# Local and Network-Wide Time Scales of Delay Propagation in Air Transport: A Granger Causality Approach

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Real Operational Data

#### 2.2. Delay Propagation Assessment: The Granger Causality

#### 2.3. Network Reconstruction and Analysis

- Link density is the fraction of the potential edges in the network that are active, i.e., that have passed the statistical significance test.
- Diameter is the greatest distance between any pair of nodes in the network. Note that such a distance is defined as the number of links in the shortest path connecting the nodes and not the physical distance between them. Thus, it indicates how many intervening airports are needed to disseminate the delays throughout the whole network in the worst possible case.
- Transitivity measures the existence of triangles in the network and represents the propensity of nodes to form clusters. It is defined as the ratio between the number of closed triangles and of connected triplets of nodes. A high density of triplets of airports that are strongly connected (a high transitivity) means that a delay in one of them is easily propagated to the other airports of the group.
- Assortativity is the propensity of links to connect nodes of similar degrees (i.e., with a similar number of connections) [52].
- Efficiency measures how easily the network can move information within it and is defined as the inverse of the harmonic mean of the distances between pairs of nodes [53,54]. The efficiency of a network thus represents how easily information (here, delays) can move between two nodes and is inversely proportional to the number of intervening nodes needed on average to reach the destination of the propagation.
- The information content (IC) metric evaluates the presence of regularities in the adjacency matrix. It is defined as the quantity of information lost when pairs of nodes are iteratively merged [55]. Small values of IC indicate complex topological patterns, while large values correspond to random-like structures.
- Out-degree centrality is the number of edges coming out from a node. It represents how strongly an airport can influence other.
- In-degree centrality is the number of edges arriving to a node. It represents how strongly an airport is influenced by its neighbours.
- Betweenness centrality represents the fraction of times a node is included in the shortest path between two other nodes. Thus, it depicts the amount of influence a node has over the flow of information (delays) in a graph [56].

## 3. Synthetic Model of Delay Propagation

## 4. Analysis of Real Delay Propagation Patterns

#### 4.1. Methodological Viewpoint: Best Time Scale for Detecting Delay Propagation

#### 4.2. Operational Viewpoint: Delay Propagation Time

#### 4.3. Network Viewpoint: Propagation Network and Its Structure

## 5. Resampling Validation

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Information on the 50 airports considered in this study, including their 4-letter ICAO code, number of landing flights and percentage of flights delayed more than 10 and 30 min.

Rank | Name | ICAO | # Flights | % Delayed > 10 min. | % Delayed > 30 min. |
---|---|---|---|---|---|

1 | Frankfurt Airport | EDDF | 23,061 | 30.65% | 5.06% |

2 | Amsterdam Airport Schiphol | EHAM | 22,350 | 52.77% | 11.55% |

3 | Paris Charles de Gaulle Airport | LFPG | 22,275 | 33.76% | 5.39% |

4 | London Heathrow | EGLL | 19,407 | 68.39% | 20.39% |

5 | Munich Airport | EDDM | 18,588 | 25.22% | 2.75% |

6 | Adolfo Suárez Madrid-Barajas Airport | LEMD | 18,345 | 38.50% | 7.44% |

7 | Josep Tarradellas Barcelona-El Prat Airport | LEBL | 15,817 | 44.39% | 10.14% |

8 | Rome-Fiumicino International Airport | LIRF | 13,930 | 14.45% | 2.84% |

9 | Milan Malpensa Airport | LIMC | 13,383 | 12.67% | 2.84% |

10 | London Gatwick Airport | EGKK | 12,985 | 61.29% | 18.78% |

11 | Palma de Mallorca Airport | LEPA | 12,616 | 32.13% | 10.03% |

12 | Vienna International Airport | LOWW | 12,482 | 42.16% | 7.34% |

13 | Copenhagen Kastrup Airport | EKCH | 11,811 | 18.72% | 2.24% |

14 | Zürich Airport | LSZH | 11,426 | 41.24% | 5.56% |

15 | Oslo Airport | ENGM | 11,203 | 12.79% | 1.11% |

16 | Athens Intl Eleftherios Venizelos | LGAV | 10,808 | 25.54% | 3.22% |

17 | Dublin Airport | EIDW | 10,731 | 26.94% | 4.63% |

18 | Stockholm Arlanda Airport | ESSA | 10,585 | 22.57% | 2.51% |

19 | Brussels Airport | EBBR | 10,313 | 40.79% | 6.73% |

20 | Düsseldorf Airport | EDDL | 10,178 | 33.52% | 4.87% |

21 | Humberto Delgado Airport | LPPT | 9852 | 45.98% | 8.10% |

22 | Manchester Airport | EGCC | 9635 | 39.81% | 6.58% |

23 | Paris Orly Airport | LFPO | 9283 | 30.09% | 4.73% |

24 | London Stansted Airport | EGSS | 8634 | 41.52% | 6.24% |

25 | Berlin Tegel “Otto Lilienthal” Airport | EDDT | 8545 | 27.55% | 3.25% |

26 | Warsaw Chopin Airport | EPWA | 8434 | 10.91% | 1.67% |

27 | Václav Havel Airport Prague | LKPR | 7263 | 24.03% | 3.33% |

28 | Geneva Airport | LSGG | 7108 | 25.63% | 4.47% |

29 | Nice Côte d’Azur Airport | LFMN | 6703 | 20.50% | 3.43% |

30 | Málaga-Costa del Sol Airport | LEMG | 6660 | 35.47% | 4.76% |

31 | Hamburg Airport | EDDH | 6531 | 21.45% | 2.56% |

32 | Cologne Bonn Airport | EDDK | 6423 | 26.48% | 3.30% |

33 | London Luton Airport | EGGW | 6103 | 46.53% | 8.65% |

34 | Stuttgart Airport | EDDS | 6066 | 28.70% | 3.07% |

35 | Edinburgh Airport | EGPH | 5843 | 21.80% | 2.57% |

36 | Boryspil International Airport | UKBB | 5620 | 22.94% | 2.74% |

37 | Budapest Ferenc Liszt International Airport | LHBP | 5452 | 17.59% | 2.27% |

38 | Bucharest Henri Coandă International Airport | LROP | 5390 | 13.36% | 1.78% |

39 | Lyon-Saint Exupéry Airport | LFLL | 5127 | 26.41% | 3.49% |

40 | Alicante-Elche Miguel Hernández Airport | LEAL | 4946 | 34.17% | 5.14% |

41 | Birmingham Airport | EGBB | 4891 | 20.40% | 2.25% |

42 | Venice Marco Polo Airport | LIPZ | 4720 | 27.58% | 5.76% |

43 | Francisco Sá Carneiro Airport | LPPR | 4443 | 38.06% | 10.44% |

44 | Orio al Serio International Airport | LIME | 4417 | 16.32% | 2.04% |

45 | Marseille Provence Airport | LFML | 4329 | 23.56% | 2.73% |

46 | Toulouse-Blagnac Airport | LFBO | 4159 | 17.62% | 1.90% |

47 | Naples International Airport | LIRN | 4041 | 27.86% | 3.07% |

48 | Glasgow Airport | EGPF | 3732 | 28.27% | 3.99% |

49 | Catania-Fontanarossa Airport | LICC | 3660 | 14.64% | 1.53% |

50 | Bologna Guglielmo Marconi Airport | LIPE | 3519 | 25.09% | 2.90% |

Metric | Definition | Range |
---|---|---|

Link density | ${l}_{d}=\frac{L}{{N}^{2}}$, | $0\le {l}_{d}\le 1$ |

L being the total number of active links in the network and N the number of nodes. | ||

Diameter | $D={max}_{i,j}{d}_{i,j}$, | $0\le D\le N$ |

${d}_{i,j}$ being the distance between nodes i and j | ||

Transitivity | $T=\frac{3{N}_{\u25b5}}{{N}_{3}},$ | |

${N}_{\u25b5}$ being the number of closed triangles $3{N}_{\u25b5}=\sum _{k>i>j}{a}_{i,j}{a}_{i,k}{a}_{j,k}$, and ${N}_{3}$ the number of connected triplets ${N}_{3}=\sum _{k>i>j}({a}_{i,j}{a}_{i,k}+{a}_{j,i}{a}_{j,k}+{a}_{k,i}{a}_{k,j})$. | $0\le T\le 1$ | |

Assortativity | $r=\frac{1}{{\sigma}_{q}^{2}}{\sum}_{jk}jk({e}_{jk}-{q}_{j}{q}_{k})$, | $-1\le r\le 1$ |

j and k being the degrees of nodes at each end of a link; ${q}_{k}$ the distribution of the remaining degree, i.e., of the degree without the link under study; ${\sigma}_{q}^{2}$ the variance of the distribution ${q}_{k}$; and ${e}_{jk}$ the joint probability distribution of the remaining degrees of the two vertices at either end of a randomly chosen link. | ||

Efficiency | $E=\frac{1}{N(N-1)}\sum _{i\ne j}\frac{1}{{d}_{ij}}$, | $0\le E\le 1$ |

${d}_{ij}$ being the distance between the nodes i and j. | ||

Information Content | See [55] | $0\le IC$ |

**Table A3.**Formal definition of the centrality metrics considered in this study. Note that these metrics are calculated for a single node (here denoted as w), as opposed to the whole network; nodes can then be ranked in importance accordingly.

Centrality Metric | Definition |
---|---|

Out-degree centrality | ${c}_{O}\left(w\right)\propto {\sum}_{j}{a}_{w,j}$, |

where ${a}_{w,j}$ is equal to one if a link exists between nodes w and j, and zero otherwise. | |

In-degree centrality | ${c}_{I}\left(w\right)\propto {\sum}_{j}{a}_{j,w}$, |

where ${a}_{j,w}$ is equal to one if a link exists between nodes j and w, and zero otherwise. | |

Betweenness centrality | $c}_{B}\left(w\right)\propto \sum _{s,t\in V}\frac{{P}_{w}(s,t)}{P(s,t)$, |

where V is the set of nodes, $P(s,t)$ the number of shortest paths between s and t, and ${P}_{w}(s,t)$ the number of shortest paths between s and t that includes w. |

**Figure A1.**Scatter plot of the length of the optimal window for each pair of airports as a function of the number of operations recorded in the data set for the source (X axis) and destination (Y axis) airports.

**Table A4.**Coefficients of determination ${R}^{2}$ obtained by fitting linear models to recover the best window size for each pair of airports, using the corresponding features listed in the left columns; a: source airport of the causality (i.e., the cause); b: destination airport of the causality (i.e., the consequence).

Input Time | Metric | ${\mathit{R}}^{2}$ | |||||
---|---|---|---|---|---|---|---|

Series | a | b | $\mathit{a}\xb7\mathit{b}$ | ${log}_{2}\mathit{a}/\mathit{b}$ | Max | Min | |

Landing times | MAD | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $0.05$ | $0.02$ |

Hurst exponent | $0.00$ | $0.01$ | $0.01$ | $0.00$ | $0.02$ | $0.00$ | |

Landing separation | Mean | $0.01$ | $0.01$ | $0.03$ | $0.00$ | $0.02$ | $0.02$ |

Standard deviation | $0.00$ | $0.04$ | $0.03$ | $0.01$ | $0.02$ | $0.04$ | |

Linear correlation | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | |

MAD | $0.01$ | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $0.01$ | |

Delays | Mean | $0.01$ | $0.02$ | $0.01$ | $0.00$ | $0.02$ | $0.01$ |

Standard deviation | $0.01$ | $0.02$ | $0.02$ | $0.00$ | $0.02$ | $0.01$ | |

Linear correlation | $0.01$ | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $0.00$ | |

MAD | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | |

Hurst exponent | $0.02$ | $0.01$ | $0.03$ | $0.00$ | $0.02$ | $0.02$ | |

$\Delta $ consecutive delays | Mean | $0.00$ | $0.01$ | $0.00$ | $0.00$ | $0.01$ | $0.01$ |

Standard deviation | $0.00$ | $0.01$ | $0.01$ | $0.00$ | $0.01$ | $0.01$ | |

Linear correlation | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | |

MAD | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ |

**Table A5.**List of the ten pairs of airports with the longest propagation time (in hours); a: source airport of the causality (i.e., the cause); b: destination airport of the causality (i.e., the consequence).

Airport a | Airport b | Distance (NM) | Prop. Time $\mathit{a}\to \mathit{b}$ |
---|---|---|---|

EBBR | EDDH | 482 | 8.0 |

LIME | LEMG | 1549 | 8.0 |

EDDH | EDDS | 552 | 7.5 |

EDDT | LPPR | 2077 | 7.3 |

LEBL | LHBP | 1523 | 7.1 |

EBBR | EGBB | 463 | 7.1 |

LIMC | LOWW | 657 | 7.1 |

LIMC | LKPR | 646 | 7.1 |

LIME | LOWW | 588 | 7.1 |

EGBB | EBBR | 463 | 7.1 |

**Table A6.**List of the ten pairs of airports with the shortest propagation time (in hours); a: source airport of the causality (i.e., the cause); b: destination airport of the causality (i.e., the consequence).

Airport a | Airport b | Distance (NM) | Prop. Time $\mathit{a}\to \mathit{b}$ |
---|---|---|---|

LFPO | EDDS | 502 | 0.8 |

UKBB | LGAV | 1485 | 1.0 |

EDDS | LFPO | 502 | 1.2 |

EHAM | EDDH | 379 | 1.2 |

LIML | ENGM | 1645 | 1.4 |

LGAV | UKBB | 1485 | 1.8 |

EDDS | EDDH | 552 | 1.8 |

LFML | ENGM | 1906 | 2.0 |

ENGM | EDDS | 1286 | 2.0 |

LSZH | EPWA | 1031 | 2.2 |

**Table A7.**List of the ten pairs of airports with the most asymmetrical propagation time, defined as the difference between the forward (reported in the fourth column) and backward time (fifth column); a: source airport of the causality (i.e., the cause); b: destination airport of the causality (i.e., the consequence). Propagation times in hours.

Airport a | Airport b | Distance (NM) | Prop. Time $\mathit{a}\to \mathit{b}$ | Prop. Time $\mathit{b}\to \mathit{a}$ |
---|---|---|---|---|

EDDH | EDDS | 552 | 7.5 | 1.8 |

LIME | LEMG | 1549 | 8.0 | 3.7 |

EBBR | EDDH | 482 | 8.0 | 4.0 |

LICC | LSGG | 1223 | 6.5 | 4.2 |

EKCH | ESSA | 546 | 7.0 | 5.2 |

LIMC | EDDS | 342 | 6.8 | 5.0 |

LGAV | LPPR | 2803 | 7.0 | 5.7 |

LIMC | EHAM | 795 | 7.0 | 5.8 |

LIMC | LOWW | 657 | 7.1 | 6.0 |

LIME | LOWW | 588 | 7.1 | 6.0 |

**Figure A2.**Probability distributions of delay propagation times. Violin plots report the distribution of the delay propagation times in hours for all 50 considered airports for outgoing (i.e., delays propagated by an airport, top panels) and incoming (i.e., delays received by an airport, bottom panels) links.

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**Figure 1.**Analysis of the synthetic model—see Section 3 for details. (

**Top**panels) Example of the synthesis of one time series, with the calculation of the global delay trend using a Lorenz system (

**top left**panel); synthesis of individual landing events (

**top central**panel); and reconstruction of the evolution of the average delay (

**top right**panel); (

**Middle**panels) Analysis of the Granger causality test as a function of the traffic volume: evolution of the fraction of significant tests as a function of the window size w and of the event separation time $\eta $; (

**left middle**panel); and size of the best window size, i.e., the one maximising the fraction of significant tests for different values of $\eta $ (

**right middle**panel); (

**Bottom**panels) Analysis of the Granger causality test as a function of the traffic regularity: evolution of the fraction of significant tests as a function of w and of the asymmetry exponent e (

**left bottom**panel); and value of w maximising that fraction as a function of e, as obtained from the quadratic fits (

**right bottom**panel). In the middle and bottom panels, coloured points in the right panels correspond to the line of the same colour in the left ones.

**Figure 2.**Best time scale for detecting real delay propagation patterns. (

**Top left**) window length minimising the p-value of the Granger causality test for each pair of source–destination airports; note that results are only reported for those pairs for which the test is statistically significant and that airports are sorted in decreasing number of operations (see Table A1); (

**Top right**) histogram of the best window lengths w; (

**Middle right**) box plot depicting the distribution of the best window lengths for each airport, considering the links it causes, with airports sorted by decreasing number of operations; boxes indicate the interquartile range (Q3–Q1) and blue horizontal lines the median of the distribution; (

**Bottom**) scatter plots of the four features that best predict the length of the optimal window as a function of the latter for each statistically significant causality link; the black lines represent the best linear fit. L. sep.: landing separation; L. time: landing time. See main text and Table A4 for details on the metrics.

**Figure 3.**Delay propagation time. The left panel reports the delay propagation time in hours between each pair of airports. Only pairs with a statistically significant propagation are reported, and airports are sorted in decreasing number of operations (see Table A1). The right panel represents a scatter plot of the propagation time in hours as a function of the distance between each pair of airports. The diagonal dotted line approximates the flight time as a function of the distance.

**Figure 4.**Graphical representation of the most extreme delay propagation times. Red and green arrows indicate the five pairs of airports with, respectively, the largest and smallest propagation times; blue lines indicate those pairs with the most asymmetrical ones. See Table A5, Table A6 and Table A7 for numerical values.

**Figure 5.**Violin plots reporting the distributions of propagation times as a function of the size of the airports at each end of the link (

**left**panel) and of the distance between them (

**right**panel). L: large; S: small.

**Figure 6.**Evolution of the six topological metrics described in Section 2.3 (blue lines, left Y axes), as a function of the size of the window used for calculating the Granger causality. Aqua lines (right Y axes) depict the reliability of each value, i.e., the fraction of pairs of airports having operations in at least $50\%$ of the windows used to calculate the causality. The horizontal dotted lines indicate the values of the metrics when the optimal window size is used for each link.

**Figure 7.**Evolution of the centrality ranking as a function of the window size used to calculate the Granger causality. From left to right, the three panels depict the evolution of the ranking according to the out-degree, in-degree and betweenness centrality. In each case, the five reported airports are those with the maximal centrality in the networks reconstructed with optimal window sizes.

**Figure 8.**Probability distribution of $\delta p$, i.e., the difference between the p-values obtained with the original and the resampled time series. Each violin plot corresponds to a different resampling, and its values to all pairs of airports with a statistically significant causality (according to the original time series). The water green line (right Y axis) shows the percentage of pairs for which the resampled time series yield a smaller p-value (that is, a more statistically significant relationship) than the one obtained with the original series.

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## Share and Cite

**MDPI and ACS Style**

Pastorino, L.; Zanin, M.
Local and Network-Wide Time Scales of Delay Propagation in Air Transport: A Granger Causality Approach. *Aerospace* **2023**, *10*, 36.
https://doi.org/10.3390/aerospace10010036

**AMA Style**

Pastorino L, Zanin M.
Local and Network-Wide Time Scales of Delay Propagation in Air Transport: A Granger Causality Approach. *Aerospace*. 2023; 10(1):36.
https://doi.org/10.3390/aerospace10010036

**Chicago/Turabian Style**

Pastorino, Luisina, and Massimiliano Zanin.
2023. "Local and Network-Wide Time Scales of Delay Propagation in Air Transport: A Granger Causality Approach" *Aerospace* 10, no. 1: 36.
https://doi.org/10.3390/aerospace10010036