Journal of Advanced Transportation

Volume 2019, Article ID 4576961, 12 pages

https://doi.org/10.1155/2019/4576961

## Route Redundancy-Based Network Topology Measure of Metro Networks

^{1}The Key Laboratory of Road and Traffic Engineering, Ministry of Education, Tongji University, Shanghai 201804, China^{2}Shanghai Shentong Metro Group, Shanghai 201103, China

Correspondence should be addressed to Xiangdong Xu; nc.ude.ijgnot@uxgnodgnaix

Received 4 April 2019; Accepted 28 May 2019; Published 3 July 2019

Academic Editor: David Llorca

Copyright © 2019 Weiwei Jing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The metro system plays a very important role in the urban multimodal transportation system, yet it is susceptible to accidents. A well-designed metro system needs to provide alternative routes to travellers both in the disruptive events and the normal operating conditions for providing rerouting opportunities and balancing crowded lines. This paper provides a new dimension of assessing metro network performance—travellers’ route redundancy (or route diversity), which is defined as the number of behaviourally effective routes between each origin-destination (O-D) pair in the network. The route redundancy of metro network is evaluated by statistical indicators of the distribution of the O-D-level number of effective routes. Compared with the existing connectivity and accessibility measures of topology network performance, route redundancy is also based on the topology network, but it takes the travellers’ route choice into consideration. Specifically, the effective routes between each O-D pair would provide disaggregated information from the travellers’ perspective. Case studies in four metropolises in the world, i.e., Shanghai, Beijing, London, and Tokyo, are conducted to examine the predisaster preparedness of the four metro networks explicitly from the perspective of route redundancy. The results indicate that the London metro network has the best route redundancy performance in terms of the statistical indicators of the distribution of the O-D level number of effective routes. Furthermore, the results of route redundancy are compared with typical measures of topology network performance in terms of measuring connectivity and accessibility of metro networks. Their differences are attributed to the fact that the route redundancy measure considers the travellers’ O-D-level route choice beyond the pure network topology and the shortest path considerations of the existing measures. The route redundancy proposed in this paper could assist in evaluating the predisaster preparedness of current or planning metro networks from O-D level to network level.

#### 1. Introduction

The metro system is becoming a priority choice to mitigate the traffic pressure in many cities, due to its promising advantages such as large capacity, high efficiency, low energy consumption, low pollution, and land resource saving [1]. As of July 2018, as summarized in Wikipedia [2–4], about 180 cities in 54 countries have opened their metro systems, and 40 cities are planning to open a metro system in the future. With the development of economy and technology, metro systems are developing towards high density, high efficiency, and networking [5]. In China, metro systems have received a rapid development in past decades, and 35 cities have opened their metro systems with a total distance of 4,898 kilometres by July 2018. For example, in Shanghai, there are 16 metro lines and 395 stations in operation with the total mileage of 673 kilometres. On the other hand, the metro ridership also continues to increase significantly. For example, in the Beijing metro system, the average daily ridership has a breakneck growth since 2008, and the maximum daily ridership has exceeded 13 million by July 2018.

Accidents (e.g., collapse, leak, terrorist attack, fire, and suicide) frequently occur in metro networks [6]. When an accident causes the failure of a station, it would affect not only the individual metro line but also multiple lines or even the whole network [7]. Since the metro network is susceptible to disruptions, measuring its performance under uncertainties has attracted a lot of attention from both researchers and practitioners. Recently,* network resilience* is being increasingly considered to be an important aspect of network performance or network behaviour following disruptions [8–10]. Resilience is often used in association with several threatening events, which may show critical and catastrophic phases such as terrorist attacks and natural disasters [11]. The White House [12] defined resilience as the ability to prepare for and adapt to changing conditions and withstand and recover rapidly from disruptions. Resilience engineering offers a much broader sociotechnical framework to cope with infrastructure threats and disruptions by focusing on three aspects: readiness and preparedness, response and adaptation, and recovery and adjustment [13].

According to Bruneau* et al*. [14], resilience can be characterized by the four “R’s” concept: redundancy, robustness, resourcefulness, and rapidity.* Redundancy is one of the key dimensions of resilience* [15], which is defined as “the extent to which elements, systems, or other units of analysis exist that are substitutable”.* A resilient network should be redundant as redundancy reflects the predisaster preparedness of a network*. In other words, a redundant network would provide alternative choices to reduce the impact of disruptions [16, 17]. Note that* the redundancy evaluation of the metro network is different from the vulnerability analysis*, which focuses on the consequences caused by disruptions or incidents. Interested readers are directed to the book by Taylor [10] which charts the development and a comprehensive overview of transportation network vulnerability analysis. However,* very limited attention has been paid to the redundancy of public transportation networks,* as shown by the more detailed reviews to be presented in Section 2.

This paper provides a new dimension of metro network performance assessment—travellers’ route redundancy, which is measured by the distribution characteristics of the number of effective routes (or paths) between each O-D pair in a metro network. The route redundancy describes the O-D effective connections explicitly from the users’ perspective. We should point out that route redundancy is still based on the topology network, but it could offer more behavioural information than the typical topology network-based measures such as the measures aggregated from the node level (e.g., degree, clustering coefficient) and the measures based on the shortest paths (e.g., diameter, network efficiency). Specifically, it can not only provide disaggregated information from O-D level, but also quantify the number of behaviourally effective alternatives considering that the travellers may not always choose the shortest paths in reality. By looking at four metro networks, i.e., Shanghai, Beijing, London, and Tokyo, we examine the performance of metro networks based on the concept of route redundancy. Furthermore, the differences and relationships between the route redundancy-based measure and the typical topology network measures will be discussed based on the four metro networks.

Redundancy reflects the predisaster preparedness of a network for combatting vulnerability. Therefore, with the networking development of metro systems, a well-designed metro network needs to provide alternative routes for travellers as much as possible under the occurrence of accidents. Also, alternative routes are needed to split the passenger flow of congested segments/lines under normal operating conditions. Hence, it is necessary to have a deep understanding of the route redundancy (or route diversity) of metro networks. Following Xu* et al*. [18], we customize the definition of route redundancy to metro networks as* the number of behaviourally effective routes (or paths) available for passengers between any two stations in the metro network*. The assessment of route redundancy could help to evaluate the predisaster preparedness of the current metro network or the planned scheme, and also offer the information of alternative routes to assist metro managers in rerouting passengers in a highly congested or disruptive event. Note that this paper only focuses on the route redundancy from the travellers’ perspective, i.e., the first dimension of transportation network redundancy proposed by Xu* et al*. [18]. The second dimension (i.e., network spare capacity from the planners’ perspective) of metro network redundancy will be related to the station capacity and line capacity, which itself is more complex than the first dimension. When the operation and scheduling data are available, the second dimension could also be further measured.

The reminder of this paper is organized as follows. The studies on redundancy and network topology are introduced in Section 2. In Section 3, we provide the definition and computational method of route redundancy. In Section 4, we examine the route redundancy of four metropolitan metro networks and discuss the differences and relationships between the route redundancy and the typical topology network measures. Conclusions are summarized in Section 5.

#### 2. Literature Review

The studies on redundancy and network topology of transportation networks are reviewed in this section.

##### 2.1. Redundancy

In a broad picture, redundancy is an important topic in systems engineering. In reality, many infrastructure and safety-critical systems require redundancy design such as the water distribution networks [19] and the aircraft door management systems [20]. The concept of redundancy has also been applied in the transportation systems as one of the measures of resilience. Berdica [21] defined redundancy in transportation systems as the existence of numerous optional routes/means of transport between origins and destinations that can result in less serious consequences in case of a disturbance in some part of the system. Immers* et al*. [22] described the redundancy of road network as sufficient spare capacity which could avoid the degradation in the quality of service. Freckleton* et al*. [23] defined redundancy as the ability of a traveller to adjust routes as necessary to detour around the affected sections of the network under disruptions. Actually,* the concept of redundancy has been widely studied but there is no consensus definition for transportation networks*, which comes down to two aspects: the number of alternatives and the network spare capacity (e.g., [10, 18, 24, 25]). For quantifying the redundancy of transportation networks, El-Rashidy and Grant-Muller [25] proposed a redundancy index of various nodes in road networks covering the static aspect of redundancy (i.e., alternative paths) and the dynamic feature of redundancy (i.e., the availability of spare capacity under different network loading and service levels). In their redundancy model, the entropy concept was adopted to measure the configuration of a system and to model the uncertainties inherent in road networks. However, the entropy concept may not be intuitive to travellers, who do care about the existence of optional routes/means. Xu* et al*. [18] developed two network-based measures for systematically characterizing the redundancy of transportation networks: one is travel alternative diversity from travellers’ perspective to address the question of “*how many effective redundant alternatives are there for travellers?”* and the other one is network spare capacity from planners’ perspective to address the question of “*how much redundant capacity does the network have?*”

*The above redundancy studies are mainly conducted for road networks*. For public transportation networks, Jenelius and Cats [26] developed a methodology for evaluating the value for robustness and redundancy of extending the public transportation network. The value of redundancy of the network extension was defined as the value of robustness (i.e., the change in passenger welfare under network disruptions in the extended network compared to the baseline network) minus the difference in welfare under normal conditions in the extended network compared to the baseline network. However,* in case of disruptions, the travellers’ inconvenience significantly depends on the availability of alternative travel options*, i.e., the amount of redundancy in the network. Yang* et al*. [7] applied the route diversity index (i.e., the simple average number of reasonable routes between all O-D pairs) of Xu* et al*. [18] to analyse the route diversity of the Beijing metro network and to identify the vulnerable stations. Yang* et al*. [7] simply considered the reasonable/efficient routes in defining the route diversity. For improving the realism of route redundancy in metro networks, in this paper, we consider not only efficient routes but also not-too-long routes (termed as effective routes). To a certain degree, this modelling captures both network topology and travellers’ route choice behaviours (via the route cost constraint). In addition, the transfer costs at transfer stations are explicitly modelled in metro networks, which is a significant difference between road networks and metro networks.

##### 2.2. Measures of Topology Network Performance

In the past decades, many scholars (e.g., [27–30]) have examined the metro network performances from the viewpoint of graph theory and complex network by focusing on the topological properties of metro networks. The small-world property and scale-free pattern are considered as significant network topological properties, which are also well-studied in metro networks. Small-world effect is referred to as a high clustering and small average shortest path length ([31, 32]), and a scale-free network is defined as the network with a nodal degree distribution following the power law distribution ([32]). For example, Latora and Marchiori [33] analysed the small-world property of Boston subway network; Derrible and Kennedy [34] demonstrated that most metros are small-worlds and scale-free by looking at 33 metro systems.

Besides, many metrics have been proposed to measure the performances of topology networks. According to Grubesic* et al*. [35] and Zhang* et al*. [36], these metrics could be divided to* connectivity and accessibility measures of the network topology*. Typical network performance measures based on topology network are summarized in Table 1. On the one hand, the connectivity measures (e.g., Alpha, Beta, and Gamma indices, average degree, cyclomatic number, and clustering coefficient) are used to assess the connectedness. For these measures, Alpha, Beta, and Gamma indices represent the connectivity and complexity of a network. Specifically, Alpha index is expressed by the ratio of the number of cycles to the maximum number of cycles; Beta index represents the relationship between the number of links and the number of nodes; and Gamma index quantifies the relationship between the number of links and the maximum possible number of links [36]. The node degree measures the number of links converging to each node [37]. The cyclomatic number calculates the number of cycles (or loops), which has been proposed as a topology metric to evaluate the total number of alternatives from the aggregated perspective of the whole network ([34, 36]). The clustering coefficient, known as transitivity, represents an alternative possibility as it measures the overall probability for the network to have interconnected adjacent nodes ([25, 37]). On the other hand, the accessibility measures of network topology, such as diameter and average shortest path length, are directly related to the shortest paths. Network efficiency measures the travel efficiency of passengers between two nodes ([38]). Node betweenness characterizes the centrality of nodes, which could reflect the role of nodes in the network ([39]). These measures have been widely extended to consider passenger flows, route choice, etc. and applied to the robustness and vulnerability analysis of metro networks ([34, 40–46]).