Network analysis

The question of the non-linear propagation of delay has not extensively been studied in the literature, as much of the focus has been centred, for example, on the delay multiplier metric - a metric designed to detect linear correlations [BHBR99]. We will show in Section 6.1 that the function driving propagation is more complex than expected, especially because it presents a high heterogeneity across airports, and that non-linearities might be triggered when high magnitude delays are suffered. Specifically, we will show how some airports present a supralinear growth of the outbound delay for severe perturbations, while others seem to have enough operational buffers to control extreme situations. High delays are therefore more propitious to explode, as they can be non-linearly propagated; ergo the importance of detecting the non-linear propagation patterns.

By applying the NNC causality test to each pair of airports and their corresponding stationarised delay time series, we are able to construct a functional network analog to Fig. 5.3 -representing non-linear propagations within the system (Fig. 5.14). Note the yet strong pres-ence of LMML (Malta airport), reinforcing the idea - suggested earlier by the GC Network - of its central role in delay propagation⁵.

Does LMML propagates delays in a linear fashion as suggested by GC? Or in a non-linear fashion as suggested by the NCC network? The answer lies within the characteristics of both causality metrics. First, GC is able to detect non-linear couplings⁶ under small perturbations, i.e. small delay magnitude. Secondly, NNC (with default parameters) does not detect linear causation if the relation is weak (see results in Annex A). Finally, both successfully detect

5NNC metric results are confirmed through a validation process by performing a shuffling of the time series (see Annex A for more details).

6Such is done considering the linear part of a non-linear function (which is equivalent to extracting the Jacobian of that function) as a good approximation for small magnitudes.

LTAI

Figure 5.14: The left panel represents the complete NNC network, the right one the network corresponding to the 10 biggest airports. In all cases, the size of nodes is proportional to their degree.

non-weak linear relationships or strong non-linear causation for small delays. As such, the deactivation in the NNC network (Fig. 5.14) of the links present in the GC network (Fig.

5.3, left panel) yields a representation of the non-linear propagation network (see blue network in Fig. 5.15). Similarly, deactivating the NNC links in the GC network only leaves weak linear relationships - see green network in Fig. 5.15). Finally, the network composed of links present both in NNC and GC networks represents linear and small perturbation’s non-linear propagations (see yellow network in Fig. 5.15).

The LMML airport’s utter centrality, being spotted by both NNC and GC, suggests either a strong linear propagation of LMML delays or their systematical non-linear propagation when their magnitude is low. The second option, i.e. non-linear propagation of small delays, is in accordance with the EE network’s results. Indeed, the low magnitude of the delays at LMML combined with their low variance results in a stationarised time series with high kurtosis, and subsequently to the deficiency of extreme unexpected delay, thus explaining the inexistent role of the LMML airport in EE propagation.

LEZL

Figure 5.15: Linear, non-linear and low magnitude delay propagation networks, respectively in green, blue and yellow. See main text for more details. Size of nodes is proportional to the node’s number of connections.

Also, big airports seem to propagate their delays in a non-linear way. It can be appreciated how the connections between the 10 busiest airports are more densde in Fig. 5.14 than in Fig.

5.4. The connections between big airports are therefore non-linear, but also convey important (i.e. high magnitude) perturbations ⁷. To fully characterise the centrality and the role of the airports in both the weak linear (green network) or non-linear (blue network) networks, we have to analyse the ratio of inbound per outbound delay propagation connections. Fig. 5.16, left and right panels, displays such ratios respectively for the weak linear network and the non-linear one, as a function of the number of operations registered at the airports. It is easily observed how large airports have significantly much more inbounds than outbounds connections when

7Otherwise GC would be able to detect them

0 20000 40000 60000 80000 100000 120000

0 20000 40000 60000 80000 100000 120000

Number of operations

Figure 5.16: NNC Network links analysis. Left and right panels represents the number inbound link per outbound one for, respectively, the weak linear (green network) and non-linear (blue network) propagation networks.

the propagation is non-linear (right panel). Inversely, small airports have more outbounds than inbounds links. This is further corroborated by a positive linear fit (R² = 0.518). On the other hand, the ratio of inbounds and outbounds connections seems to be decorrelated from the airport size for the weak linear propagation network (R² = 0.021, left panel). This distribution suggests a tendency of delays to propagate non-linearly from smaller to bigger airports while small linear propagation channels appear more randomly within the network. Such difference between both networks is further confirmed by the lower IC (more mesoscale structure) and the higher dissortativity (i.e. the tendency of nodes to link with nodes of different degree) of the non-linear network, as reported in Tab. 5.6. Both networks are nevertheless characterised - in different measures - by a high transitivity and a low efficiency.

Metric Weakly Linear Network Non-Linear Network

Link Density 0.076 0.110

Transitivity 0.163 (22.363) 0.146 (5.290) Efficiency 0.178 (-2.613) 0.123 (-5.021)

Assortativity –0.184 -0.357

Information Content 0.786 0.651

Table 5.6: Resume of topological results.