Higher-order properties of multi-modal public transit networks

Over the years, research advances in public transit networks (PTNs) are restricted to pairwise interactions, while providing limited understanding of higher-order PTNs. In particular, exploring the higher-order interactions of multi-modal PTNs (MPTNs) can not only facilitate a more efficient and convenient way to meet the travel demands of passengers, but also improve the carrying capacity and attractiveness of public transit systems. Consequently, this paper characterizes the characteristic metrics of higher-order MPTNs and the node importance identification method considering higher-order cliques. The construction method of MPTNs and the metrics of higher-order network properties are first introduced. Based on investigating the characteristics of higher-order cliques, we then further design a node importance measure to explore the degree of station core of higher-order MPTNs. Finally, taking the Beijing public transit system as an example, three different types of MPTNs (including the bus network, bus-metro network, and bus-metro-taxi/ride-hailing network) are constructed, and their higher-order properties are statistically analyzed. The empirical study finds that the Beijing MPTNs with higher-order interactions have a large number of complex clique and cavity structures, which are typical of scale-free networks. In terms of node rankings, our proposed node importance measure incorporates rich information about the higher-order properties compared to well-known benchmark indices. Correspondingly, the measure performs overall better than other benchmarks in identifying the most vulnerable nodes under intentional attacks. The paper provides a reference for future exploration of the dynamic evolution mechanism of higher-order MPTNs, and can serve the actual network planning and design of urban MPTNs.