Complexity of Public Transport Networks

doi:10.1016/S1007-0214(07)70029-9

Tsinghua Science & Technology

Volume 12, Issue 2, April 2007, Pages 204-213

https://doi.org/10.1016/S1007-0214(07)70029-9 Get rights and content

Abstract

The theory of complex networks was used to classify public transport networks into public transportation route networks, public transportation transfer networks, and bus station networks. The practical significance of the network parameters was then analyzed. The public transport networks in Langfang, Jining, and Dalian were then chosen as specific research cases. The results show that the public transportation networks have the characteristics of complex networks. In addition, the urban transportation network parameters all significantly affect the accessibility, convenience, and terrorist security capability of the urban public transportation network. The results link the findings with the actual situations to explore means to solve transportation system problems.

Introduction

A network is composed of many vertexes and edges. Many complex systems in the real world can be described as networks with the individuals in the real system represented by vertexes and the mutual relationships among individuals represented by edges. For example, neural systems, electrical power grids, the World Wide Web, and transportation systems can all be represented as networks.

Watts and Strogatz^[1] published the first article on small world networks in Nature in 1998. Since then, the importance of small world networks has been recognized by many experts and scholars from many fields. Research on complex networks has been a hot topic in the international academic community since then.

Many studies have focused on the complexity of networks. In 1998, Watts and Strogatz^[1] introduced a small world network model (the so-called WS model) and analyzed its characteristics. The model exhibits a high clustering coefficient and a short average distance. In 1999, Newman and Watts developed the NW network model^[2] which is slightly different from the WS model. Their study shows that an NW network and a WS network exhibit similar characteristics when the number of vertexes is large enough and the connecting probability is small enough. In 1999, Barabasi and Bonabeau^[3] proposed a BA model, which is a scale-free network model. They took the two important properties of real networks, growth and preferential attachment, into consideration. The BA model is a major breakthrough in the study of complex networks. Later on, many other models were developed based on the BA model^[4], such as the local-world evolving network model, the weighted evolving network model and the deterministic scale-free network model. These models have enabled researchers to pay more attention to various physical processes and network dynamics, such as spread and synchronization, and how the network structure impacts the dynamic behavior, with many interesting but perhaps not rigorous interpretations^[5].

However, there are few papers directly related to transportation networks. Latora and Marchiori6, 7 introduced the definition of efficiency coefficient and applied it to a case study of the Boston subway. Sen et al.^[8] found that India's railway network exhibited small world properties and predicted that other national railway networks would also exhibit small world properties. Similar properties were reported by Seaten and Hackett^[9] in a study of railway networks in Boston and Vienna. Jiang and Claramunt^[10] concluded that the topological networks of streets in big cities exhibited small world properties but were not scale-free networks through analysis of the topological networks in three cities. Li and Cai^[11] showed that the topological structure of the air traffic network of China (ANC) had two key characteristics of small world networks, a short average path length and a high degree of clustering, with the cumulative degree distributions of both the directed and undirected ANC obeying two-regime power laws with different exponents. Guimera et al.^[12] showed that the worldwide air traffic network was a scale-free and small world network. Unlike other real networks, the world's air traffic network is influenced by geography and politics. Wu et al.^[13] used statistical analysis to show that the urban transit system in Beijing is a scale-free network through. Sienkiewicz and Holyst^[14] analyzed 22 public transportation networks in Poland. Ferber et al.^[15] found that large public transportation networks are more likely to have a scale-free structure. Holme^[16] investigated the relationship between centrality measures and the traffic density for simple particle hopping models on networks with emerging scale-free degree distributions and studied how the speed of the dynamics was affected by the underlying network structure. Wu et al.^[17] simulated the spread of traffic jams with the susceptible infective recovered (SIR) model based on the assumption that the rates of infection and recovery were unchanged.

These studies show that some transportation networks (city street networks, air traffic networks, railway networks, and public transportation networks^[18]) exhibit some characteristics of complex networks. Therefore, research on complex network will provide a new way to analyze traffic problems.

Although researchers have analyzed the complexity of transportation networks using different examples, general common characteristics have not yet been identified and the assumptions have not been confirmed. In addition, conclusions cannot be compared because of different statistical methodologies. Research on the complexity of transportation networks is still exploring theory, so the analyses have not yet been able to solve practical problems.

Section snippets

Network parameters

Notations used throughout this paper are listed in Table 1.

The degree of a vertex defines the number of connections between this vertex and other vertexes. The average degree of the network is represented by k. The degree distribution of the vertexes in a network is characterized by a probability density function, $p (x = k)$ , or the cumulative probability distribution function, $p (x \geq k)$ . $p (x = k)$ means the probability that a vertex has degree k, which is equivalent to the ratio of the

Properties of public transportation networks

Public transportation networks consist of two basic elements, bus routes and bus stations. A route is formed by a number of bus stations. Under normal circumstances, passengers can travel from bus station A to bus station B along a certain route and also travel from bus station B to bus station A along the same route. Therefore, public transportation networks are usually treated as undirected graphs.

Public transportation networks generally exhibit growth and preferential attachment properties.

Conclusions

The analysis in this paper shows that public transportation networks exhibit some of the characteristics of complex networks. Public transportation transfer networks obviously exhibit the small world effect, with degree distributions following power law or exponential distributions. Developed bus station networks show small world effect, with degree distributions between exponential and power law distributions, and the shortest path length distribution following a Gamma distribution.

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View full text

Complexity of Public Transport Networks

Abstract

Introduction

Section snippets

Network parameters

Properties of public transportation networks

Conclusions

Phys. Letter. A

Physica A

Physica A

Collective dynamics of “small-world” networks

Nature

Emergence of scaling in random networks

Science

Summary of researches on theory and application of complex network

System Engineering

The construction and analysis of complex network model [Dissertation]

Efficient behavior of small-world Networks

Physical Review letters

Small-world properties of the Indian railway network

Topological analysis of urban street networks

Environment and Planning B