Overview and classification of air network metrics

In the previous section, we put forth a conceptual definition of transport connectivity and accessibility. In this section, we use the definition to classify metrics proposed in the literature. We focus on passenger transport-centric metrics and disregard network structure indices such as hub connectivity measures (see Redondi and Burghouwt, 2010), network concentration indices (e.g. the Gini concentration index, the Freeman centrality index and the entropy function) and network geometry features (the diameter and the clustering coefficient) (Reggiani et al., 2009). This is because the latter metrics may provide important operational insights at the airline level but are only weakly linked to the quality of air transport links with regard to the societal value of the air transport network for specific locations. The latter can be inferred from Wang et al. (2011), who find an inverted U-shaped relationship between the clustering coefficient and degreeness, indicating that these indices reflect inherently different properties of the air networks.

Our review of the available literature finds that existing air connectivity and accessibility metrics differ mosdy by the approaches for capturing transport costs. In particular, we find earlier metrics to rely on number-based measures (e.g. number of segments), whereas more-recent measures apply time-based or even fare-based approaches, for which reliability depends heavily on the correct specification of a functional form and on sufficient empirical calibration. For measures which consider destination quality, the measures of destination quality vary from economic opportunities (e.g. population and income) to traffic-based measures (e.g. traffic intensity).

Table 2.1 provides a classification scheme of existing metrics, and Table 2.2 gives an overview of all models discussed. A brief technical summary of all metrics is shown in a summary at the end of this chapter.

Connectivity metrics for air transport

Centrality-type connectivity measures

Early work on air transport networks relies on measures of centrality, as outlined by Freeman (1978). The concept of centrality was developed in social network theory, where graph theory was used to study topological (i.e. aspatial) networks. Link attributes, such as the operating airline or schedule coordination, are not considered in these metrics. Freeman (1978) discusses two relevant indices of centrality: degreeness and closeness.

Table 2.1 Overview of different air connectivity, accessibility and spatial metrics

Source: Extended and adapted by authors from Burghouwt and Redondi (2013)

Table 2.2 Overview of airport connectivity measures in recent air transport research

(Continued)

Table 2.2 (Continued)

Source: Extended and adapted by authors from Burghouwt and Redondi (2013)

Nieminen (1974) defined degreeness of a node (in this case, an airport) as the number of other nodes with which it is (directly) connected. Thus, the concept is inherently aspatial. Degreeness metrics have been applied broadly for aviation, especially to analyse the structure of different (intercity) passenger air transportation networks (Bagler, 2008; Guida and Maria, 2007; Guimera et al., 2005; Wang et al., 2011; Xu and Harriss, 2008). The term gross vertex, as defined by Ivy (1993) and Ivy et al. (1995), extends the degreeness concept to capture indirect connections. More specifically, it aims to measure the number of airports connected by itineraries with three or fewer legs. In addition, Cristea and Danila (2017) were the first to suggest a market connectivity index calculated by weighting degreeness of each destination with direct-flight frequency and with the national market share of the destination airport, thereby introducing a traffic-based measure of destination quality into the degreeness analysis.

Closeness-type connectivity metrics focus on analysing connectivity through understanding the number of steps required to reach destinations. Like the degreeness concept, closeness is an inherently aspatial concept. While the required counts of steps are specific to origin-destination pairs, they can be averaged to further gain insights into the average closeness of destinations. Sabi-dussi (1966) introduced an early implementation of a closeness metric, which is calculated as the number of destinations (minus one) over the summed number of edges (link between nodes in a network) on the shortest path between a specific origin node and all possible destination nodes (see Wang et al. [2011] for an example in aviation). The shortest path length, calculated as the inverse of closeness, represents the average number of legs to reach any destination in the network (see summary at the end of this chapter for details). Early work on shortest-path-length connectivity in the field of aviation was conducted by Shaw (1993) and Shaw and Ivy (1994), and later work was presented by Malighetti et al. (2008) and Paleari et al. (2010).²

Freeman (1978) notes that both degreeness and closeness are structural properties of centrality. In many analyses, degreeness metrics and closeness metrics are therefore considered simultaneously (e.g. Reggiani et al., 2009; Shaw, 1993; Wang et al., 2011). This is because the concepts complement each other. Degreeness provides insights into the number of destinations, and closeness helps understand the number of steps required to reach destinations. Combining the two metrics can subsequently provide insights into both the availability of connections and the ease of making trips. This conclusion is in line with empirical data: Burghouwt and Redondi (2013) report imperfect but high correlation between degreeness and closeness at 0.72 among European airports.