Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 2045838, 10 pages https://doi.org/10.1155/2020/2045838 Research Article Space-Time Evolution of Subway Vulnerability Based on Hypergraph Space Kaihua Lu 1,2 and Xiaoyan Gu 3 1Nanjing University of Aeronautics and Astronautics, Nanjing 210000, China 2Nanjing Public Engineering Construction Center, Nanjing 210000, China 3Jinling Institute of Technology, Nanjing 210000, China Correspondence should be addressed to Xiaoyan Gu; [email protected] Received 1 June 2020; Revised 18 September 2020; Accepted 27 September 2020; Published 14 October 2020 Academic Editor: Eric Florentin Copyright © 2020 Kaihua Lu and Xiaoyan Gu. +is 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. How to qualitatively and quantitatively evaluate the ability of the whole subway system to maintain the maximum normal operation has become a noteworthy topic. In this study, the hypergraph space structure of Nanjing subway network is used for research. +is study uses the method of time-space evolution mechanism to dynamically and quantitatively analyze the gradual change characteristics of Nanjing subway system vulnerability under the influence of time and space. In addition, this study also makes static and dynamic comparative analysis on the vulnerability of a subway system by using connectivity rate and trip efficiency index, which provides new analysis methods and ideas for the study of a subway system network. +e results show that although the overall structure of Nanjing’s subway network shows a loose trend, as Nanjing’s subway is still in a rapid development stage, it can be seen from the long-term planning of Nanjing’s subway that the construction of Nanjing’s subway ring line will be strengthened after the completion of the connection between the main city and the suburbs. +erefore, the vulnerability of the subway system should be reduced and the reliability will be improved. 1. Introduction complexity of network dynamics. After analysis, they ob- tained the influence of network size on dynamics research Subway, as an indispensable mode of transportation in [3]. Alexander et al. deduced the complexity estimates of two urban transportation, is favored by people because of its types of deterministic networks. +is breakthrough made it safety, comfort, large carrying capacity, energy saving, en- possible to create networks with the best complexity [4]. An vironmental protection, convenience, and rapidity. How- et al. optimized the layout of traffic detectors using complex ever, with the busy operation of the urban subway, subway networks, which extended the practical application of failures occur frequently, which are often directly manifested complex networks in the field of transportation [5]. In in a station or a section of the subway network that cannot addition, Karunakaran and Selvaganesh introduced the work normally. According to statistics, the number of concept of neighborhood matrix. After that, the “neigh- subway failures in Nanjing is about 1,000 a year [1]. borhood matrix” has become a new model that uses vertex +erefore, how to qualitatively and quantitatively evaluate neighborhood sets to represent graphs [6]. Beranek and the ability of the whole subway system to maintain the Remes analyzed the intensity distribution and behavior maximum normal operation has become a noteworthy topic. characteristics of selected nodes and selected participants in In general network research, C. Pozrikidis presents a the tripartite network, which further realized the systematic method to calculate spanning tree and uses binomial dis- analysis of more complex tripartite networks [7]. Mensah tribution to describe the node degree distribution of com- et al. got rid of the limitations of small-scale networks and plete network spanning tree [2]. Gorjão et al. analyzed the proposed an efficient and accurate shortest path algorithm 2 Mathematical Problems in Engineering for large-scale networks [8]. Dzamiˇ c´ et al. introduced a new short turn model. And through this model, they strived to community structure detection standard for complex net- alleviate the current situation of overcrowding in dense works. +is new detection standard provides a new path for traffic demand effective methods [26]. Nian et al. proposed a community detection [9]. Latora and Marchiori proposed a quantitative evaluation method. +is method can be used to complicated subway network model based on transportation evaluate the performance of urban subway networks under efficiency [10]. Accordingly, Mrvar et al. studied the be- various interference conditions [27]. Crucitti et al. made use tweenness of the complex subway network [11]. +ese an- of the network efficiency concept to study scale-free net- alyses of complex networks are often limited to general works, and the conclusion which global efficiency can better networks and cannot reflect the quantitative and other reflect the response of complex networks to external factors characteristics of subway networks. than characteristic path lengths can be drawn [28]. Based on With the development of social economy, various space traffic flow redistribution, Su et al. proposed a cascading structures have become increasingly complicated, so the failure model and applied it to the transportation network analysis of indicators such as vulnerability of spatial models studies [29]. has also been cited in many aspects. Moreover, the vul- With the acceleration of China’s urbanization process, nerability of complex networks was applied to the logistics the pressure brought by the increase of urban population on service supply chain by Ma et al. [12]. Abdelaty et al. urban traffic has become increasingly obvious. However, the established a static robustness index by studying the ro- speed of urbanization must not be restricted by traffic bustness of the public transport network [13]. Polishchuk pressure. +erefore, underground subway traffic, which et al. introduced the concepts of parameter and betweenness. corresponds to the traditional aboveground traffic, has be- +ese two concepts can be used to determine the negative come a new channel to relieve the traffic pressure in cities. In impact of complex network structure on internal and ex- the future, 43 cities in China will build subways. It can be ternal [14]. Furthermore, Sun et al. considered the impact of said that China is in a period of rapid development of passenger flow on the vulnerability of Shanghai’s subway subway. Moreover, with the continuous increase of new network [15]. Derrible and Kennedy established vulnera- subway lines, the overall transportation network of the bility evaluation indicators such as network redundancy and Nanjing subway system will be gradually improved. applied them to complex networks [16]. However, these analyses are only limited to the static analysis of network 2. Hypergraph Space of Nanjing Subway space and do not take into account the changes of dependent variables caused by the changes of time variables and space Claude Berge, a French mathematician known as the father variables. of modern graph theory, expanded the edges of graphs to A dynamic model of transportation network evolution superedges. +e expansion of this small step has triggered was proposed by Malinetskii and Stepantsov. At the same the development of a large field. time, they also concluded that the dynamic behavior of the +e edges of hypergraphs are defined as superedges, transportation network is the result of system self-organi- which can be connected with any number of vertices. zation [17]. Furthermore, Kim et al. proposed a fast time Suppose V � fv1; v2; v3; ... ; vng is a finite set and simulation method of air traffic network based on space- E � fe1; e2; e3; ... ; emg is a set family of a series of subsets of v. time parallelism. Using this method, we can obtain greater If ei ≠ ∅ (i � 1, 2, ..., m) and ∪ ei � V(i � 1; 2; ... ; m), acceleration in the traffic network [18]. It is worth noting then set family H � (V: e1; e2; e3; ... ; em) is considered to that Hosseini and Talebpour used space-time diagrams to be a hypergraph of finite set V. Among them, ∪ ei is the obtain better prediction results [19]. On this basis, Su et al. union of e1; e2; e3; ... ; em. applied time-space correlation analysis to road traffic con- Elements v1; v2; v3; ... ; vn are called vertices of hyper- gestion [20]. Backaliˇ c´ et al. found the coordinates of the graphs, while sets e1; e2; e3; ... ; em are called superedges of traffic accident by the space-time coordinate system. More hypergraphs. Each superedge is a subset of V, and E � {e1, e2, specifically, they proposed a new method to study the time e3,..., em} is the set of hyperedges and superedges. interval of traffic accidents [21]. Additionally, Qu and He used the spatiotemporal net- 2.1. Matrix Representation of Hypergraphs work model to predict and control station traffic congestion 1; vi ∈ ej; caused by emergencies. At the same time, they also verified H(V; E) � ( (1) the feasibility of the model [22]. Kang et al. conducted many 0; 0; vi ∉ ej: long-term studies on the mechanism of how the temporal and spatial characteristics of traffic accidents for the elderly If i � 5, j � 4; the hypergraph matrix is in Seoul change over time [23]. Based on the research of the predecessors, some new methods were born as follows. e1 e2 e3 e4 Specifically, Zawack and +ompson developed a spatio- v1 1000 v temporal network to represent the traffic flow of the road 2 1111. v 0011 (2) traffic system over time [24]. On the basis of considering 3 v 0100 only points or lines in the past, Sun and Guan proposed to 4 v 0000 measure the vulnerability of the subway network from the 5 perspective of line operation [25]. Ding et al. established a +e corresponding hypergraph is Mathematical Problems in Engineering 3 subway is compatible with the city’s economic development.