Quantification and Comparison of Hierarchy in Unimodal Public Transport Networks

Abstract

Hierarchy is a network property that identifies the organisation and importance of network elements. Public transport systems consist of stops organised by connections, which can be regarded as networks. In a Public Transport Network (PTN) with a high hierarchy, the number of elements gradually decreases as their importance increases, where the majority of elements have low importance and a few high-important elements. The hierarchical organisations in PTNs contribute to the network performance by efficiently allocating resources based on the elements' importance. The hierarchy has been studied in transport with different methods and data sources. However, the topology-related quantification and comparison methodologies of PTN hierarchy and its mode-wise and continent-wise effects are scarce. A unified PTN hierarchy definition and the quantification methodology are necessary to enable PTN comparisons in terms of the network organisations and reflect the relative PTN performance.

Based on the current research gaps, this study develops topology-based PTN hierarchy quantification and comparison methodologies. First, six topological characteristics of PTN hierarchy are identified from the element scale (vertex accessibility, element intermediacy and vertex cluster importance) and network scales (scale-free structures, high-clustering structures and vertex connection pattern). Second, six element-based or network-based topological indicators are selected to quantify the topological characteristics. For element-based indicators (vertex degree centrality, closeness centrality, betweenness centrality and eigenvector centrality), the coefficient of determination (R square) of the indicator's probability density distribution fitting the skewed normal distribution represents the PTN hierarchy. For network-based indicators (modularity coefficient and assortativity coefficient), the quantified indicator values represent the PTN hierarchy. Next, the radar chart is developed for comprehensively assessing the normalised six-dimension PTN hierarchy.

To evaluate the performance of the methodology, a database based on the GTFS data containing topological information of 63 high-capacity unimodal PTNs worldwide is applied with the hierarchy quantification methodology as a case study. In the PTN hierarchy comparison, the hierarchy in the closeness centrality and betweenness centrality are prone to be high and have higher importance for PTN operation, reflecting the hierarchical organisations of stops' accessibility and the traffic intermediacy on infrastructures. PTN hierarchy in the eigenvector centrality and network modularity dimensions reflect the mono-centric or multi-centric network structures. In the vertex degree centrality and network assortativity dimensions, the identified PTN hierarchy topological characteristics are less significant. In mode-wise effects analysis, the order of modes having PTN hierarchy from high to low is metro, tram and BRT networks. The continent-wise effects show that the European PTNs usually have a higher hierarchy than North American PTNs.

Overall, the research offers a unified topology-based PTN hierarchy as a network property. This study's quantification and comparison methodologies bring a comprehensive multi-dimension and intuitive perspective for PTN hierarchy with the radar chart representation. With the case study, the six-dimension PTN hierarchy, the mode-wise and continent-wise effects are analysed. The methodology benefits the universal PTN performance comparison with basic topological information.