Vulnerability effects of passengers' intermodal transfer distance preference and subway expansion on complementary urban public transportation systems

Author’s Accepted Manuscript Vulnerability effects of passengers’ intermodal transfer distance preference and subway expansion on complementary urban public transportation systems Liu Hong, Yongze Yan, Min Ouyang, Hui Tian, Xiaozheng He www.elsevier.com/locate/ress

PII: DOI: Reference:

S0951-8320(16)30624-X http://dx.doi.org/10.1016/j.ress.2016.10.001 RESS5648

To appear in: Reliability Engineering and System Safety Received date: 13 February 2016 Revised date: 31 August 2016 Accepted date: 13 October 2016 Cite this article as: Liu Hong, Yongze Yan, Min Ouyang, Hui Tian and Xiaozheng He, Vulnerability effects of passengers’ intermodal transfer distance preference and subway expansion on complementary urban public transportation s y s t e m s , Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j.ress.2016.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Vulnerability effects of passengers’ intermodal transfer distance preference and subway expansion on complementary urban public transportation systems Liu Hong a, b, Yongze Yana, Min Ouyang a, b*, Hui Tian a, Xiaozheng He c a

School of Automation, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, PR China

b

Key Lab. For Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, PR China

c

NEXTRANS Center, Purdue University, W. Lafayette, IN 47906, USA

Abstract The vulnerability studies on urban public transportation systems have attracted growing attentions in recent years, due to their important role in the economy development of a city and the well-beings of its citizens. This paper proposes a vulnerability model of complementary urban public transportation systems (CUPTSs) composed of bus system and subway system, with the consideration of passengers’ intermodal transfer distance preference (PITDP) to capture different levels of complementary strength between the two systems. Based on the model, this paper further introduces a CUPTSs-aimed vulnerability analysis method from two specific aspects: (a) vulnerability effects of different PITDP values, which facilitate the design of policies to change PITDP to reduce system vulnerability; (b) vulnerability effects of different subway expansion plans, which facilitate the vulnerability investigation of current expansion plan and the identification of the optimal expansion plan from the system vulnerability perspective. The proposed CUPTSs-aimed vulnerability analysis method is applied to investigate the complementary bus and subway systems in the city of Wuhan, China. The insights from this study are helpful to analyze other CUPTSs for valuable planning suggestions from the vulnerability perspective. Key words: Vulnerability, Urban public transportation systems, Complementary Strength, Subway expansion

1. Introduction Urban public transportation systems, which can offer low-cost, efficient and safe shared transport services for the general public, play an important role in the economy development of a city and the well-beings of its citizens. In most of big cities in the world, urban public transportation systems are mainly composed of bus systems and subway systems, which provide alternative transport services to carry passengers to their desired places. Due to their importance, any disruption or incident in these systems will reduce the citizens’ mobility in urban area and largely affect their daily life. Hence, modeling and analyzing the vulnerability of urban public *

Corresponding author: [email protected] 1

transportation systems have attracted growing attentions in recent years (Lv, Yan et al. 2015). Many of the previous studies mainly focused on the vulnerability modeling and analyzing of single urban public transportation system (Feng and Hsieh 2009, Sullivan, Aultman-Hall et al. 2009, Wang, Chan et al. 2014, He and Peeta 2016). Wu et al. (WU, GAO et al. 2006) studied the topological characteristics of Beijing bus system and analyzed its vulnerability under random failures and intentional attacks. Dribble and Kennedy (Derrible and Kennedy 2010) analyzed the vulnerability of 33 subway systems in the world using complex network theory. Zhang et al. (Zhang, Xu et al. 2011) studied the topological characteristics of Shanghai subway system and analyzed its vulnerability. Wang et al. (Wang, Li et al. 2013) analyzed the vulnerability of urban rail systems in San Francisco and Boston, and identified the most vulnerable rail segments. Rodríguez-Núñez and García-Palomares (Rodríguez-Núñez and García-Palomares 2014) presented a methodology to analyze the criticality and vulnerability of public transportation system, and used it to analyze the vulnerability of Madrid subway system. Kpotissan Adjetey-Bahun et al. (Adjetey-Bahun, Birregah et al. 2016) proposed a novel model to quantify the resilience of mass railway transportation systems where the passenger delay and passenger load were used as system performance indicators, and the subsystems and interdependencies between them were also considered in the model. These studies have improved our understanding on the vulnerability of different types of urban public transportation systems, but from the perspective of passengers, these systems are not independent: they are connected to each other at some places by nearby stations that belong to different systems and together provide intermodal services for the public. There are mainly two types of relationships between different transport modes: competition and complementary. For example, during off-peak hours, these systems compete by price or some other convenient characteristics to attract commuters, and during on-peak hours, they complement each other to transport commuters to their destinations. Many studies concerned about the competition relationship between different transport modes (Adler, Pels et al. 2010, van der Weijde, Verhoef et al. 2013). Some studies used historical passenger data to empirically discuss the market competition between these systems. Behrens and Pels (Behrens and Pels 2012) studied inter- and intra-modal competition of high-speed rail and airline in the London-Paris passenger market during the period of 2003-2009. Dobruszkes (Dobruszkes 2011) discussed the competition between high-speed railway and airline in Western Europe from a supply-oriented perspective. Other studies use game theory to analyze the competition relationship. Hsu et al. (Hsu, Lee et al. 2010) developed a game theoretical model to discuss the completion between high-speed and conventional railway systems. Yang and Zhang (Yang and Zhang 2012) discussed the effects of high-speed rail and airline competition on prices, profits and welfares. When considering public transportation system vulnerability, the complementary relationship is more important than the competition relationship, because the former can reduce system vulnerability by providing alternative transport service. For example, in the case of interruption or incapacity of 2

transportation system A at a place, if some nearby stations that belong to another transportation system B can provide alternative service, then the transport demand of A at that place can be fulfilled by B and the vulnerability of A will be reduced. Some studies addressed the vulnerability of transportation systems with the consideration of complementary relationship between them. Cats and Jenelius (Cats and Jenelius 2012) analyzed the vulnerability of public transportation systems consisting of subway lines, trunk bus lines and light rail train lines, where a dynamic and stochastic notion of public transport network vulnerability was developed with considering the interactions between supply and demand dynamics. Berche et al. (Berche, Ferber et al. 2009) studied the vulnerability of urban public transportation systems in fourteen major cities around the world under random failures and intentional attacks, where the stations in different transportation systems were regarded as the homogeneous nodes in the network modeling. Ferber et al. (Ferber, Berche et al. 2012) compared the vulnerability of public transportation systems in London and Paris under random failures and intentional attacks, in which the public transportation systems including bus, subway and tram were all modeled by L-space model. Jin et al. (Jin, Tang et al. 2014) discussed how to enhance subway network resilience through localized integration of bus services and a two-stage stochastic programming model was developed to assess the intrinsic subway network resilience and optimize the localized integration strategy. Yang et al. (Yang, Huang et al. 2014) studied the statistic properties and cascading failures of complementary bus and subway systems in Beijing based on complex network theory. Recently, Ouyang et al. (Ouyang, Pan et al. 2015) proposed a network-based approach to analyze the vulnerability of complementary Chinese railway and airline systems, and introduced a complementary strength metric to describe complement strength between these two systems from the vulnerability perspective. Although the aforementioned studies analyze the positive effects of complementary relationship between transportation systems to reduce their vulnerability, they ignore an important factor in CUPTSs, the passengers’ intermodal transfer distance preference (PITDP), which will seriously affect the complementary relationship modeling between different urban transport modes and further affect the system vulnerability. The PITDP means the longest distance passengers would like to accept when they plan to transfer from one transport mode to another, which is varied by people according to their age, weight, sex or other personal characteristics. For example, the older people and children usually are less willing to walk a long distance to transfer from one transport mode to another than the young people, which will lead to the situation that two nearby stations that belong to different transport modes can be regarded as complementary stations by young people but not by older people and children. Increasing the average PITDP can strengthen the complementary relationship between different transportation systems in CUPTSs, and further reduce the systems vulnerability. Especially, in the case of an emergency, the government or related infrastructure stakeholders can issue some policies to increase the value of PITDP for reducing CUPTS vulnerability. For example, providing free bicycles, transit vehicles around the 3

subway stations or building interesting shelter galleries between bus stations and nearby subway stations are all useful methods to increase PITDP. Furthermore, existing vulnerability studies in the literature do not discuss the effects of transportation system expansion on the CUPTSs vulnerability. The topologies of urban public transportation systems are always changing and expanding to meet the rapid increase of urban travel demand. For example, the urban subway systems are experiencing rapid expansion in many Chinese big cities, and in each of recent years there has and will have one new subway line opened in several cities. The CUPTSs vulnerability is always changing during the systems expansion process. Few studies analyze the CUPTSs vulnerability with considering the system expansion process and discuss the optimal expansion plan to minimize the CUPTSs vulnerability. This paper proposes a vulnerability model of CUPTSs with the consideration of PITDP, and based on this model, introduces a CUPTSs-aimed vulnerability analysis method to investigate the effects of different PITDP values and different subway expansion sequences on the CUPTSs vulnerability. The rest of the paper is organized as follows: Section 2 proposes a network-based vulnerability model for CUPTSs; Section 3 introduces the CUPTSs-aimed vulnerability analysis method from two specific aspects: vulnerability effects of different PITDP values and different subway expansion sequences; Section 4 uses the complementary bus and subway systems in the city of Wuhan in China as an example for CUPTSs vulnerability analysis; Section 5 discusses the findings and extensions and provides conclusions and future research directions.

2. Network-based vulnerability model for CUPTSs This paper interprets vulnerability associated with a disruptive event, and the vulnerability to this event is quantified as the system performance drop (Ouyang and Wang 2015). According to this vulnerability definition, the proposed network-based vulnerability model for CUPTSs contains four parts: network-based description of CUPTSs, performance metric selection for CUPTSs, disruptions modeling of CUPTSs and CUPTSs vulnerability assessment.

2.1 Network-based description of CUPTSs For the vulnerability analysis of engineering infrastructure systems, including public transportation systems, there exist many approaches about system modeling in the literature, such as empirical approaches (Utne, Hokstad et al. 2011), agent based approaches (Bompard, Napoli et al. 2009), system dynamics based approaches (Santella, Steinberg et al. 2009), economic theory based approaches (Haimes and Jiang 2001), network based approaches and others (Hong, Ouyang et al. 2015). A detailed review of these approaches is provided in the reference (Ouyang 2014). To model different types of transportation systems and their complementary relationship, it requires the topological and geographical information of each involved system. Hence, a network based approach is adopted in this paper. 4

As complementary urban public transportation systems consist of multiple systems and each system can be described by a network, then the multi-layer network approach can be applied in this paper (Boccaletti, Bianconi et al. 2014). This type of approaches models each system or each character of a multiplex system by a network and describes the relationships between those networks by inter-layer links for analyzing system topological characteristics and dynamic behaviors, and has been extensively used to model the multiplex character of many real-world systems in recent years (Kivelä, Arenas et al. 2014), including international trade system (Barigozzi, Fagiolo et al. 2010) in which different countries and their import and export relationships for each type of commodities is modeled as a layer, language system (Liu and Cong 2014, Martinčić-Ipšić, Margan et al. 2016) in which semantic, syntax, co-occurrence and characters subsystems are modeled as four layers, railway system (Kurant and Thiran 2006) in which stations and railway tracks are modeled as physical layer and stations and train routes are modeled as logical layer, and so on. The CUPTSs, including the road system, bus system and subway system, can be modeled as a three-layer network, represented as 𝐺 = {𝐺 𝑏 , 𝐺 𝑠 , 𝐺 𝑟 , 𝑀, 𝐶}, where 𝐺 𝑏 , 𝐺 𝑠 , 𝐺 𝑟 represent the three systems in CUPTSs respectively, 𝑀 represents the mapping between road segments in 𝐺 𝑟 , system edges and bus lines in 𝐺 𝑏 , 𝐶 represents the complementary relationship between 𝐺 𝑏 and 𝐺 𝑠 . In order to describe how to model CUPTSs by multilayer network approach and assess their vulnerability, this section next will take the road, bus and subway systems in the city of Wuhan in China as an example for illustrative purpose. Wuhan is the capital city of Hubei Province and is also the political, economic, financial, cultural, educational and transportation center of central China, with a population of 10,338,000 and a land area of 8,494.41 square kilometers, and the famous Yangzi river flows across the city (Wikipedia 2016.01.01). The public transport modes in Wuhan include bus, subway, taxi and ferry, and the total number of passengers in 2013 is more than 2.17 billion, most of them are transported by the bus (about 1.5 billion) and subway (about 0.24 billion) systems. With the rapid expansion of Wuhan subway system, the subway passengers will increase sharply in the next a few years. Similar to Wuhan, the public transportation systems in most of big cities in China also mainly consist of bus and subway systems. Considering the needs of vulnerability analysis, three networks are constructed based on the public transportation systems in Wuhan: the bus network, the subway network and the road network. The bus and subway networks and their complementary relationship are used to assess system performance under disruptions, while the road network is used to locate the disruptions in the bus system, find the corresponding failed bus lines and further assess the system vulnerability.

5

(a) Bus network

(b) Subway network

(c) Road network Figure 1. (a) Network-based description of bus system in Wuhan, which is shown by ArcGIS with stations geographical information. (b) Network-based description of subway system in Wuhan, in which solid lines represent the operational subway lines and the dash ones represent the planned subway lines. (c) Network-based description of the road system in Wuhan, which is shown by ArcGIS with intersections and road segments geographical information. Bus network The bus system in Wuhan, which has 1833 bus stations and 646 bus lines, is modeled as a network𝐺 𝑏 = (𝑁 𝑏 , 𝐸 𝑏 , 𝐿𝑏 ), where bus stations are represented as nodes and two nodes are connected by an edge if they are served by at least one bus line without other bus stations in between

(Ouyang,

Zhao

set 𝑁 𝑏 = {𝑛𝑏1 , 𝑛𝑏2 , … , 𝑛

𝑏𝑘 𝑏

et

al.

2014).

This

network

} , an edge set𝐸 𝑏 = {𝑒 𝑏1, 𝑒 𝑏2 , … , 𝑒

𝑏𝑝𝑏

is

defined

by

a

node

}, and a bus line set 𝐿𝑏 =

𝑏

{𝑙 𝑏1 , 𝑙 𝑏2 , … , 𝑙 𝑏𝑞 } , where 𝑘 𝑏 denotes the number of nodes, 𝑝𝑏 denotes the number of edges, 𝑞𝑏 denotes the number of bus lines in 𝐺 𝑏 , 𝑘𝑏 = 1833, 𝑝𝑏 = 3172, 𝑞𝑏 = 646. A bus line in 𝐿𝑏 is composed of sequential edges in 𝐸 𝑏 and usually one bus edge 𝑙 𝑏 contains several 𝑒 𝑏 . Figure1 (a) shows the network-based description of Wuhan bus system with each stations’ geographical 6

information. Subway network The subway system in Wuhan has 75 subway stations and three subway lines in operation in 2014 and other six new lines will be opened one by one from 2015 to 2020, Table 1 presents the subway lines in operation from 2014 to 2020 and the red bold italic words mean the subway line newly opened in each year. The subway system in 2020 is modeled as a network 𝐺 𝑠 = (𝑁 𝑠 , 𝐸 𝑠 , 𝐿𝑠 ), where subway stations are represented as nodes and rail tracks between stations are represented as 𝑠

edges. This network is defined by a node set 𝑁 𝑠 = {𝑛 𝑠1 , 𝑛 𝑠2 , … , 𝑛 𝑠𝑘 }, a rail track set 𝐸 𝑠 = 𝑠

𝑠

{𝑒 𝑠1 , 𝑒 𝑠2 , … , 𝑒 𝑠𝑝 }, and a subway lines set 𝐿𝑠 = {𝑙 𝑠1 , 𝑙 𝑠2 , … , 𝑙 𝑠𝑞 }, where 𝑘 𝑠 denotes the number of nodes, 𝑝 𝑠 denotes the number of edges, and𝑞 𝑠 denotes the number of subway lines in 𝐺 𝑆 , 𝑘 𝑠 = 166, 𝑝 𝑠 = 189, 𝑞 𝑠 = 9. Figure 1 (b) shows the network-based description of Wuhan subway system in 2020, the solid edges represent the three existing lines in 2014 and the dash edges represent the six planned lines. Table 1. The subway lines in operation in Wuhan city from 2014 to 2020. Year 2014 2015 2016 2017 2018 2019 2020

Subway lines Line 1, Line 2, Line 4 Line 1, Line 2, Line 4, Line 3 Line 1, Line 2, Line 4, Line 3, Line 6 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7, Line 8 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7, Line 8, Line 5 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7, Line 8, Line 5, Line 9

* The red bold italic words mean the subway line newly opened in each year For the vulnerability analysis of transportation system, the stations, the road segments, rail tracks and the crossroads are usually regarded as attacked components. The attacked components in bus system are different with that in subway system. In the subway system, the network-based description is coincident with the physical rail track network and can be used to discuss the disruptions to stations or rail tracks and compute the subway system performance after disruptions. While, in the bus system, the attacked components are usually road segments or crossroads, not directly toward the edges in bus network 𝐺 𝑏 . Because the edges in 𝐺 𝑏 represent the connection relationship between two stations and not the bus lines or road segments. One edge in 𝐺 𝑏 can contain several road segments while one bus line can contain several edges in 𝐺 𝑏 . Hence, it requires using the road segments as the attacked components rather than the edges in 𝐺 𝑏 . At the same time, attacking bus stations or bus vehicles can also be regarded as disruptions in the corresponding road segments or intersections. So it is necessary to model the road system separately for vulnerability analysis of transportation systems.

7

Road network The road system in Wuhan is model as a network 𝐺 𝑟 = (𝑁 𝑟 , 𝐸 𝑟 ), where the intersections, the corners of sharply bended streets and the dead-end of streets are represented as nodes, and the road segments between them are represented as edges. This network is defined be a node set 𝑁 𝑟 = 𝑟

𝑟

{𝑛𝑟1 , 𝑛𝑟2 , … , 𝑛𝑟𝑘 } and an edge set 𝐸 𝑟 = {𝑒 𝑟1 , 𝑒 𝑟2 , … , 𝑒 𝑟𝑝 }, where 𝑘 𝑟 denotes the number of nodes and 𝑝𝑟 denotes the number of edges in 𝐺 𝑟 , 𝑘 𝑟 = 1023, 𝑝𝑟 = 1474. The mapping between edges in 𝐸 𝑟 , 𝐸 𝑏 ,and 𝐿𝑏 are established by the geographical information of the networks and the bus and subway schedule tables, including the road segments subset in 𝐸 𝑟 which are used by each edge in𝐸 𝑏 as well as the edges subset in 𝐸 𝑏 which are used by each bus line in 𝐿𝑏 . The mapping between these edges will be used to find out the corresponding failed bus lines when some road segments or intersections are disrupted. Figure 1 (c) shows the network-based description of Wuhan road system, where only main roads in Wuhan which have bus lines pass through are selected for analysis. Note that the Yangzi River divides the three networks into two main parts which are connected by bridges and tunnels in Figure 1. Complementary relationship The bus system and the subway system can provide alternative transport services for passengers in urban commuting. Usually there are several bus stations around a subway station for intermodal transfer. In normal case, passengers can choose any one of the two transport modes according to their preference; in the case of some stations in one mode are unavailable, the passengers can choose the nearby stations that belong to another transport mode for their trips. This complementary relationship between bus and subway systems can reduce the vulnerability of each single system and provide more robust transport services to the passengers. In this paper, two nearby stations that belong to different transport modes to provide alternative transport services are called complementary stations. To describe the complementary relationship in CUPTSs, the complementary stations should be identified first, which are related with the longest walking distance that passengers would like to accept to transfer from one mode to another, different types of passengers may have different preferences. If the walking distance between a bus station and a nearby subway station is not larger than the longest distance a passenger would like to accept, then these two stations can be regarded as complementary stations for this passenger. Here, we define a metric, passengers’ intermodal transfer distance preference (PITDP) to represent the acceptable walking distance for passengers. As mentioned above, the value of PITDP is varied for different people. In this study, the PITDP represents the average value for all the passengers. According to the definition of complementary stations, PITDP and the stations’ geographical information in CUPTSs, the complementary station pairs can be established through the steps listed below: Step 1, set the value of PITDP; Step 2, obtain the distances (Dis) from each bus station in 𝑁 𝑏 to all the subway stations in 8

𝑁 𝑠 by measuring the walking distance in road network 𝑁 𝑟 between the two stations by ArcGIS; Step 3, choose a bus station 𝑛𝑖𝑏 in 𝑁 𝑏 , check the Dis between this bus station and all the subway stations in 𝑁 𝑠 , and label the bus station 𝑛𝑖𝑏 and a subway station 𝑛𝑗𝑠 as complementary stations if the value of Dis between these two stations is not larger than PITDP, 𝐷𝑖𝑠(𝑛𝑖𝑏 , 𝑛𝑗𝑠 ) ≤ 𝑃𝐼𝑇𝐷𝑃, 𝑖 ∈ [1, 𝑘 𝑏 ], 𝑗 ∈ [1, 𝑘 𝑠 ]; Step 4, establish a complementary edge between the bus station 𝑛𝑖𝑏 and all of its complementary subway station(s), and the complementary edge between 𝑛𝑖𝑏 and 𝑛𝑗𝑠 is denoted as𝑒 𝑏𝑖↔𝑠𝑗 , 𝑒 𝑏𝑖↔𝑠𝑗 ∈ 𝐸 𝑏↔𝑠 ; Step 5, repeat Step 2, Step 3 and Step 4 for all the other bus stations in 𝑁 𝑏 . Through these five steps, all the complementary station pairs in CUPTSs can be identified and the bus network 𝐺 𝑏 and subway network 𝐺 𝑠 are connected by these complementary edges between complementary station pairs. These station pairs are denoted by a complementary matrix 𝐶𝑘 𝑐×2 , 𝑘 𝑐 denotes the number of complementary station pairs. Each row of 𝐶𝑘 𝑐 ×2 corresponds to a pair of complementary bus station ID and subway station ID. If the value of PITDP is set to 800m and only considering the three existing subway lines in 2014, there are 533bus stations with complementary subway stations and 73 subway stations with complementary bus stations, 𝑘 𝑐 =647; if considering the total nine subway lines in 2020, there are 989 bus stations having complementary subway stations and 163 subway stations having complementary bus stations, 𝑘 𝑐 =1301. In the CUPTSs of Wuhan, one subway station may have several complementary bus stations and this is a common situation in many CUPTSs all over the world because of larger carrying capacity of subway system.

2.2 Performance metric selection for CUPTSs Based on the aforementioned vulnerability definition, to assess system vulnerability, it needs a performance metric. Many researchers use service level to assess infrastructure system performance. For example, the number of served customers was used to assess the performance of power system (Ouyang and Dueñas-Osorio 2014) and water supply system (Chang S. E. 2004). Transportation system performance can also be assessed by the number of or the percentage of served passengers. But at now stage, it is hard to get passenger flow data (Li, Jiang et al. 2016), especially in CUPTSs. Accessibility is a key performance metric in transportation domain (Guo, Agrawal et al. 2016), which can be define as the ease with which people can reach their destinations or activity sites. Lei and Church provided a comprehensive review on accessibility (Lei and Church 2010). Based on how to define the ease of accessibility, the related analysis in transportation domain can be grouped into three types, topology-based measure, distance-based measure and time-based measure. The topology-based accessibility mainly focuses on the physical connectivity in transportation system, in which two stations are regarded as accessible if they can be reached by taking one or several transport modes, and distance-based and time-based measures 9

are concerned more about how far it is and how long it takes to travel between two stations. Note that when the travel time based measure is used as a performance metric, it should also consider the walking time between two complementary stations and the waiting time when passengers change the transport mode. The methods about how to calculate the accessibility of a transportation system have been discussed in many research works, but few of them analyze how to calculate the accessibility of CUPTSs composed of multiple transportation systems (Adjetey-Bahun, Planchet et al. 2016). This paper will introduce a procedure to compute the accessibility of CUPTS, which not only can calculate the topology-based accessibility, but also can estimate the travel distance-based and time-based accessibility. In this procedure, an adjacent matrix of all bus stations and subway stations are constructed to calculate and record the accessibility between any two stations in CUPTSs, denoted as 𝐴(𝑘 𝑏 +𝑘 𝑠 )×(𝑘 𝑏 +𝑘 𝑠) = {𝑎𝑖𝑗 }, where𝑖, 𝑗 ∈ [1, 𝑘 𝑏 + 𝑘 𝑠 ]. In 𝐴(𝑘 𝑏 +𝑘𝑠 )×(𝑘 𝑏 +𝑘 𝑠 ), the nodes whose indexes that belong to [1, 𝑘 𝑏 ] denote the bus stations in 𝐺 𝑏 and the nodes whose indexes that belong to [𝑘 𝑏 + 1, 𝑘 𝑏 + 𝑘 𝑠 ] denote the subway stations in 𝐺 𝑠 . The value of topology-based, distance-based and time-based accessibility can be calculated in a unified way by the steps listed below: Step 1: Initialize 𝐴(𝑘 𝑏 +𝑘𝑠 )×(𝑘 𝑏 +𝑘 𝑠 ) If two stations 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 are connected by a bus line or a subway line (this means these two stations belong to the same transport mode), then: (1) for topology-based accessibility, 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦

𝑎𝑖𝑗

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 equals to 1; (2) for distance-based accessibility, 𝑎𝑖𝑗 equals to the travel distance in

𝐺 𝑟 between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 by bus or subway, which can be obtained by ArcGIS based on 𝑡𝑖𝑚𝑒 𝐸 𝑟 , 𝐸 𝑏 , 𝐺 𝑟 and the mapping between them; (3) for time-based accessibility, 𝑎𝑖𝑗 contains two

parts, one part is the waiting time in bus system or subway system where these waiting time are assumed as constant value which can be obtained from historical statistic data, another part is the travel time between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 by the bus line or subway line which can be extracted from bus and subway schedule tables. If two stations 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 are complementary stations in CUPTSs, which means the walking distance between these two stations are not larger than PIDTP, then: (1) for 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦

topology-based accessibility, 𝑎𝑖𝑗

equals to 1; (2) for distance-based accessibility,

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑖𝑗 equals to the walking distance between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 ; (3) for time-based accessibility, 𝑡𝑖𝑚𝑒 𝑎𝑖𝑗 equals to the walking time between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 .

Otherwise, 𝑎𝑖𝑗 equals to+∞. Step 2: Calculate the value of 𝐴(𝑘 𝑏 +𝑘 𝑠 )×(𝑘 𝑏 +𝑘 𝑠) Use Floyd-Warshall algorithm (Cormen, Leiserson et al. 1990) to calculate the shortest path between any two stations in CUPTSs based on the initial 𝐴(𝑘 𝑏 +𝑘𝑠 )×(𝑘 𝑏 +𝑘𝑠 ) obtained in Step 1 and update the value of 𝑎𝑖𝑗 in 𝐴(𝑘 𝑏 +𝑘 𝑠)×(𝑘𝑏 +𝑘𝑠 ). 10

If 𝑎𝑖𝑗 equals to +∞, it means there is no route between stations 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 by bus or 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦

subway, otherwise, (1) for topology-based accessibility, the value of 𝑎𝑖𝑗

must be positive

integer, which means the minimal number of travel legs in a route between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 in CUPTSs, these travel legs can be bus line, subway line or walking path between complementary stations (Hoogendoorn-Lanser, Nes et al. 2005). In this paper, the topology-based accessibility only considers whether or not two stations can be connected by transportation systems in CUPTSs, 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦

not accounting about the exact travel legs which connect the two stations. So, if 𝑎𝑖𝑗 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦

then set 𝑎𝑖𝑗

𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦

= 0, otherwise set 𝑎𝑖𝑗

= +∞,

= 1. (2) for distance-based accessibility, the value

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 of 𝑎𝑖𝑗 equals to the shortest travel distance in CUPTSs between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 . Usually, the 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 value of 𝑎𝑖𝑗 will increase after disruptions, and if using the average shortest distance as the

performance metric and according to the vulnerability definition, the vulnerability under disruptions will be negative values, which are still valid, but provide non-intuitive results. In this paper, the reciprocal of the shortest distance is used to represent the distance-based accessibility 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 , then if the shortest distance is infinity, set 𝑎𝑖𝑗 = 0. (3) for time-based 𝑡𝑖𝑚𝑒 accessibility, 𝑎𝑖𝑗 means the shortest travel time in CUPTSs between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 . Because of 𝑡𝑖𝑚𝑒 the same reason with that of distance-based accessibility, the reciprocal of 𝑎𝑖𝑗 is used to

represent the time-based accessibility between 𝑛𝑖𝑏𝑠 and 𝑛𝑗𝑏𝑠 , then if the shortest travel time is 𝑡𝑖𝑚𝑒 infinity, set 𝑎𝑖𝑗 = 0.

After these two steps the topology-based, distance-based and time-based accessibilities between any station pairs in CUPTSs can be obtained. The transportation system performance based on the three types of accessibilities can be calculated through 𝐴(𝑘 𝑏 +𝑘 𝑠 )×(𝑘 𝑏 +𝑘 𝑠) . In order to compare the transportation system vulnerability with and without complementary system, the performance of bus system and subway system are calculated separately. Note that the computation complexity of the above procedure mainly depends on Floyd-Warshall algorithm, which has a complexity of O((𝑘𝑏 + 𝑘 𝑠 )3). For a bus station 𝑛𝑖𝑏 , considering the complementary subway system, its accessibility to all 1

𝑏

the other bus stations can be represented as 𝑘 𝑏 −1 ∑𝑘𝑗=1 𝑎𝑖𝑗 , 𝑖 ≠ 𝑗, 𝑖, 𝑗 ∈ [1, 𝑘 𝑏 ] . Then the performance of bus system with complementary subway system before disruption can be calculated as: 1

𝑏

𝑏

𝐹𝑏←𝑠 = 𝑘 𝑏 −1 ∑𝑘𝑖=1 ∑𝑘𝑗=1 𝑎𝑖𝑗 , 𝑖 ≠ 𝑗, 𝑖, 𝑗 ∈ [1, 𝑘 𝑏 ]

(1)

Similar, the performance of subway system with complementary bus system before disruption can be calculated as 1

𝑏

𝑠

𝑏

𝑠

𝑘 +𝑘 𝑏 𝑏 𝑠 𝐹𝑠←𝑏 = 𝑘 𝑠 −1 ∑𝑘𝑖=𝑘+𝑘 𝑏 +1 ∑𝑗=𝑘 𝑏 +1 𝑎𝑖𝑗 , 𝑖 ≠ 𝑗, 𝑖, 𝑗 ∈ [𝑘 + 1, 𝑘 + 𝑘 ]

(2)

This performance calculation method for transportation systems can be used to assess the performance of CUPTSs as a whole and also can be extended to discuss the performance of 11

CUPTSs that contain more than two transport modes.

2.3 Disruptions modeling of CUPTSs Three types of disruptions are discussed in this paper: the random failures, and the intentional attacks including degree based and betweenness based attacks. The random failures are used to model the random happened disruptions in CUPTSs, such as vehicle accidents and local flood events; while the intentional attacks are used to model terrorist attacks at important stations and road segments or subway tracks. For the bus network 𝐺 𝑏 , the random node failures mean randomly choosing nodes in 𝐺 𝑟 to set them as failed, and the bus lines using these failed components are also set as failed too; the random edge failures mean randomly choosing edges in 𝐺 𝑟 as failed, and the bus lines using these failed road segments are set as failed too; similarly, the intentional attacks mean choosing components (nodes or edges) in 𝐺 𝑟 according to their topological metrics (degree or betweenness) as failed, where components with high topological metric value are attacked in priority. Because the bus stations are usually within the road segments and not at the crossroads, the attacks on the bus stations can be regarded as the corresponding edge attacks in the road network. The random failures and the intentional attacks on subway system can be defined in a similar way.

2.4 CUPTSs vulnerability assessment To assess system vulnerability, it requires calculating system accessibility before and after disruptions. Note that the post-event accessibility is affected by the duration of the disruptive events. For the transportation systems in Wuhan, any two stations can be reached within two hours in CUPTSs, and if the disruption duration is less than two hours, the shorter the event duration, the more buses or trains with their departure time after the event would not be interrupted, and then the post-event metrics are non-decreasing function of the event duration; when the disruptive event lasts for more two hours, the affected buses or trains in the next two hours immediately after the event would be deterministic and then the post-event accessibility metrics are independent of event duration. Hence, for illustrative purposes, this paper does not consider the effects of event duration on post-event performance and vulnerability measures, and simply assumes all events will last for at least two hours. Also, the bus drivers cannot change the operational route without permission from the bus company in most cities in China. The bus vehicles are all equipped with GPS devices and are monitored by the company all the time. The driver will be punished if he/she changes the route without permission. Hence, the re-routes can be ignored within the next two hours immediately after the events. This section uses node attacks to bus system with complementary subway system as an example to illustrate how to assess the CUPTSs vulnerability after disruptions. If a disruption initially causes the failures of some crossroad, the corresponding nodes and 12

edges connected to these nodes in 𝐺 𝑟 are all set as failed. Based on the mapping between 𝐸 𝑏 , 𝐿𝑏 and 𝐸 𝑟 , the corresponding nodes (bus stations) and edges in 𝐺 𝑏 and bus lines in 𝐿𝑏 are removed from the bus system. The post-disruption bus network 𝐺 𝑏 is reconstructed by the remained bus stations and bus lines. When only considering the bus system, the system performance will be measured by the accessibility of post-disruption bus network, represented as 𝐹𝑏𝑑 . When considering the bus system with the complementary subway system, if a removed bus station has at least one complementary edge, then the corresponding bus station node will be 𝑑 remained in 𝐺 𝑏 . The post-disruption performance of bus system is represented as 𝐹𝑏←𝑠 = 1

𝑏

𝑘 𝑏 −1

𝑏

𝑑 𝑑 ∑𝑘𝑖=1 ∑𝑘𝑗=1 𝑎𝑖𝑗 , 𝑖 ≠ 𝑗, 𝑖, 𝑗 ∈ [1, 𝑘 𝑏 ] , where 𝑎𝑖𝑗 denotes the post-disruption accessibility

between two bus stations 𝑛𝑖𝑏 and 𝑛𝑗𝑏 , which is calculated by the accessibility calculation method proposed in Section 2.2 based on the post-disruption bus network, the subway network and the complementary edges between these two networks. Based on the aforementioned vulnerability definition, when only considering the bus system, the system vulnerability under disruption d, can be represented as 𝑉𝑏𝑑 =

𝐹𝑏 −𝐹𝑏𝑑 𝐹𝑏

=1−

𝐹𝑏𝑑 𝐹𝑏

; when

considering the bus system with the complementary subway system, the bus system vulnerability 𝑑 under disruption d, denoted by 𝑉𝑏←𝑠 , can be assessed as: 𝑑 𝑉𝑏←𝑠 =

𝑑 𝐹𝑏←𝑠 −𝐹𝑏←𝑠

𝐹𝑏←𝑠

=1−

𝑑 𝐹𝑏←𝑠

𝐹𝑏←𝑠

𝑏

=1−

𝑏

𝑘 𝑑 ∑𝑘 𝑖=1 ∑𝑗=1 𝑎𝑖𝑗 𝑏 𝑘𝑏 ∑𝑘 𝑖=1 ∑𝑗=1 𝑎𝑖𝑗

, 𝑖 ≠ 𝑗, 𝑖, 𝑗 ∈ [1, 𝑘 𝑏 ]

(3)

In a similar way, the vulnerability of subway system with the complementary bus system 𝑑 under disruption d, denoted by𝑉𝑠←𝑏 , can be assessed as: 𝑑 𝑉𝑠←𝑏

=

𝑑 𝐹𝑠←𝑏 −𝐹𝑠←𝑏

𝐹𝑠←𝑏

𝑑 𝐹𝑠←𝑏

= 1−𝐹

𝑠←𝑏

=1−

𝑏 +𝑘𝑠 𝑏 𝑠 ∑𝑘 +𝑘 𝑎𝑑 𝑖=𝑘𝑏 +1 𝑗=𝑘𝑏 +1 𝑖𝑗 𝑏 𝑠 𝑏 𝑠 ∑𝑘 +𝑘 ∑𝑘 +𝑘 𝑎 𝑖=𝑘𝑏 +1 𝑗=𝑘𝑏 +1 𝑖𝑗

∑𝑘

, 𝑖 ≠ 𝑗, 𝑖, 𝑗 ∈ [𝑘𝑏 + 1, 𝑘 𝑏 + 𝑘 𝑠 ]

(4)

3. CUPTSs-aimed vulnerability analysis method Different from complementary railway and airline systems which serve the passengers at national level for long distance travel (Ouyang, Pan et al. 2015), the CUPTSs mainly serve the urban passengers for short distance commuting in urban areas, and this leads to different characteristics which will affect the system vulnerability: First, they have different network topologies, the complementary railway and airline systems at the national level should consider the network coverage and construction cost, so usually the network topologies show a hub-spoke like style, especially for the airline system; while in the CUPTSs, there are few hub nodes and the networks usually are mesh style. Second, the complementary railway and airline systems at the national level usually have a relatively stable topology (after large-scale construction period) compared to the CUPTSs which might change or expand their topologies frequently to meet urban development, so the CUPTSs vulnerability during the system expansion process should be time-varied. Third, the complementary styles at national level and urban level are different, the 13

railway stations and airports in the same city are assumed complementary without considering the distance between them, while the complementary relationship for subway and bus stations in the urban area depends on the value of PITDP. All these different characteristics make it necessary to study the complementary systems at the urban level to enrich the knowledge of complementary system vulnerability analysis. Based on the CUPTSs vulnerability model proposed in Section 2, the CUPTSs-aimed vulnerability analysis method will be discussed from two aspects: vulnerability effects of PITDP and vulnerability effects of subway expansion.

3.1 Vulnerability effects of PITDP As aforementioned in Section 1, the value of PITDP will affect the transportation system vulnerability, because the number of complementary stations in CUPTSs depends on the value of PIDTP and usually the more the complementary stations exist the lower the system is. If the distance between a bus station and a subway station is larger than a passenger’s PITDP, these two stations cannot be regarded as complementary stations to this passenger. One way to increase the number of complementary stations in CUPTSs is to shorten the distance between subway stations and its nearby bus stations in system designing stage beforehand. After system construction process, it is hard to change the distance between stations. But it is possible to increase passengers’ PITDP by some policies and thereby to increase the number of complementary stations in CUPTSs and reduce the system vulnerability, especially in some emergency situations. For example, if free bicycles can be provided to the passengers where the distance between a subway station and a bus station is beyond their PITDP, some passengers would perhaps be willing to take the free bicycle to transfer between the two stations. Another way is to build shelter gallery or commercial corridor between subway stations and bus stations, which can be used to attract passengers to walk through these places to transfer between subway and bus stations. Actually, in some big cities in China, similar policies have been established to increase passengers’ PITDP. For example, a bicycle system has been established to connect the subway stations to nearby bus stations in Wuhan city. A passenger can borrow a bicycle for free from a station and return it at another station. The effects of those policies and how the value of PITDP affects the CUPTSs vulnerability will be addressed in this paper. To analyze the effects of PITDP on the CUPTSs vulnerability, different PITDP values are used in complementary relationship modeling and then the vulnerability of bus system and subway system without or with complementary system are assessed under the three types of disruptions. The results will illustrate how different PITDP values affect the CUPTSs vulnerability. Further, based on the CUPTSs vulnerability analysis results, the complementary strength defined in (Ouyang, Pan et al. 2015) between bus and subway systems are also calculated respectively to see how the value of PITDP affects the static and dynamic complementary strength. As mentioned in Section 2, different passengers may have different PITDP values, the 14

discussion about transportation systems vulnerability under different PITDP can reflect different CUPTSs vulnerability from the perspectives of different types of passengers.

3.2 Vulnerability effects of subway expansion The subway systems in many big cities in China are experiencing rapid expansion in recent years. Almost in each year, one new subway line will be opened in these cities. For example, in Wuhan, there will be a new subway line in each year from 2015 to 2020 and the total subway mileage will increase from 73 km at 2014 to 385 km at 2020. The expansion of subway system changes the topologies of CUPTSs and also affects system vulnerability. A raised question is how the newly opened subway lines affect the CUPTSs vulnerability during the subway expansion process and whether the opening sequences of new lines can be optimized from the system vulnerability perspective. With the opening of new subway lines year by year, new subway stations are added into CUPTSs and the number of bus stations that have complementary subway stations will accordingly increase. As mentioned in Section 2, the increase of complementary stations will reduce the transportation system vulnerability. During the subway expansion process, the topology of subway system will remain stable for the time period between the opening of two new subway lines and during this stable topology period the CUPTSs vulnerability (depended on the topologies of existing bus and subway systems) will not change. Different opening sequences of new subway lines lead to different complementary stations during each period of stable topology and further affect the transportation system vulnerability during the whole subway system topology expansion process. Following from the above question, a metric to assess the dynamic vulnerability of CUPTSs during the subway system topology expansion process, as well as the method to optimize the opening sequence of new subway lines to minimize the transportation system vulnerability during the process should be addressed. This paper proposes a metric to measure the dynamic vulnerability of transportation systems during their expansion process, called dynamic accumulating vulnerability (DAV). The DAV is related with two factors, the system vulnerability and the time period of the system under this vulnerability state, which is defined as 1

1

𝑇

𝐷𝐴𝑉𝑑,𝑇 = 𝑇 ∫0 ∫0 (𝐸(𝑉𝑑,𝑝,𝑡 ) ∗ 𝑓(𝑝))𝑑𝑝𝑑𝑡

(5)

where 𝑉𝑑,𝑝,𝑡 denotes the system vulnerability under disruption d when the percentage of failed system components is 𝑝 at time t, 𝑝 ∈ [0,1] 𝑎𝑛𝑑 𝑡 ∈ [0, 𝑇], 𝐸(𝑉𝑑,𝑝,𝑡 ) denotes the expectation of 𝑉𝑑,𝑝,𝑡 , 𝑓(𝑝) denotes the probabilistic distribution of 𝑝 percentage of failed components. The effects of different subway expansion sequences on CUPTSs vulnerability can be analyzed by checking their DAV values which are calculated based on the equation (5) and vulnerability model of CUPTSs in Section 2. When the number of newly opened subway lines is small, the best expansion sequence can be identified through exhausting the DAV values of all 15

possible expansion sequences. When the number of newly opened subway lines is large which will lead to a heavy computation burden, a greedy algorithm is proposed to find the optimal system expansion sequence to minimize the CUPTSs vulnerability during the process. Let 𝑙 𝑠𝑖 denotes the planned subway line in the future, 𝑙 𝑠𝑖 ∈ 𝐿𝑠 , i∈ {1,2, … , 𝐼}, where I is the number of planned subway lines. It is assumed that a new subway line will be opened at each time period and 𝑙𝑗𝑠𝑖 denotes i-th subway line will be opened at j-th time period, j∈ {1,2, … , 𝐼}. Let X= {𝑥1 , 𝑥2 , … , 𝑥𝐼 }, where 𝑥𝑗 is used to record the label of optimal newly opened subway line at time period j, 𝐺 denotes the initial state of CUPTSs at the beginning and 𝐺 𝑗 denote the state at the j-th time period. The greedy algorithm is described as below: Step 1, Use equation (5)to calculate the DAV value of bus system based on 𝐺 + 𝑙 𝑠𝑖 for each planned subway line respectively, where 𝑉𝑑,𝑝,𝑡 is calculated by the method introduced in Section 2.4 with the consideration of complementary subway system based on equation (3); Choose the minimal DAV among these values and the corresponding subway line is set as the newly opened 𝑥

line at the first time period, represented as 𝑙1 1 , 𝑥1 ∈ {1,2, … , 𝐼}; if there are more than one subway lines that lead to the minimal DAV value, then randomly choose one of these lines and set it as the newly opened one. Step 2, Use equation (5)to calculate the DAV value of bus system based on 𝐺 1 + 𝑙 𝑠𝑖 for each planned subway lines respectively except 𝑙 𝑥1 , where 𝐺 1 = 𝐺 + 𝑙 𝑥1 ; Choose the minimal DAV among these values and the corresponding subway line is set as the newly opened line at the 𝑥

second time period, represented as 𝑙2 2 , 𝑥2 ∈ {1,2, … , 𝐼}; If there are more than one subway lines that lead to the minimal DAV value, then randomly choose one of these lines and set it as the newly opened one. 𝑥

Using the similar process, 𝑙𝐼 𝐼 can be identified at Step I and the sequence of 𝑥

𝑥

𝑥

{𝑙1 1 , 𝑙2 2 , … , 𝑙𝐼 𝐼 } is the optimal opening sequence, where 𝑥1 , 𝑥2 , … , 𝑥𝐼 ∈ {1,2, … , 𝐼}.

4. Experiment results The complementary bus and subway systems (nine subway lines in 2020) are used as an example to analyze the CUPTSs vulnerability using the proposed vulnerability model and CUPTSs-aimed vulnerability analysis method. In order to compare the transportation system vulnerability with and without the complementary relationship, the vulnerability of bus system and subway system are assessed respectively in this section. The edge-based attacks and topology-based accessibility are adopted as experiment settings in this paper.

16

4.1 CUPTSs vulnerability in Wuhan

Figure 2. Accessibility-based bus system vulnerability as a function of road segments failure fraction 𝑃𝑟 in the case of with (𝑉𝑏←𝑠 ) and without (𝑉𝑏 ) the complementary subway system under random failures, degree-based and betweenness-based attacks.

Figure 3. Accessibility-based subway system vulnerability as a function of subway edges failure fraction 𝑃𝑠 in the case of with (𝑉𝑠←𝑏 ) and without (𝑉𝑠 ) the complementary bus system under random failures, degree-based and betweenness-based attacks. The value of PITDP is set as 800m in the experiment (López and Monzón 2010). Figure 2 and Figure 3 show the accessibility-based bus system and subway system vulnerability in case of with and without considering the complementary system under three types of disruptions. This paper uses the bus system vulnerability assessment in Figure 2 as an example to illustrate the experiment process: for the random road segment failures, a fraction 𝑃𝑟 of edges in 𝐺 𝑟 is randomly selected and assumed to fail, where 𝑃𝑟 ∈ [0,1] and the increasing step of 𝑃𝑟 is 0.02, the 17

vulnerability of bus system with and without complementary subway system are assessed for each 𝑃𝑟 . Similarly, for the degree-based or betweenness-based attacks, a fraction 𝑃𝑟 of edges with the largest degree (the edge degree is the average degree of its end-point two nodes) or betweenness values are assumed to fail, and the bus system vulnerability with and without complementary subway system are assessed for each 𝑃𝑟 . The results in Figure 2 and Figure 3 are averaged over 10,000. In Figure 2 and Figure 3, with the increase of the failure fraction of edges, the vulnerability curves are all increasing functions. For single bus system or subway system without complementary system, the system vulnerability value reaches almost 1 when the edge failure fraction is larger than 0.2 in all the three types of disruption modes, this implies that when 20% of edges are failed in 𝐺 𝑟 or 𝐺 𝑠 the transportation systems are fully disconnected. The subway system vulnerability increases much faster than the bus system because it has less nodes and edges and less redundant than the bus system. In Figure2 and Figure 3, the vulnerability curves in the case of considering complementary system are always below the curve without considering it. This implies the complementary relationship can reduce the vulnerability of the transportation system under disruptions. Comparing the results in Figure 2 and Figure 3, the effect of complementary relationship from bus system to subway system is more significant than that from subway system to bus system. For example, when 𝑃𝑟 =0.4, considering the complementary subway system, the bus system vulnerability decreases by 30%; when 𝑃𝑠 =0.4, considering the complementary bus system, the subway system vulnerability decreases by 96%. Considering the vulnerability curve under betweenness-based attacks in Figure 3, it is found that when 𝑃𝑠 increases from 0 to 0.08, the subway system vulnerability with the consideration of the complementary bus system is equal to 0, this means that even removing the 15 highest betweenness subway tracks will not affect the transport services provided by the bus and subway systems. This phenomenon shows that in Wuhan city, there are sufficient bus services around important subway facilities to provide redundant transport services for the passengers even when a large fraction of subway components is under terrorist attacks or natural hazards. In the extreme case that all road segments are failed, the accessibility of bus system totally depends on the complementary subway system, and the vulnerability of bus system reaches its 𝑑 maximum value 0.7090, denoted as 𝑚𝑎𝑥𝑉𝑏←𝑠 , which can be calculated based on the definition of

vulnerability, presented in the equation (6). Note that the maximum vulnerability value under three types of disruptions are the same, this is because when vulnerability reaches its maximum value the original transportation system is fully collapsed, and the accessibility depends on the topology of the complementary system. Similarly in Figure 3, the vulnerability curve of subway system 𝑑 with the complementary bus system reaches its maximum value 0.0359, denoted as 𝑚𝑎𝑥𝑉𝑠←𝑏 , 𝑑 which can be calculatedbased on the equation (7). The value of 𝑚𝑎𝑥𝑉𝑠←𝑏 is very small, this

implies even all the subway stations are failed, the complementary bus stations can also provide 18

sufficient commuting service for the passengers near the subway stations. 1

0

𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠 =1−

1833 1832

𝑑 𝑚𝑎𝑥𝑉𝑠←𝑏 =1−

166 165

1

(

(

0

∗ (1833 − 989) +

∗ (166 − 163) +

162 165

988 1832

∗ 989) = 0.7090

(6)

∗ 163) = 0.0359

(7)

4.2 Effects of PITDP on CUPTSs vulnerability Many studies use a walking distance thresholds of 0.25 miles (400m) for accessing bus stations (López and Monzón 2010, Munoz-Raskin 2010) and 0.5 miles (800m) for subway or railway stations (El-Geneidy, Grimsrud et al. 2014). In this section, the value of PITDP is set from 0 to 2000m with an increasing step of 50m, and the effects of different PITDP value on CUPTSs vulnerability are discussed in the experiment.

4.2.1 PITDP-depended CUPTSs vulnerability analysis Degree-based Attacks

1

1

0.9

0.9

0.8

0.8

0.7

0.7

Bus Vulnerability

Bus Vulnerability

Random Attacks

0.6 0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0

500

1000 PITDP(m) 1500

2000

0.2

0

0.4

0.6

0.8

0 0

1

Pr

500

1000 PITDP(m)

1500

2000

0

0.2

0.4 Pr

0.6

0.8

1

Betweenness-based Attacks

1 0.9 0.8

Bus Vulnerability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000 PITDP(m)

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Figure 4. The vulnerability of bus system with complementary subway system at the PITDP value from 50m to 2000m under random failures, degree-based and betweenness-based attacks. Figure 4 shows the vulnerability of bus system with complementary subway system at different PIDTP values (0~2000m). In Figure 4, with the increase of PITDP, the vulnerability of bus system decreases because larger PITDP makes more subway and bus stations complementary to provide alternative service. Figure 4 also shows when 𝑃𝑟 is around 0.2, the bus system vulnerability almost reaches its maximum value for all the three types of disruptions; even the PITDP increases to a large value, the bus system still have high vulnerability. Table 2 lists the 19

number of complementary station pairs ( 𝑘 𝑐 ), the number of bus stations which have complementary subway station(s) ( 𝑁 𝑏←𝑠 ), the number of subway stations which have 𝑑 complementary bus station(s) (𝑁 𝑠←𝑏 ), the maximum bus system vulnerability (𝑚𝑎𝑥𝑉𝑏←𝑠 ), the 𝑑 maximum subway system vulnerability (𝑚𝑎𝑥𝑉𝑠←𝑏 ) at the PITDP value from 50m to 2000m with

an increasing step of 50m. Table 2. Vulnerability related metrics under the PITDP value from 50m to 2000m with an increasing step of 50m PITDP 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000

𝑘𝑐 41 75 116 162 225 281 344 405 482 570 680 787 909 1019 1165 1301 1458 1599 1762 1929 2113 2308 2490 2681 2881 3065 3280 3515 3752 3964 4190 4443 4684 4960 5236 5516 5797 6071 6321 6614

𝑁 𝑏←𝑠

𝑁 𝑠←𝑏

41 75 116 162 225 281 342 403 477 553 648 726 792 859 930 989 1045 1088 1131 1171 1217 1249 1275 1306 1334 1356 1380 1403 1425 1441 1457 1475 1482 1500 1515 1526 1532 1542 1547 1555

38 64 89 113 137 147 155 158 159 160 161 161 161 162 162 163 163 163 164 164 164 164 164 164 164 164 165 165 165 165 165 165 165 165 165 165 165 165 165 166

𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠 0.9995 0.9983 0.9960 0.9922 0.9850 0.9766 0.9653 0.9518 0.9324 0.9091 0.8751 0.8433 0.8134 0.7805 0.7427 0.7090 0.6751 0.6478 0.6194 0.5920 0.5593 0.5358 0.5163 0.4925 0.4705 0.4528 0.4333 0.4142 0.3957 0.3821 0.3683 0.3526 0.3464 0.3304 0.3170 0.3070 0.3015 0.2924 0.2878 0.2804

𝑑 𝑚𝑎𝑥𝑉𝑠←𝑏 0.9487 0.8528 0.7141 0.5379 0.3198 0.2164 0.1285 0.0943 0.0828 0.0712 0.0595 0.0595 0.0595 0.0478 0.0478 0.0359 0.0359 0.0359 0.0240 0.0240 0.0240 0.0240 0.0240 0.0240 0.0240 0.0240 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0000

20

𝑆𝐶𝑆 𝑏←𝑠

𝑆𝐶𝑆 𝑠←𝑏

𝑏←𝑠 𝐷𝐶𝑆𝑚𝑖𝑛

𝑠←𝑏 𝐷𝐶𝑆𝑚𝑖𝑛

0.0224 0.0409 0.0633 0.0884 0.1227 0.1533 0.1866 0.2199 0.2602 0.3017 0.3535 0.3961 0.4321 0.4686 0.5074 0.5396 0.5701 0.5936 0.6170 0.6388 0.6639 0.6814 0.6956 0.7125 0.7278 0.7398 0.7529 0.7654 0.7774 0.7861 0.7949 0.8047 0.8085 0.8183 0.8265 0.8325 0.8358 0.8412 0.8440 0.8483

0.2289 0.3855 0.5361 0.6807 0.8253 0.8855 0.9337 0.9518 0.9578 0.9639 0.9699 0.9699 0.9699 0.9759 0.9759 0.9819 0.9819 0.9819 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 1.0000

0.0005 0.0017 0.0040 0.0078 0.0150 0.0234 0.0347 0.0482 0.0676 0.0909 0.1249 0.1567 0.1866 0.2195 0.2573 0.2910 0.3249 0.3522 0.3806 0.4080 0.4407 0.4642 0.4837 0.5075 0.5295 0.5472 0.5667 0.5858 0.6043 0.6179 0.6317 0.6474 0.6536 0.6696 0.6830 0.6930 0.6985 0.7076 0.7122 0.7196

0.0513 0.1472 0.2859 0.4621 0.6802 0.7836 0.8715 0.9057 0.9172 0.9288 0.9405 0.9405 0.9405 0.9522 0.9522 0.9641 0.9641 0.9641 0.9760 0.9760 0.9760 0.9760 0.9760 0.9760 0.9760 0.9760 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 1.0000

From Figure 4 and Table 2, it is found that the effect of increasing PITDP value to reduce bus 𝑑 system vulnerability is changing during the process. Table 2 illustrates show 𝑚𝑎𝑥𝑉𝑏←𝑠 changes 𝑑 with the increase of PITDP. When the PITDP is small, the effect of 𝑚𝑎𝑥𝑉𝑏←𝑠 mitigation is not 𝑑 significant, for example, when PITDP increases from 50m to 100m, the reduction of 𝑚𝑎𝑥𝑉𝑏←𝑠 is 𝑑 0.0012; when the PITDP increases from 700m to 750m, the reduction of 𝑚𝑎𝑥𝑉𝑏←𝑠 reaches the

value 0.0378; after that, with the increase of PITDP, the mitigation effect becomes smaller. When the PITDP increases from 1800 to 1850m, the reduction of 𝑚𝑎𝑥𝑉𝑑𝑏−𝑠 is only 0.0055. At that time, increasing PITDP is not an efficient way to reduce bus system vulnerability.

Figure 5. The reduction of bus system vulnerability as a function of PITDP value The curves in Figure 5 show how the vulnerability reduction effect changes with the increase of PITDP for the bus system with the complementary subway system. According to the curves in Figure 5, the maximal reduction of bus system vulnerability is reached when PITDP is around 550m. In Figure 5, the curves fluctuate with the increase of PITDP, this is because the number of increased complementary stations varies for each 50m increase of PITDP, which depends on the topological and geographical characteristics of bus and subway systems. From a monetary perspective, the government or the stakeholders should find the most efficient plan to reduce the system vulnerability and the results in this section will support them to make more reasonable decision.

21

Figure 6. The vulnerability of subway system with complementary bus system at the PITDP value from 50m to 2000m under random failures, degree-based and betweenness-based attacks. Figure 6 shows the vulnerability of subway system with complementary bus system at different PITDP value (0-2000m). Similar with the bus system, with the increase of PITDP, the vulnerability of subway system decreases, and when 𝑃𝑠 is around 0.2, the subway system vulnerability almost reaches its maximum value. Different with the bus system, the value of subway system vulnerability becomes very small when the PITDP is not large. For example, when 𝑑 PITDP increases to 400m, 𝑚𝑎𝑥𝑉𝑠←𝑏 is less than 0.1. From Figure6 and Table 2, when the PITDP 𝑑 increases from 200m to 250m, the reduction of 𝑚𝑎𝑥𝑉𝑠←𝑏 reaches the value 0.2182; after that, the

vulnerability reduction becomes smaller for each 50m increase of PITDP. According to the curves in Figure 7, the maximal reduction for all the three types of disruptions is obtained when PITDP increases from 200m to 250m.

22

Figure 7. The reduction of subway system vulnerability as a function of PITDP value Comparing the results in Figure 4 and Figure 6, it is found that the vulnerability of subway system decreases more sharply than that of bus system when the PITDP increases. This implies the increase of PITDP affects the subway system much more than the bus system. For example, when PITDP increases from 200m to 800m, the maximum bus system vulnerability decreases from 0.9922 to 0.7096 with a reduction ration of 28%; meanwhile, the maximum subway system vulnerability decreases from 0.5379 to 0.0359with a reduction ration of 93%. The vulnerability of subway system is particularly sensitive to the change of PITDP. Based on the above discussion, from the perspective of bus system vulnerability, it is better to build new subway stations within 550m around existing bus stations; while from the perspective of subway system vulnerability, it is better to build new bus stations within 250m around existing subway stations. These results can be used to support transportation system expansion decisions.

4.2.2PITDP-dependedcomplementary strength analysis in CUPTSs The static complementary strength is used to measure the degree of complementary relationship between two systems from the system topological perspective. The static complementary strength 𝑆𝐶𝑆 𝑆1←𝑆2 of one system S2 on the other system S1 can be quantified by𝑆𝐶𝑆 𝑆1←𝑆2 =

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑖𝑛 𝑆1 𝑤ℎ𝑖𝑐ℎ ℎ𝑎𝑣𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑖𝑛 𝑆2 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑖𝑛 𝑆1

. The value of

SCS provides an intuitive quantification on how many nodes in one system have another type of complementary nodes (Ouyang, Pan et al. 2015). Figure 8 shows the static complementary strength (SCS) between bus system and subway system at the PITDP value from 50m to 2000m with an increasing step of 50m.

23

1

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0.8

0.6

SCSb-s SCSs-b

0.4

0.2

0

0

400

800

1200

1600

2000

PITDP(m)

Figure 8. The SCS between bus system and subway system in Wuhan city with the PITDP value from 50m to 2000m. Figure 8 shows that the value of static complementary strength 𝑆𝐶𝑆 𝑠←𝑏 and 𝑆𝐶𝑆 𝑏←𝑠 are all increasing with the increase of PITDP, and the 𝑆𝐶𝑆 𝑠←𝑏 curve is always above that of 𝑆𝐶𝑆 𝑏←𝑠 at different PITDP. This implies the complementary strength from bus to subway is stronger than that from subway to bus from the topological perspective. Also, the increase magnitude of 𝑆𝐶𝑆 𝑠←𝑏 is larger than that of 𝑆𝐶𝑆 𝑏←𝑠 . The values of 𝑆𝐶𝑆 𝑠←𝑏 and 𝑆𝐶𝑆 𝑏←𝑠 at different PITDP from 50m to 2000m with an increase step of 50m are listed in Table 2. Figure 8 and Table 2 both show that 400m is an important value of PITDP to 𝑆𝐶𝑆 𝑠←𝑏 , because when 𝑃𝐼𝑇𝐷𝑃 > 400m, the effect for increasing static complementary strength decreases with the increase of PITDP. Figure 9 and Figure 10 show how the dynamic complementary strength (DCS) between bus system and subway system changes at different PITDP from 50m to 2000m. The definition of dynamic complementary strength can be found in (Ouyang, Pan et al. 2015), which can reflect how one system in case of its disruption can be substituted by another system. With the increase of 𝑏←𝑠 𝑠←𝑏 PITDP, the dynamic complementary strength increases. The value of 𝐷𝐶𝑆𝑚𝑖𝑛 and 𝐷𝐶𝑆𝑚𝑖𝑛 at

different PITDP are listed in Table 2. DCS reaches its minimum value when one system is totally destroyed and the accessibility is completely provided by its complementary system. According to the definition of DCS, the 𝐷𝐶𝑆𝑚𝑖𝑛 value of one system for all the three types of disruptions is the 𝑏←𝑠 𝑠←𝑏 same and the sum of 𝐷𝐶𝑆𝑚𝑖𝑛 and 𝑚𝑎𝑥𝑉 𝑑 equals to 1. So the value of 𝐷𝐶𝑆𝑚𝑖𝑛 and 𝐷𝐶𝑆𝑚𝑖𝑛 𝑑 𝑑 can be calculatedbased on the value of 𝑚𝑎𝑥𝑉𝑏←𝑠 and 𝑚𝑎𝑥𝑉𝑠←𝑏 . Comparing Figure 9 and Figure

10, the value of 𝐷𝐶𝑆𝑑𝑠←𝑏 is always larger than 𝐷𝐶𝑆𝑑𝑏←𝑠 , and this result is coincident with the result in SCS analysis. The 𝐷𝐶𝑆𝑚𝑎𝑥 values at different PIDTP under three types of disruptions are shown in Figure 11. Comparing the 𝐷𝐶𝑆𝑚𝑎𝑥 in the three disruptions modes, it is found that 𝐷𝐶𝑆𝑚𝑎𝑥 under degree and betweenness based attacks is larger than that under random failures at most of PITDP values, especially for the degree-based attacks. This implies complementary relationship plays a more important role in mitigating CUPTSs vulnerability toward intentional attacks than random failures. This is partly because the critical areas in Wuhan always are transport hubs for transportation systems.

24

Random Attacks

Degree-based Attacks

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Dynamic Complementary Strength

DCS

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1200 0.2 0

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Figure 9. The dynamic complementary strength from subway system to bus system at different PITDP values under random failures, degree-based and betweenness-based attacks. Random Attacks

Degree-based Attacks

Dynamic Complementary Strength

Dynamic Complementary Strength

DCS

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Figure 10. The dynamic complementary strength from bus system to subway system at different PITDP values under random failures, degree-based and betweenness-based attacks. 25

1000

Figure 11. The value of 𝐷𝐶𝑆𝑚𝑎𝑥 at different PIDTP under three types of disruptions

4.3Effects of subway expansion on CUPTSs vulnerability In Wuhan city, the network topology of bus system is relatively stable, while the topology of subway system is expanding every year recently. With the opening of new subway lines, the vulnerability curves for each year are shown in Figure 12. The maximum vulnerability value of 𝑑 bus system 𝑚𝑎𝑥𝑉𝑏←𝑠 at each year are listed in the left part of Table 3, and the vulnerability

decreases year by year with the opening of new subway lines. But the maximum vulnerability value is not sufficient to measure the dynamic vulnerability of CUPTSs during the subway expansion process, because it only considers the system vulnerability in extreme cases at a particular year and cannot reflect the accumulation effect of system vulnerability.

Figure 12. The vulnerability of bus system with complementary subway system as a function of road segment failure fraction 𝑃𝑟 under random failure during the planned expansion process of subway system in Wuhan from 2014 to 2020. 26

Table 3. The maximum vulnerability value of bus system with complementary subway system of each year during the expansion process of subway system under original and optimal opening sequences. Year 2014 2015 2016 2017 2018 2019 2020

Planned opening sequence 𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠

0.8993 0.8572 0.8218 0.7871 0.7697 0.7270 0.7090

New subway line None Line 3 Line 6 Line 7 Line 8 Line 5 Line 9

Optimal opening sequence 𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠 0.8993 0.8551 0.8153 0.7780 0.7460 0.7264 0.7090

New subway line None Line 6 Line 3 Line 5 Line 7 Line 9 Line 8

In the case of subway expansion in Wuhan, the six subway lines will be opened one by one at the beginning of each year from 2015 to 2020, so in a particular year, the topologies of CUPTSs will keep stable and the accessibility-based vulnerability will not change in this year. Then, the DAV of bus system with complementary subway system from 2014 to a specific year 201X can be calculated as the integral of each bus system vulnerability curve from 2014 to 201X in average according to equation (5) in section 3.2.Because of limited traffic related data, the occurrence probability of 𝑃𝑟 is not obtained, and then it is assumed that 𝑓(𝑃𝑟 ) are equal for any value of 𝑃𝑟 . The left part of Table 4 presents the DAV value of bus system with complementary subway system according to the planned subway opening sequence at each year from 2014 to 2020. Table 4 illustrates that with the opening of new subway lines, the value of DAV decreases year by year. Table 4. DAV of bus system with complementary subway system in planned and optimal subway opening sequences under the three types of disruptions Year

DAV in planned opening sequence Random Degree Betweenness

2014 2015 2016 2017 2018 2019 2020

0.8328 0.8111 0.7920 0.7742 0.7602 0.7438 0.7297

0.8217 0.7984 0.7787 0.7603 0.7457 0.7296 0.7157

0.8294 0.8077 0.7890 0.7710 0.7566 0.7397 0.7250

DAV in optimal opening sequence Random Degree Betweenness

0.8328 0.8098 0.7888 0.7692 0.7513 0.7363 0.7232

None Line 6 Line 3 Line 5 Line 7 Line 9 Line 8

0.8217 0.7978 0.7779 0.7583 0.7402 0.7250 0.7118

None Line 3 Line 5 Line 6 Line 7 Line 9 Line 8

0.8294 0.8064 0.7846 0.7649 0.7467 0.7313 0.7179

None Line 6 Line 5 Line 3 Line 7 Line 9 Line 8

In this paper, a greedy algorithm aforementioned in section 3.2 is used to find the optimal subway expansion sequence to minimize the bus system dynamic vulnerability. This paper uses the bus system under random failures as an example to illustrate how to apply the algorithm: First, set PITDP as 800m. Second, assess the bus system vulnerability with the complementary subway system which has three operational subway lines (Line 1, 2, 4) at 2014. Third, assess the bus system vulnerability with all possible combinations of four subway lines (including Line 1, 2, 4, 3; Line 1, 2, 4, 6; Line 1, 2, 4, 7; Line 1, 2, 4, 8; Line 1, 2, 4, 5; Line 1, 2, 4, 9), and choose the 27

combination whose DAV value is minimum, the result is Line 1, 2, 4, 6, then set the Line 6 as the newly opened subway line at 2015. Fourth, based on the four lines (Line 1, 2, 4, 6) identified above, the bus system vulnerability with all possible combinations of five subway lines are assessed respectively and the minimum DAV combination is chosen, the result is Line 1, 2, 4, 6, 3 and Line 3 is set as the newly opened subway line at 2016. Through the similar process, the combination of six, seven, eight, nine subway lines can be identified, and the optimal subway opening sequence is Line 1, 2, 4, 6 (2015), 3 (2016), 5 (2017), 7 (2018), 9 (2019), 8 (2020). 𝑑 The right part of Table 3 shows 𝑚𝑎𝑥𝑉𝑏←𝑠 at each year based on the optimal subway system

opening sequence. Comparing the planned sequence and optimal sequence in Table 3, the values 𝑑 of 𝑚𝑎𝑥𝑉𝑏←𝑠 at 2014 and 2020 are the same. This is because the topology-based accessibility is

only related with transportation system topology and the topologies of bus system with complementary subway system of these two years are the same. This result also indicates that the maximum vulnerability is not sufficient to measure the system dynamic vulnerability during a 𝑑 time period. It is found in Table 3, the 𝑚𝑎𝑥𝑉𝑏←𝑠 of each year in planned sequence is always

larger than that in optimal sequence from 2015 and 2019. This implies the optimal subway lines opening sequence will make the bus system less vulnerable during the subway expansion process. The DAV value in Table 4 for the optimal opening sequence also supports this result. The DAV value of bus system based on optimal sequence is always less than that of the planned sequence under all the three types of disruptions in each year from 2015 to 2020 in Table 4. At the same time, in Table 4, the DAV value under degree-based and betweenness-based attacks are always less than the DAV under random failures, which is coincident with the results obtained in section 4.2.2 and indicates that complementary relationship plays a more important role in reducing CUPTSs vulnerability toward intentional attacks than random failures. The optimal sequences under the three types of disruptions are listed in Table 4. It shows that the optimal opening sequences of the three types of disruptions are different. The difference is related with the topologies of bus and subway systems. Comparing the optimal opening sequences in Table 4, Line 6, Line 5 and Line 3 have higher priority than other lines in all the three types of disruptions. It is partly because Line 6, Line 5 and Line 3 have much stations and mileages than Line 7, Line 9 and Line 8. This implies it is better to build subway lines with large number of stations and mileages in priority to reduce the CUPTSs vulnerability. In the experiment, the bus system vulnerability is used to assess the effects of different opening sequences of subway lines. If taking the minimization of the whole CUPTSs vulnerability as the objective function, the optimal opening sequence of subway lines can be similarly identified. Note that the opening sequence of planned subway lines depends on many factors (López and Monzón 2010), such as the passenger demand, the construction budget, and the regional development plan and so on. The dynamic accumulate vulnerability consideration can be regarded as one of the factors provided to the government or other stakeholders when they design or expand 28

the urban public transportation system. When the effects of other factors are the same or similar, then the dynamic accumulate vulnerability can be used as a key judgment factor.

5. Conclusions and future works This paper proposes a vulnerability model of CUPTSs composed of bus system and subway system, and based on this model a CUPTSs-aimed vulnerability analysis method is introduced to analyze the vulnerability effects of different values of passengers’ intermodal transfer distance preference (PITDP) and different subway expansion strategies. Three networks, including bus network, subway network and road network, are used to describe CUPTSs. The complementary station pairs are defined to represent the complementary relationship between bus and subway systems, which depends on the value of PITDP. A metric, called dynamic accumulating vulnerability (DAV), is proposed to measure the dynamic vulnerability of CUPTSs during the subway expansion and the optimal expansion sequence is identified based on the DAV value through a greedy algorithm. Then the proposed vulnerability model and CUPTSs-aimed vulnerability analysis method are used to analyze the CUPTSs in the city of Wuhan in China, including PITDP-depended CUPTSs vulnerability, PITDP-depended complementary strength and the optimal subway expansion sequence. The experiment shows that: (1) the complementary relationship can significantly reduce the CUPTSs vulnerability; (2) the complementary relationship contributes more vulnerability mitigation on subway system than that for bus system, for example, when PITDP =50m, the minimal dynamic complementary strength from bus system to subway system is about 100 times that from subway system to bus system; (3) increasing PITDP will reduce the CUPTSs vulnerability and the subway system vulnerability is more sensitive to PITDP, when the PITDP increases from 200m to 800m, the decease percentage of subway system vulnerability reaches 93%; (4) from the perspective of monetary and efficiency, it is better to build new subway stations within 550m around existing bus stations and build new bus stations within 250m around existing subway stations; (5)complementary relationship plays an more important role in reducing CUPTSs vulnerability toward intentional attacks than random failures; (6) the optimal subway expansion sequence is affected by the disruption modes, and it is better to build subway lines with large number of stations and mileages in priority to reduce the CUPTSs vulnerability during the subway expansion. In this study, based on the available data, the system performance is measured by accessibility which is only related with system topological characteristics. If more specific information related to the traffic flows is available, such as the passenger throughput for each station, number of passenger flow in each road segment or subway track, the CUPTSs performance can be assessed more practically. Similarly, if more detailed disruption data is available and 𝑓(𝑝) in equation (2) can be estimated, the dynamic accumulating vulnerability can be assessed more precisely to approach the practice. Further, this study only discusses three types of disruptions. Wuhan is a city prone to flood hazards because of the Yangzi River. So, it is 29

necessary to analyze the CUPTSs vulnerability under other types of disruptions in the future, such as floods, earthquakes and so on. Also, this paper uses the multilayer network approach to describe the CUPTSs, but their topological characteristics, such as degree, closeness, betweenness, clustering, modular structures, and their purely topology-based dynamic behavior, are not systemically addressed. Collecting pertinent CUPTSs data in different cities to make such a study by using multi-layer network approaches is also a direction for future research.

Acknowledgments This work is jointly supported by National Natural Science Foundation of China (61572212, 51208223, 61433006, 61304175, 61503166 and 71571076), and the U.S. Department of Transportation through the NEXTRANS Center, the USDOT Region 5 University Transportation Center. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors.

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33

List of Tables

Table 1. The subway lines in operation in Wuhan city from 2014 to 2020. Table 2. Vulnerability related metrics under the PITDP value from 50m to 2000m with an increasing step of 50m Table 3. The maximum vulnerability value of bus system with complementary subway system of each year during the expansion process of subway system under original and optimal opening sequences. Table 4.DAV of bus system with complementary subway system in planned and optimal subway opening sequences under the three types of disruptions

34

Table 1. The subway lines in operation in Wuhan city from 2014 to 2020. Year 2014 2015 2016 2017 2018 2019 2020

Subway lines Line 1, Line 2, Line 4 Line 1, Line 2, Line 4, Line 3 Line 1, Line 2, Line 4, Line 3, Line 6 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7, Line 8 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7, Line 8, Line 5 Line 1, Line 2, Line 4, Line 3, Line 6, Line 7, Line 8, Line 5, Line 9

35

Table 2. Vulnerability related metrics under the PITDP value from 50m to 2000m with an increasing step of 50m PITDP 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000

𝑘𝑐 41 75 116 162 225 281 344 405 482 570 680 787 909 1019 1165 1301 1458 1599 1762 1929 2113 2308 2490 2681 2881 3065 3280 3515 3752 3964 4190 4443 4684 4960 5236 5516 5797 6071 6321 6614

𝑁 𝑏←𝑠

𝑁 𝑠←𝑏

41 75 116 162 225 281 342 403 477 553 648 726 792 859 930 989 1045 1088 1131 1171 1217 1249 1275 1306 1334 1356 1380 1403 1425 1441 1457 1475 1482 1500 1515 1526 1532 1542 1547 1555

38 64 89 113 137 147 155 158 159 160 161 161 161 162 162 163 163 163 164 164 164 164 164 164 164 164 165 165 165 165 165 165 165 165 165 165 165 165 165 166

𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠 0.9995 0.9983 0.9960 0.9922 0.9850 0.9766 0.9653 0.9518 0.9324 0.9091 0.8751 0.8433 0.8134 0.7805 0.7427 0.7090 0.6751 0.6478 0.6194 0.5920 0.5593 0.5358 0.5163 0.4925 0.4705 0.4528 0.4333 0.4142 0.3957 0.3821 0.3683 0.3526 0.3464 0.3304 0.3170 0.3070 0.3015 0.2924 0.2878 0.2804

𝑑 𝑚𝑎𝑥𝑉𝑠←𝑏 0.9487 0.8528 0.7141 0.5379 0.3198 0.2164 0.1285 0.0943 0.0828 0.0712 0.0595 0.0595 0.0595 0.0478 0.0478 0.0359 0.0359 0.0359 0.0240 0.0240 0.0240 0.0240 0.0240 0.0240 0.0240 0.0240 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0000

36

𝑆𝐶𝑆 𝑏←𝑠

𝑆𝐶𝑆 𝑠←𝑏

𝑏←𝑠 𝐷𝐶𝑆𝑚𝑖𝑛

𝑠←𝑏 𝐷𝐶𝑆𝑚𝑖𝑛

0.0224 0.0409 0.0633 0.0884 0.1227 0.1533 0.1866 0.2199 0.2602 0.3017 0.3535 0.3961 0.4321 0.4686 0.5074 0.5396 0.5701 0.5936 0.6170 0.6388 0.6639 0.6814 0.6956 0.7125 0.7278 0.7398 0.7529 0.7654 0.7774 0.7861 0.7949 0.8047 0.8085 0.8183 0.8265 0.8325 0.8358 0.8412 0.8440 0.8483

0.2289 0.3855 0.5361 0.6807 0.8253 0.8855 0.9337 0.9518 0.9578 0.9639 0.9699 0.9699 0.9699 0.9759 0.9759 0.9819 0.9819 0.9819 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 0.9940 1.0000

0.0005 0.0017 0.0040 0.0078 0.0150 0.0234 0.0347 0.0482 0.0676 0.0909 0.1249 0.1567 0.1866 0.2195 0.2573 0.2910 0.3249 0.3522 0.3806 0.4080 0.4407 0.4642 0.4837 0.5075 0.5295 0.5472 0.5667 0.5858 0.6043 0.6179 0.6317 0.6474 0.6536 0.6696 0.6830 0.6930 0.6985 0.7076 0.7122 0.7196

0.0513 0.1472 0.2859 0.4621 0.6802 0.7836 0.8715 0.9057 0.9172 0.9288 0.9405 0.9405 0.9405 0.9522 0.9522 0.9641 0.9641 0.9641 0.9760 0.9760 0.9760 0.9760 0.9760 0.9760 0.9760 0.9760 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 0.9880 1.0000

Table 3. The maximum vulnerability value of bus system with complementary subway system of each year during the expansion process of subway system under original and optimal opening sequences. Year 2014 2015 2016 2017 2018 2019 2020

Planned opening sequence 𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠 0.8993 0.8572 0.8218 0.7871 0.7697 0.7270 0.7090

New subway line None Line 3 Line 6 Line 7 Line 8 Line 5 Line 9

37

Optimal opening sequence 𝑑 𝑚𝑎𝑥𝑉𝑏←𝑠 0.8993 0.8551 0.8153 0.7780 0.7460 0.7264 0.7090

New subway line None Line 6 Line 3 Line 5 Line 7 Line 9 Line 8

Table 4.DAV of bus system with complementary subway system in planned and optimal subway opening sequences under the three types of disruptions Year

DAV in planned opening sequence Random Degree Betweenness

2014 2015 2016 2017 2018 2019 2020

0.8328 0.8111 0.7920 0.7742 0.7602 0.7438 0.7297

0.8217 0.7984 0.7787 0.7603 0.7457 0.7296 0.7157

0.8294 0.8077 0.7890 0.7710 0.7566 0.7397 0.7250

DAV in optimal opening sequence Random Degree Betweenness

0.8328 0.8098 0.7888 0.7692 0.7513 0.7363 0.7232

38

None Line 6 Line 3 Line 5 Line 7 Line 9 Line 8

0.8217 0.7978 0.7779 0.7583 0.7402 0.7250 0.7118

None Line 3 Line 5 Line 6 Line 7 Line 9 Line 8

0.8294 0.8064 0.7846 0.7649 0.7467 0.7313 0.7179

None Line 6 Line 5 Line 3 Line 7 Line 9 Line 8

Vulnerability effects of passengers’ intermodal transfer distance preference and subway expansion on complementary urban public transportation systems Liu Hong a, b, Yongze Yan, Min Ouyang a, b*, Hui Tian a, Xiaozheng He c a

School of Automation, Huazhong University of Science and Technology, 1037 Luoyu

Road, Wuhan 430074, PR China b

Key Lab. For Image Processing and Intelligent Control, Huazhong University of

Science and Technology, Wuhan 430074, PR China c

NEXTRANS Center, Purdue University, W. Lafayette, IN 47906, USA

Research Highlights 

We model complementary urban public transportation systems’ (CUPTSs) vulnerability.



We use a PITDP metric to capture different levels of complementary relationship.



We study vulnerability under different PITDP and different subway expansion plans.



*

We analyze dynamic vulnerability of CUPTSs during their expansion process.

Corresponding author: [email protected] 39

List of Figures Figure 1. (a) Network-based description of bus system in Wuhan, which is shown by ArcGIS with stations geographical information. (b) Network-based description of subway system in Wuhan, in which solid lines represent the operational subway lines and the dash ones represent the planned subway lines. (c) Network-based description of the road system in Wuhan, which is shown by ArcGIS with intersections and road segments geographical information.

Figure 2. Accessibility-based bus system vulnerability as a function of road segments failure fraction 𝑃𝑟 in the case of with (𝑉𝑏←𝑠 ) and without (𝑉𝑏 ) the complementary subway system under random failures, degree-based and betweenness-based attacks.

Figure 3. Accessibility-based subway system vulnerability as a function of subway edges failure fraction 𝑃𝑠 in the case of with ( 𝑉𝑠←𝑏 ) and without ( 𝑉𝑠 ) the complementary

bus

system

under

random

failures,

degree-based

and

betweenness-based attacks.

Figure 4. The vulnerability of bus system with complementary subway system at the PITDP value from 50m to 2000m under random failures, degree-based and betweenness-based attacks.

Figure 5. The reduction of bus system vulnerability as a function of PITDP value

Figure 6. The vulnerability of subway system with complementary bus system at the PITDP value from 50m to 2000m under random failures, degree-based and betweenness-based attacks.

Figure 7. The reduction of subway system vulnerability as a function of PITDP value

Figure 8. The SCS between bus system and subway system in Wuhan city with the 40

PITDP value from 50m to 2000m.

Figure 9. The dynamic complementary strength from subway system to bus system at different PITDP values under random failures, degree-based and betweenness-based attacks.

Figure 10. The dynamic complementary strength from bus system to subway system at

different

PITDP

values

under

random

failures,

degree-based

and

betweenness-based attacks.

Figure 11. The value of 𝐷𝐶𝑆𝑚𝑎𝑥 at different PIDTP under three types of disruptions

Figure 12. The vulnerability of bus system with complementary subway system as a function of road segment failure fraction 𝑃𝑟 under random failure during the planned expansion process of subway system in Wuhan from 2014 to 2020.

(a) Bus network

(b) Subway network

41

(c) Road network

Figure 1. (a) Network-based description of bus system in Wuhan, which is shown by ArcGIS with stations geographical information. (b) Network-based description of subway system in Wuhan, in which solid lines represent the operational subway lines and the dash ones represent the planned subway lines. (c) Network-based description of the road system in Wuhan, which is shown by ArcGIS with intersections and road segments geographical information.

Figure 2. Accessibility-based bus system vulnerability as a function of road segments failure fraction 𝑃𝑟 in the case of with (𝑉𝑏←𝑠 ) and without (𝑉𝑏 ) the complementary subway system under random failures, degree-based and betweenness-based attacks.

42

Figure 3. Accessibility-based subway system vulnerability as a function of subway edges failure fraction 𝑃𝑠 in the case of with ( 𝑉𝑠←𝑏 ) and without ( 𝑉𝑠 ) the complementary

bus

system

under

random

betweenness-based attacks.

43

failures,

degree-based

and

Random Attacks

1 0.9 0.8

Bus Vulnerability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000 PITDP(m) 1500

2000

0

0.2

0.4

0.6

1

0.8

Pr

Degree-based Attacks

1 0.9 0.8

Bus Vulnerability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000 PITDP(m)

1500

2000

0.2

0

0.4 Pr

0.6

0.8

1

Betweenness-based Attacks

1 0.9 0.8

Bus Vulnerability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000 PITDP(m)

1500

2000

0

0.2

0.4

0.6

0.8

1

Pr

Figure 4. The vulnerability of bus system with complementary subway system at the PITDP value from 50m to 2000m under random failures, degree-based and betweenness-based attacks.

44

Figure 5. The reduction of bus system vulnerability as a function of PITDP value

45

Figure 6. The vulnerability of subway system with complementary bus system at the PITDP value from 50m to 2000m under random failures, degree-based and betweenness-based attacks.

Figure 7. The reduction of subway system vulnerability as a function of PITDP value

46

1

Static Complementary Strength SCS

0.8

0.6

SCSb-s SCSs-b

0.4

0.2

0

0

400

800

1200

1600

2000

PITDP(m)

Figure 8. The SCS between bus system and subway system in Wuhan city with the PITDP value from 50m to 2000m.

47

Random Attacks

Degree-based Attacks

Dynamic Complementary Strength

Dynamic Complementary Strength

DCS

1

DCS

1

0.8

0.6

0.4

0.2

0 1

0.8 0.6

0.4

1200 0.2 0

0

400

Pr

1600

2000

0.8

0.6

0.4

0.2

0 1

0.8 0.6

800

0.4

1200 0.2 0

Pr

PITDP(m)

0

400

1600

2000

800 PITDP(m)

Betweenness-based Attacks

Dynamic Complementary Strength

DCS

1

0.8

0.6

0.4

0.2

0 1

0.8 0.6

0.4

1200 0.2 0

Pr

0

400

1600

2000

800 PITDP(m)

Figure 9. The dynamic complementary strength from subway system to bus system at different PITDP values under random failures, degree-based and betweenness-based attacks.

48

Random Attacks

Degree-based Attacks

Dynamic Complementary Strength

Dynamic Complementary Strength

DCS

1

DCS

1

0.8

0.6

0.4

0.2

0 1

0.8 0.6

0.4

600 0.2 0

0

200

Ps

800

1000

0.8

0.6

0.4

0.2

0 1

0.8 0.6

400

0.4

600 0.2 0

Ps

PITDP(m)

0

200

800

1000

400 PITDP(m)

Betweenness-based Attacks

Dynamic Complementary Strength

DCS

1

0.8

0.6

0.4

0.2

0 1

0.8 0.6

0.4

600 0.2 0

0

200

Ps

800

1000

400 PITDP(m)

Figure 10. The dynamic complementary strength from bus system to subway system at

different

PITDP

values

under

random

betweenness-based attacks.

49

failures,

degree-based

and

Figure 11. The value of 𝐷𝐶𝑆𝑚𝑎𝑥 at different PIDTP under three types of disruptions

Figure 12. The vulnerability of bus system with complementary subway system as a function of road segment failure fraction 𝑃𝑟 under random failure during the planned expansion process of subway system in Wuhan from 2014 to 2020.

50