Abstract
Purpose
The purpose of this paper is to attempt to realize the optimization of cascading failure process of urban transit network based on Load‐Capacity model, for better evaluating and improving the operation of transit network.
Design/methodology/approach
Robustness is an essential index of stability performance for urban transit systems. In this paper, firstly, the static robustness of transit networks is analyzed based on the complex networks theory. Aiming at random and intentional attack, a concrete algorithm process is proposed on the basis of Dijstra algorithm. Then, the dynamic robustness of the networks, namely cascading failure, is analyzed, and the algorithm process is presented based on the Load‐Capacity model. Finally, the space‐of‐stations is adopted to build the network topology of Foshan transit network, and then the simulation analyses of static and dynamic robustness are realized.
Findings
Results show that transit network is robust to random attack when considering static robustness, but somewhat vulnerable to intentional attack. For dynamic robustness analysis, a large‐scale cascade of transit network may be triggered when the tolerance parameter α is less than a value, so that the robustness of transit network can be improved through some reasonable measures.
Practice implications
The results of this study provide useful information for urban transit network robustness optimization.
Originality/value
An effective method for analyzing the static and dynamic robustness of transit network is provided in this paper.
Keywords
Citation
Zou, Z., Xiao, Y. and Gao, J. (2013), "Robustness analysis of urban transit network based on complex networks theory", Kybernetes, Vol. 42 No. 3, pp. 383-399. https://doi.org/10.1108/03684921311323644
Publisher
:Emerald Group Publishing Limited
Copyright © 2013, Emerald Group Publishing Limited
1 Introduction
Since the complex networks theory was put forward in 1960s by Erdös and Rényi (1960), the complex scientific problems have exploded after the works of Watts and Strogatz (1998) as well as Barabási and Albert (1999) and Barabási et al. (1999), and a lot of real‐world networks have been examined. The complex networks theory has been widely used to describe systems in various areas, such as society, biology, transportation, communication, etc. (Albert and Barabasi, 2002; Dorogovtsev and Mendes, 2002; Newman, 2003; Pastor‐Satorras and Vespignani, 2004). Little attention was initially paid to transportation networks. With further development of complex networks theory, several public transportation systems have been investigated using various concepts of statistical physics for complex networks (Amaral et al., 2000; Crucitti et al., 2004; Guimera et al., 2005; Li and Cai, 2004; Strogatz, 2001) during the past few years.
Although massive fundamental studies on complex networks theories have been conducted by scholars, the research results about transportation network complex system are still quite few. The existing research results mainly concentrate on the analysis of airport networks, subway networks, street networks, public transit (PT) networks, and so on. In Amaral et al. (2000) work, the topology structure of airport network was presented. Latora and Marchiori (2001, 2002) and Marchiori and Latora (2000) studied in detail a network by simplifying the Boston subway as an undirected weighted graph. The results indicated that this network had the small‐world property. Jiang and Claramunt (2004) explored the city street networks by calculating the network connectivity, the average path length and the clustering coefficient. The results confirmed that the city street networks had the small‐world property rather than scale‐free property. In previous works (Amaral et al., 2000; Guimera et al., 2005), a survey on the world‐wide airport network was presented. Von Ferber et al. (2005) investigated the public transport networks of Berlin, DUS and Paris. The results showed that the node degree distribution of large PT networks obeyed the Zipf law and hence a PT network may constitute an example of a scale‐free network.
On the basis of previous studies, this paper takes urban PT networks as the object to analyze the characteristics. Urban PT system is the important transportation infrastructure on which urban social and economic activities rely. The developing experience of various countries shows that it is the only scientific approach to solve urban transportation problems by laying out public transport network rationally and developing efficient public transport system with priority. But the operation of the PT network is interfered by many uncertain factors, which makes the public transport network sometimes very vulnerable, thus, how to improve the security and stability of the PT network is the hot topic concerned in transportation area.
Urban public transport network, having complex network characteristics, is a typical complex network. It is important to improve the robustness of the transit network by investigating the robustness of transit network and presenting innovation and optimization approaches based on the analysis.
2 The complex network properties of urban transit network
Complex network is the abstraction of a considerable amount of complicated systems, which can describe a variety of interactions or relationship within the system. The basic models include regular network model topology, Erdös and Rényi (ER) random model (Erdös and Rényi, 1960), small‐world network model (Watts and Strogatz, 1998) and scale‐free network model (Barabási and Albert (BA) model) (Barabási and Albert, 1999). The PT network is composed of nodes representing individuals in real systems and the edges standing for the relationships among individuals. Both domestic and overseas scholars have confirmed the complexity of urban PT network (Houli et al., 2010), which has its unique features compared with other types of networks. The studies at home and abroad reveal that the topology of PT network lies between the regular network and the random network, and has the small‐world (Latora and Marchiori, 2002) or scale‐free (Von Ferber et al., 2005) properties. PT network has the following characteristic (Guimera et al., 2005; Kurant and Thiran, 2006; Latora and Marchiori, 2001; Marchiori and Latora, 2000):
- •
Network size varies and difference is significant. The number of different bus stops varies from several hundred to thousands.
- •
Several bus lines pass by the same stations.
- •
It has the feature of growth. With the enlargement of city size and the development of economy, new sites are added to PT network gradually. Certainly, some bus stops are deleted from the network because their functions are not adapted to the city development and the travel demands.
- •
It has preference dependence. Similar to scale‐free networks, there are several large stops in urban PT network, called the major stations or collection points. The large stops play important roles in urban PT network, and generally new sites are often easier to be associated with them.
3 Robustness analysis of urban transit network
3.1 Topology structure establishment of urban transit network
It is the prerequisite and base to analyze the complex feature of transit network by defining the topology. Currently there are three main topology‐mapping methods: “space‐of‐stations”, “space‐of‐changes” and “space‐of‐routes”:
- 1.
Space‐of‐stations (Kurant and Thiran, 2006). In this topology, the set of nodes is defined by the set of all stations. Two stations are connected only if they are physically and directly connected (without station in between). This reflects the geographical position of all stations and the order in transit routes, which is similar to transit map.
- 2.
Space‐of‐changes (Kurant and Thiran, 2006). Nodes in space‐of‐changes are the same as those in space‐of‐stations, but the edge between two nodes here means that there is a direct bus route which links them. Obviously, the node degree in this topology is the total number of nodes reachable by using a single route, and the distance can be interpreted as the transfer time (plus one) from one stop to another.
- 3.
Space‐of‐routes (Xu et al., 2007). Nodes in space‐of‐routes represent bus lines, and two lines are connected if they intersect at the same stations. Networks mapped by space‐of‐routes reflect the relationship among bus lines. This topology concerns the number of intersecting stations among bus lines.
Since there are various ways to represent the urban transit network in terms of a graph, and the representations allow for a comprehensive analysis of various transit network properties reflecting their operating functions (Von Ferber et al., 2009), it is natural to choose the terms of representations based on the purpose of present analysis. The paper selects space‐of‐stations to define the transit network topology structure, by analyzing the network robustness to describe bus service performance, and the topology in space‐of‐stations reflects the geographical properties of the system, especially illustrates the spatial relationship of all bus stops. The topology is described as followed: set bus stops as nodes, and constitute the node‐set V(G). If there exists one or multiple routes between two nodes, a link is available, otherwise the two nodes are not connected, E(G) is named as the edge‐set. All the nodes and edges form an undirected connected graph G(V,E). In order to reveal the importance of bus stations in urban transit network, this study makes an improvement based on the previous research (Berche et al., 2009). Define the edge weight as the number of transit routes, and the weight of node is the sum of the weights of all the edges linking with this node. In other words, the edge weight is the total number of nonstop lines among stations, reflecting the transfer convenience among network nodes. As the weighted network defined, we can get the number of transit routes passing through a station clearly, and the higher weight of the node is, the more significant the station is.
3.2 Robustness analysis of urban transit network
3.2.1 Connotation of transit network robustness
The robustness refers to the persistence of a system's characteristic behavior under perturbations or conditions of uncertainty (Albert et al., 2000). Urban transit network robustness is referred to the ability of the network to maintain its function when the nodes or edges in the network suffer from random or intentional attack. There are two different ways when analyzing network robustness: static robustness and dynamic robustness, respectively.
For the static robustness of network (Albert and Barabasi, 2002), the failure of a node (edges) is assumed not to cause other nodes' (edges') failure without considering the dynamic association of nodes (edges) failure, under this assumption, the failure of a few nodes would not trigger collapse of the whole network. Generally, two approaches, random and selective attach, are adopted to interfere with the transit network so as to analyze its robustness. For random attack, each node in the network is endowed with a random probability, nodes are deleted stochastically, and the ratio of the giant connected component (GCC) and average path length are calculated till all nodes in the network are not connected. For selective attack, the network nodes are attacked selectively, starting from the nodes with the largest connected degree, and going on according to the descending order of nodes degree.
The dynamic robustness of network represents the resistance ability when the nodes (edges) analyzed are under attack under the condition of the network capacity limit. Actually, the load of the transit network is vehicle flow, which is dynamic, especially when the network structure is changed, e.g. nodes removal or addition, and the load redistribution. Because actual load‐capacity of network nodes is limited, a node failure will lead to load redistribution of the whole network, thus making the load of some nodes exceed its load‐capacity, and causing the failure of the nodes, and then these nodes could lead to failure of other nodes, which will produce the “cascading failure” (Motter and Lai, 2002). Therefore, the dynamic robustness analysis of transit network conforms to actual situations to a greater extent.
3.2.2 Indicators of transit network robustness analysis
According to the connotation of transit network robustness, by integrating complex network theory, network average efficiency and relative size of the GCC are chosen as the analysis index:
- •
Network average efficiency E. The connectivity of network will change when network nodes suffer from attack. The strength of connectivity can be reflected by E (Albert and Barabasi, 2002). The connected efficiency εij between a node pair (i,j) is assumed to be the inverse of the path length dij between the two nodes, and E is the mean of εij values of all node pairs (Fang et al., 2007): Equation 1 where, n is number of nodes in the network. The larger E is, the better connectivity of the network is.
- •
The relative size of the GCC of the network S. The stability performance can be represented by the relative size of GCC S: Equation 2 where, N and N′ are, respectively, the number of nodes in the GCC of the network before and after the attack. The value of S is between [0, 1]. When S=1, the network is fully connected; when 0<S<1, the network is still relatively integrated; when S=0, the network collapses.
3.2.3 Attack mode to transit network
The “attack” to urban transit network refers to the incidents or behavior that cause the failure of nodes or edges in transit network, and then trigger the obvious decrease of network efficiency. The attack can be divided into random and intentional attack. In practice, the transit network after attacked is performed as the cancellation of the whole bus line, and shortening of the operation distance passing through several stops of certain routes, but this condition is rather complicated. Here, some simplification is done: it is supposed that all edges connecting with the suffered node are deleted, namely single point failure.
3.3 The analysis of transit network robustness
3.3.1 The static robustness analysis of transit network
A simplified assumption should be made before the algorithm design. Set node p∈V, and there are n pairs of ODs passing through p with the shortest path marked as (i1,j1), (i2,j2), … ,(in,jn), respectively. Therefore, for the network G′(n′,e′) without node p, only (i1,j1), (i2,j2), … ,(in,jn) in the shortest paths between all node pairs changes, and the rest remains.
In this study, it is assumed that the connection relationship of network G(V,E) is designated by adjacent table w, and the length of the shortest paths between node pairs of the network is represented by distance matrix D. The steps of numerical simulation analysis of static robustness for urban transit network are presented in Table I.
According to the steps above, the simulation is performed by using the software Matlab to program to find out the characteristics of static robustness for the urban transit network.
3.3.2 The dynamic robustness analysis of transit network
Define the load of node i as li, whose physical meaning is equal to the number of shortest paths from all vertices to all others that pass through that node i (betweenness centrality). Based on the load‐capacity model (Motter and Lai, 2002) of node cascade, the capacity of a node Ci (the maximum load node can withstand) is supposed to be proportional to its initial load li: Equation 3 where, α is the tolerance parameter, and α≥0. Restricted by the factors of capital, technology, space and so on, α is not infinite.
In accordance with the definition and simplified assumptions of urban transit network dynamic robustness (cascading failure) mentioned as above, the steps of numerical simulation analysis of urban transit network dynamic robustness are presented in Table II.
According to the steps above, the simulation is performed to study the characteristics of dynamic robustness for the urban transit network.
4 Example analysis
This paper takes the transit network in Foshan, Guangdong province as the research object. The number of routes and stops of transit network in Foshan are 118 and 2,890, respectively. The space‐of‐stations is adopted to build the transit network topology of the city to analyze its robustness. In order to obtain a better performance, some assumptions are made to the practical transit network in Foshan:
- •
Transit network contains two basic elements, i.e. stations and lines, which makes the abstraction of the network to be undirected network.
- •
Stations with the same name are regarded as one node, while stations with different name are considered as different nodes, and the particular same station names but at different stopping sites are neglected.
By taking a part of the network (central of the city) as the study object, 436 nodes and 492 edges are included, and the simulation of the robustness for transit network is conducted.
4.1 Static robustness example analysis
4.1.1 Random attack experiment
Based on the transit network topology graph in Foshan, we remove nodes randomly and the corresponding edges connected with it in the network. The results of E and the relative size of maximal connected sub‐graphs S are calculated for different removing node rates f, and shown in Figures 4 and 5.
Figure 4 shows that the average efficiency E of transit network in Foshan has an obvious declining trend against random attack. The curve of f‐E drops quickly when f<0.3, changes slightly when 0.3<f<0.9, and falls to 0 when f reaches 0.9326. Figure 5 shows that S declines gradually, and the curve of f‐S has two break points when f is 0.4433 and 0.6170, respectively. The entire network nearly collapses when f arrives at 0.8617.
4.1.2 Intentional attack experiment
Remove the node and edges connected with it according to the descending order of node degree. The results of E and t S are calculated for different f values, and shown in Figures 6 and 7.
Figure 6 shows that E of Foshan transit network drops quickly when f<0.12, changes a little when 0.13<f<0.41, drops rapidly when f>0.41, and falls to 0 when f=0.49. Compared with random attack, the dropping speed of E is faster, and does not appear significant fluctuation. Figure 7 shows that, S drops quickly from 0.9965 to 0.1206 when f<0.1702. The whole network displays the vulnerability to intentional attack.
As shown in Figures 4‐7, Foshan transit network is robust to random attack, but somewhat vulnerable to intentional attack. This also manifests that urban transit network lies between regular network and random network, and has the small‐world or scale‐free characteristics.
4.2 Dynamic robustness example analysis
According to the node load of Foshan transit network, the cascading failure process is triggered by removing the node with highest load. Each time set α a value, record the value of E and S, respectively, when the system achieves stability. Figures 8 and 9 show the relationship between tolerance value α and E, and that between tolerance value α and S, respectively.
As shown in Figure 8, the value of E has a mutation when α is 0.4, and when α<0.4, E remains at the low value 5.63×10−5. This result indicates that the average efficiency of network can be increased by improving the capacity of stations. It is also observed that E has small a fluctuation when α=0.42, and reaches maximum at 1.121×10−4. Figure 9 shows that S increases with the increase of α, and also α=0.42, 0.70 are the oscillation points of the curve. S achieves maximum at 0.3007 when α=1. Therefore, the network robustness will have a distinct rising with the increase of α. In addition, the large‐scale cascading failure will be triggered only when α<0.4.
It should be noted the oscillations in Figure 9, two graphs of the network layout are generated below when α=0.40 and 0.42 to better illustrate the reasons.
When α=0.40, the number of break nodes is 210, while the number is 226 when α=0.42. Figures 10‐12 show the comparisons of topology graphs before and after the attack. The results can explain the oscillations in Figure 9.
Compared the analysis results of dynamic robustness with static robustness, in the analysis of dynamic robustness, when α=0, namely when the node capacity in the network is the initial capacity, S is 0.106 when the network achieves balanced after the node with the largest load is removed, and the whole network is close to collapse. While in the analysis of static robustness, S is 0.9839 when the node with the largest load is deleted, and the network still maintains good connectivity. Obviously, the network robustness will reduce significantly if the node capacity limit and flow redistribution is considered. The Table III lists part of nodes' variation condition in the network after the most loaded node (Pingzheng Bridge) is removed when α=0 (Figure 13).
Thus, when α=0 and the most loaded node is removed, all the nodes with large load exceed their capacity limit after flow redistribution, and also trigger these nodes failure. The network achieves stability when nodes no longer fail, but the whole network almost collapses because the value of S is only 0.106 at this moment.
5 Efficient response to the cascading failure of transit network
The analysis above indicates that urban transit network shows the vulnerable property when the spreading dynamics of disastrous events is considered. Since transit system is the basic guarantee of the whole city's normal operation, the ability to recover the functionality of damaged components promptly is crucial for determines. Many researchers have proposed efficient strategies for eliminating cascading effects in networks (Motter, 2004; Schäfer et al., 2006). Based on the results above, it is reasonable to improve the robustness of transit network by increasing the capacity of nodes. Some approaches can effectively prevent transit congestion, such as improving the design capacity of bus stops, enhancing hub stations' service radius by adding several stops around them, etc. On the other hand, it is helpful to study the properties and structure of transit network with the establishment of emergency plan.
6 Conclusion
This paper analyzes transit network robustness with the application of complex network theory. Based on the existing robustness studies, the optimization of cascading failure process is realized, the numerical simulation analysis is made to become better agreement with the actual situation, and the scientificity of the analysis is strengthened. As seen from the analysis results, the network shows the scale‐free properties of “robustness and vulnerability”. In comparison of static with dynamic robustness analysis, a relatively accurate judgment is determined about what nodes are important and widely influenced. Therefore, the comparison can provide the reference for urban transit network optimization.
Currently, in most cities of China, the construction of road infrastructures is oriented by the principle of traffic capacity priority. Awkwardly however, the wider road turns out to carry more congested traffic. One reason is the unreasonable layout of urban street networks. The linkage of arterial network and branches are unsuitable, which causes the traffic congestion problem more serious. Consequently, some measures should be taken to ensure the smooth of branches and make full use of their distributing function to connect the urban street networks closely. This can guarantee the connectivity of the entire network, thus enhancing its robustness.
About the authors
Zhiyun Zou is a Professor within the School of Civil Engineering and Mechanics in the Huazhong University of Science and Technology. His researcher activity is in the area of transportation planning and management, and traffic network analysis. His scientific activity has materialized in over 30 articles and scientific papers.
Yao Xiao is a postgraduate within the School of Civil Engineering and Mechanics in the Huazhong University of Science and Technology, whose tutor is Professor Zou. She has participated in several transportation planning programs and has done research under the guidance of Professor Zou.
Jianzhi Gao is a Lecturer within the School of Civil Engineering and Mechanics in the Huazhong University of Science and Technology. He is engaged in transportation planning and management (especially transit planning), and traffic logistics. He has published research papers in several journals. Jianzhi Gao is the corresponding author and can be contacted at: [email protected]
References
Albert, R. and Barabasi, A.L. (2002), “Statistical mechanics of complex networks”, Reviews of Modern Physics, Vol. 74, pp. 47‐97.
Albert, R., Jeong, H. and Barabasi, A.L. (2000), “Error and attack tolerance of complex networks”, Nature, Vol. 406, pp. 378‐382.
Amaral, L.A.N., Scala, A., Barthelemy, M. and Stanley, H.E. (2000), “Classes of small‐world networks”, Proceedings of the National Academy of Sciences of the United States of America, Vol. 97, pp. 11149‐11152.
Barabási, A.‐L. and Albert, R. (1999), “Emergence of scaling in random networks”, Science, Vol. 286, pp. 509‐512.
Barabási, A.‐L., Albert, R. and Jeong, H. (1999), “Mean‐field theory for scale‐free random networks”, Physica A: Statistical Mechanics and Its Applications, Vol. 272, pp. 173‐187.
Berche, B., Von Ferber, C., Holovatch, T. and Holovatch, Y. (2009), “Resilience of public transport networks against attacks”, The European Physical Journal B, Vol. 71, pp. 125‐137.
Crucitti, P., Latora, V. and Marchiori, M. (2004), “A topological analysis of the Italian electric power grid”, Physica A: Statistical Mechanics and Its Applications, Vol. 338, pp. 92‐97.
Dorogovtsev, S.N. and Mendes, J.F.F. (2002), “Evolution of networks”, Advances in Physics, Vol. 51, pp. 1079‐1187.
Erdös, P. and Rényi, A. (1960), “On the evolution of random graphs”, The Mathematical Institute of the Hungarian Academy of Science, Vol. 5, pp. 17‐61.
Fang, J., Wang, X. and Zhigang, Z. (2007), “New interdisiplinary science: network science(I)”, Progress in Physics, Vol. 27, pp. 239‐334.
Guimera, R., Mossa, S., Turtschi, A. and Amaral, L.A.N. (2005), “The worldwide air transportation network: anomalous centrality, community structure, and cities' global roles”, Proceedings of the National Academy of Sciences of the United States of America, Vol. 102, pp. 7794‐7799.
Houli, D., Zhiheng, L. and Yi, Z. (2010), “Robustness analysis model of urban transit network”, Journal of South China University of Technology, Vol. 38, pp. 70‐75, 81.
Jiang, B. and Claramunt, C. (2004), “Topological analysis of urban street networks”, Environment and Planning B: Planning and Design, Vol. 31, pp. 151‐162.
Kurant, M. and Thiran, P. (2006), “Extraction and analysis of traffic and topologies of transportation networks”, Physical Review E, Vol. 74, p. 36114.
Latora, V. and Marchiori, M. (2001), “Efficient behavior of small‐world networks”, Physical Review Letters, Vol. 87, pp. 198701‐198704.
Latora, V. and Marchiori, M. (2002), “Is the Boston subway a small‐world network?”, Physica A: Statistical Mechanics and Its Applications, Vol. 314, pp. 109‐113.
Li, W. and Cai, X. (2004), “Statistical analysis of airport network of China”, Physical Review E, Vol. 69.
Marchiori, M. and Latora, V. (2000), “Harmony in the small‐world”, Physica A, Vol. 285, pp. 539‐546.
Motter, A.E. (2004), “Cascade control and defense in complex networks”, Physical Review Letters, Vol. 93, p. 098701.
Motter, A.E. and Lai, Y.‐C. (2002), “Cascade‐based attacks on complex networks”, Physical Review E, Vol. 66, p. 065102.
Newman, M.E.J. (2003), “The structure and function of complex networks”, Siam Review, Vol. 45, pp. 167‐256.
Pastor‐Satorras, R. and Vespignani, A. (2004), Evolution and Structure of the Internet: A Statistical Physics Approach, Cambridge University Press, Cambridge.
Schäfer, M., Scholz, J. and Greiner, M. (2006), “Proactive robustness control of heterogeneously loaded networks”, Physical Review Letters, Vol. 96, p. 108701.
Strogatz, S.H. (2001), “Exploring complex networks”, Nature, Vol. 410, pp. 268‐276.
Von Ferber, C., Holovatch, Y. and Palchykov, V. (2005), “Scaling in public transport networks”, Condensed Matter Physics, Vol. 8, pp. 225‐234.
Von Ferber, C., Holovatch, T., Holovatch, Y. and Palchykov, V. (2009), “Public transport networks: empirical analysis and modeling”, The European Physical Journal B, Vol. 68, pp. 261‐275.
Watts, D.J. and Strogatz, S.H. (1998), “Collective dynamics of ‘small‐world’ networks”, Nature, Vol. 393, pp. 440‐442.
Xu, X.P., Hu, J.H., Liu, F. and Liu, L.S. (2007), “Scaling and correlations in three bus‐transport networks of China”, Physica A: Statistical Mechanics and Its Applications, Vol. 374, pp. 441‐448.