar] [CrossRef] - Hurst, M.W.; Rose, R.M. Objective job difficulty, behavioral response, and sector characteristics in air route traffic control centers. Ergonomics
**1978**, 21, 697–708. [Google Scholar] [CrossRef] - Netjasov, F.; Janic, M.; Tosic, V. Developing a generic metric of terminal airspace traffic complexity. Transportmetrica
**2011**, 7, 369–394. [Google Scholar] [CrossRef] - Prevot, T.; Lee, P.U. Trajectory-based complexity (TBX): A modified aircraft count to predict sector complexity during trajectory-based operations. In Proceedings of the IEEE/AIAA 30th Digital Avionics Systems Conference, Seattle, WA, USA, 16–20 October 2011. [Google Scholar]
- Lee, K.; Feron, E.; Pritchett, A. Describing airspace complexity: Airspace response to disturbances. J. Guid. Control Dyn.
**2009**, 32, 210–222. [Google Scholar] [CrossRef] - Lee, K.; Feron, E.; Pritchett, A. Air traffic complexity: An input-output approach. In Proceedings of the 2007 American Control Conference, New York, NY, USA, 9–13 July 2007. [Google Scholar]
- Lyons, R. Complexity analysis of the next gen air traffic management system: Trajectory based operations. Work
**2012**, 41, 4514–4522. [Google Scholar] [CrossRef] - Prandini, M.; Piroddi, L.; Puechmorel, S. Toward air traffic complexity assessment in new generation air traffic management systems. IEEE Trans. Intell. Transp. Syst.
**2011**, 12, 809–818. [Google Scholar] [CrossRef] - Prandini, M.; Putta, V.; Hu, J. Air traffic complexity in future Air Traffic Management systems. J. Aerosp. Oper.
**2012**, 1, 281–299. [Google Scholar] [CrossRef] - Rocha, E.C. Dynamics of air transport networks: A review from a complex systems perspective. Chin. J. Aeronaut.
**2017**, 2, 7–16. [Google Scholar] [CrossRef] - Sun, X.Q.; Wandelt, S. Robustness of air transportation as complex networks: Systematic review of 15 years of research and outlook into the future. Sustainability
**2021**, 13, 6446. [Google Scholar] [CrossRef] - Hossain, M.M.; Alam, S.; Symon, F.; Blom, H. A Complex Network Approach to Analyze the Effect of Intermediate Waypoints on Collision Risk Assessment. Air Traffic Control Q.
**2014**, 22, 87–114. [Google Scholar] [CrossRef] - Hossain, M.M.; Alam, S. A complex network approach towards modeling and analysis of the Australian Airport Network. J. Air Transp. Manag.
**2017**, 60, 1–9. [Google Scholar] - Complexity Metrics for ANSP Benchmarking Analysis. Available online: https://www.eurocontrol.int/publication/complexity-metrics-air-navigation-service-providers (accessed on 1 April 2006).
- Wang, H.Y.; Xu, X.H.; Zhao, Y.F. Empirical analysis of aircraft clusters in air traffic situation networks. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2017**, 231, 1718–1731. [Google Scholar] [CrossRef] - Wang, H.Y.; Song, Z.Q.; Wen, R.Y. Modeling air traffic situation complexity with a dynamic weighted network approach. J. Adv. Transp.
**2018**, 1, 1–15. [Google Scholar] [CrossRef] - Holme, P.; Saramäki, J. Temporal networks. Phys. Rep.
**2012**, 519, 97–125. [Google Scholar] [CrossRef] - Holme, P. Modern temporal network theory: A colloquium. Eur. Phys. J. B
**2015**, 88, 1–30. [Google Scholar] [CrossRef] - Rebollo, J.J.; Balakrishnan, H. Characterization and prediction of air traffic delays. Transp. Res. Part C Emerg. Technol.
**2014**, 44, 231–241. [Google Scholar] [CrossRef] - Sismanidou, A.; Tarradellas, J.; Suau-Sanchez, P. The uneven geography of us air traffic delays: Quantifying the impact of connecting passengers on delay propagation. J. Transp. Geogr.
**2022**, 98, 103260. [Google Scholar] - Balcan, D.; Colizza, V.; Goncalves, B.; Hu, H.; Ramasco, J.J.; Vespignani, A. Multiscale mobility networks and the large scale spreading of infectious diseases. Proc. Natl. Acad. Sci. USA
**2009**, 106, 21484–21489. [Google Scholar] [CrossRef] - Balcan, D.; Hu, H.; Goncalves, B.; Bajarid, P.; Poletto, C.; Ramasco, J.J.; Paolotti, D.; Perra, N.; Tizzoni, M.; Broeck, W.V.D.; et al. Seasonal transmission potential and activity peaks of the new influenza A (H1N1): A Monte Carlo Likelihood analysis based on human mobility. BMC Med.
**2009**, 7, 45. [Google Scholar] [CrossRef] - Fleurquin, P.; Ramasco, J.J.; Eguiluz, V.M. Systemic delay propagation in the US airport network. Sci. Rep.
**2013**, 3, 1159. [Google Scholar] [CrossRef] - Pyrgiotis, N.; Malone, K.M.; Odoni, A. Modelling delay propagation within an airport network. Transp. Res.
**2013**, 27C, 60–75. [Google Scholar] [CrossRef] - Li, Q.; Jing, R. Characterization of delay propagation in the air traffic network. J. Air Transp. Manag.
**2021**, 94, 102075. [Google Scholar] [CrossRef] - Zanin, M.; Belkoura, S.; Zhu, Y. Network analysis of chinese air transport delay propagation. Chin. J. Aeronaut.
**2017**, 30, 491–499. [Google Scholar] [CrossRef] - Wang, Y.; Zheng, H.; Wu, F.; Chen, J.; Hansen, M. A Comparative Study on Flight Delay Networks of the USA and China. J. Adv. Transp.
**2020**, 8, 1–11. [Google Scholar] [CrossRef] - Li, S.M.; Xie, D.F.; Zhang, X.; Zhang, Z.Y.; Bai, W. Data-Driven Modeling of Systemic Air Traffic Delay Propagation: An Epidemic Model Approach. J. Adv. Transp.
**2020**, 2020, 1–12. [Google Scholar] [CrossRef] - Cai, Q.; Alam, S.; Duong, V.N. A Spatial-Temporal Network Perspective for the Propagation Dynamics of Air Traffic Delays. Engineering
**2021**, 7, 452–464. [Google Scholar] [CrossRef] - Shao, W.; Prabowo, A.; Zhao, S.; Koniusz, P.; Salim, F.D. Predicting flight delay with spatio-temporal trajectory convolutional network and airport situational awareness map. Neurocomputing
**2022**, 472, 280–293. [Google Scholar] [CrossRef] - Jiang, J.L.; Fang, H.; Li, S.Q.; Li, W.M. Identifying important nodes for temporal networks based on the ASAM model. Phys. A: Stat. Mech. Its Appl.
**2022**, 586, 126455. [Google Scholar] [CrossRef] - Maulud, D.; Abdulazeez, A.M. A review on linear regression comprehensive in machine learning. J. Appl. Sci. Technol. Trends
**2020**, 1, 140–147. [Google Scholar] [CrossRef]

**Figure 1.**Diagram of air traffic situation temporal network. Node: aircraft; Edge: between-aircraft proximity relation.

**Figure 3.**Simple Air Traffic Scenario. (

**a**): four non-source aircraft with the same velocity; (

**b**): four non-source aircraft with the same velocity and closer distance; (

**c**): Three of the four non-source aircraft with the same velocity; Blue aircraft: source aircraft; Yellow aircraft: non-source aircraft.

**Figure 4.**Schematic diagram of aircraft group. Red square: sector boundary; Black circle: aircraft group; Green circle: between-aircraft proximity relations; Black dotted line: route.

**Figure 8.**The Propagation Process of Three Types of Aircraft with Different Propagation Capabilities.

**Figure 9.**(

**a**) I of Q1 when T increase or decrease by one standard deviation; (

**b**) IND of Q1 when T increase or decrease by one standard deviation; (

**c**) I of Q1 when ci increase or decrease by one standard deviation; (

**d**) IND of Q1 when ci increase or decrease by one standard deviation.

**Figure 10.**(

**a**) I of Q2 when T increase or decrease by one standard deviation; (

**b**) IND of Q2 when T increase or decrease by one standard deviation; (

**c**) I of Q2 when C increase or decrease by one standard deviation; (

**d**) IND of Q2 when C increase or decrease by one standard deviation.

Q1 | Q2 | Q3 | ||||
---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | |

Clustering center of infection rate | 0.6891 | 0.7016 | 0.4312 | 0.428 | 0.4622 | 0.4867 |

Clustering center of recovery rate | 0.2933 | 0.3042 | 0.1687 | 0.1571 | 0.6901 | 0.7092 |

Aircraft number | 1231 | 1126 | 2317 | 1926 | 278 | 242 |

Proportion of aircraft number | 32% | 34% | 61% | 58% | 7% | 8% |

Model | Influence Factor | Unstandardized Coefficients | Standardized Coefficients | Sig. | |||
---|---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | ||

Infection rate of Q1 | (constant) | 0.731 | 0.730 | 0.000 | 0.000 | ||

Velocity | 1.96 × 10^{−6} | 6.62 × 10^{−6} | 0.002 | 0.008 | 0.964 | 0.877 | |

Duration | −0.001 | −0.001 | −0.177 | −0.185 | 0.001 | 0.001 | |

Clustering degree | −9.71 × 10^{−6} | −9.89 × 10^{−6} | −0.072 | −0.073 | 0.181 | 0.175 | |

ci | 0.008 | 0.007 | 0.138 | 0.125 | 0.013 | 0.023 | |

c | 0.015 | 0.024 | 0.033 | 0.050 | 0.540 | 0.350 | |

R^{2} (DAY1) = 0.267R ^{2} (DAY2) = 0.251 |

Model | Influence Factor | Unstandardized Coefficients | Standardized Coefficients | Sig. | |||
---|---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | ||

Recovery rate of Q1 | (constant) | 0.346 | 0.344 | 0.000 | 0.000 | ||

Velocity | 5.73 × 10^{−5} | 6.50 × 10^{−5} | 0.084 | 0.092 | 0.094 | 0.065 | |

Duration | −0.002 | −0.002 | −0.422 | −0.424 | 0.000 | 0.000 | |

Clustering degree | −4.21 × 10^{−7} | −6.70 × 10^{−7} | −0.004 | −0.006 | 0.942 | 0.911 | |

ci | 0.006 | 0.005 | 0.116 | 0.088 | 0.023 | 0.041 | |

c | −0.022 | −0.009 | −0.054 | −0.021 | 0.281 | 0.666 | |

R^{2} (DAY1) = 0.282R ^{2} (DAY2) = 0.293 |

Model | Influence Factor | Unstandardized Coefficients | Standardized Coefficients | Sig. | |||
---|---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | ||

Infection rate of Q2 | (constant) | 0.487 | 0.484 | 0.000 | 0.000 | ||

Velocity | 9.40 × 10^{−6} | 1.30 × 10^{−5} | 0.013 | 0.018 | 0.793 | 0.714 | |

Duration | −0.001 | −0.001 | −0.140 | −0.143 | 0.003 | 0.002 | |

Clustering degree | −1.90 × 10^{−5} | −1.86 × 10^{−5} | −0.241 | −0.235 | 0.000 | 0.000 | |

ci | 0.002 | 0.002 | 0.041 | 0.038 | 0.386 | 0.412 | |

c | −0.002 | −0.001 | −0.004 | −0.002 | 0.924 | 0.964 | |

R^{2} (DAY1) = 0.277R ^{2} (DAY2) = 0.285 |

Model | Influence Factor | Unstandardized Coefficients | Standardized Coefficients | Sig. | |||
---|---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | ||

Recovery rate of Q2 | (constant) | 0.216 | 0.216 | 0.000 | 0.000 | ||

Velocity | 1.41 × 10^{−5} | 1.44 × 10^{−5} | 0.026 | 0.027 | 0.586 | 0.574 | |

Duration | −0.001 | −4.97 × 10^{−4} | −0.131 | −0.127 | 0.004 | 0.006 | |

Clustering degree | −1.38 × 10^{−5} | −1.34 × 10^{−5} | −0.239 | −0.233 | 0.000 | 0.000 | |

ci | −0.003 | −0.004 | −0.083 | −0.088 | 0.074 | 0.057 | |

c | −0.013 | −0.014 | −0.038 | −0.041 | 0.401 | 0.361 | |

R^{2} (DAY1) = 0.254R ^{2} (DAY2) = 0.232 |

Model | Influence Factor | Unstandardized Coefficients | Standardized Coefficients | Sig. | |||
---|---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | ||

Infection rate of Q3 | (constant) | 1.032 | 1.058 | 0.000 | 0.000 | ||

Velocity | −1.00 × 10^{−3} | −0.001 | −0.340 | −0.374 | 0.014 | 0.005 | |

Duration | −0.002 | −0.002 | −0.147 | −0.125 | 0.328 | 0.364 | |

Clustering degree | −1.28 × 10^{−6} | −2.11 × 10^{−5} | 0.002 | −0.040 | 0.987 | 0.765 | |

ci | 0.009 | 0.006 | 0.053 | 0.037 | 0.694 | 0.774 | |

c | 0.043 | 0.079 | 0.046 | 0.086 | 0.714 | 0.464 | |

R^{2} (DAY1) = 0.155R ^{2} (DAY2) = 0.149 |

Model | Influence Factor | Unstandardized Coefficients | Standardized Coefficients | Sig. | |||
---|---|---|---|---|---|---|---|

DAY1 | DAY2 | DAY1 | DAY2 | DAY1 | DAY2 | ||

Recovery rate of Q3 | (constant) | 0.746 | 0.786 | 0.000 | 0.000 | ||

Velocity | −1.93 × 10^{−5} | −7.41 × 10^{−5} | −0.021 | −0.085 | 0.878 | 0.532 | |

Duration | 0.001 | 0.001 | 0.153 | 0.146 | 0.327 | 0.319 | |

Clustering degree | −2.99 × 10^{−5} | −3.19 × 10^{−5} | −0.095 | −0.102 | 0.533 | 0.473 | |

ci | 0.004 | 0.002 | 0.041 | 0.018 | 0.769 | 0.893 | |

c | −0.109 | −0.072 | −0.198 | −0.132 | 0.133 | 0.291 | |

R^{2} (DAY1) = 0.094R ^{2} (DAY2) = 0.047 |

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**MDPI and ACS Style**

Wang, H.; Xu, P.; Zhong, F.
Modeling and Feature Analysis of Air Traffic Complexity Propagation. *Sustainability* **2022**, *14*, 11157.
https://doi.org/10.3390/su141811157

**AMA Style**

Wang H, Xu P, Zhong F.
Modeling and Feature Analysis of Air Traffic Complexity Propagation. *Sustainability*. 2022; 14(18):11157.
https://doi.org/10.3390/su141811157

**Chicago/Turabian Style**

Wang, Hongyong, Ping Xu, and Fengwei Zhong.
2022. "Modeling and Feature Analysis of Air Traffic Complexity Propagation" *Sustainability* 14, no. 18: 11157.
https://doi.org/10.3390/su141811157