Spatial analysis and modeling of urban transportation networks
Jingyi Lin
Doctoral Thesis in Geodesy and Geoinformatics with Specialization in Geoinformatics Royal Institute of Technology Stockholm, Sweden, 2017
TRITA SoM 2017-15 ISRN KTH/SoM/2017-15/SE ISBN 978-91-7729-613-3
Doctoral Thesis Royal Institute of Technology (KTH) Department of Urban Planning and Environment Division of Geodesy and Geoinformatics SE-100 44 Stockholm Sweden
© Jingyi Lin, 2017
Printed by US AB, Sweden 2017
Abstract
Transport systems in general, and urban transportation systems in particular, are the backbone of a country or a city, therefore play an intrinsic role in the socio- economic development. There have been numerous studies on real transportation systems from multiple fileds, including geography, urban planning, and engineering. Geographers have extensively investigated transportation systems, and transport geography has developed as an important branch of geography with various studies on system structure, efficiency optimization, and flow distribution. However, the emergence of complex network theory provided a brand-new perspective for geographers and other researchers; therefore, it invoked more widespread interest in exploring transportation systems that present a typical node-link network structure. This trend inspires the author and, to a large extent, constitutes the motivation of this thesis.
The overall objective of this thesis is to study and simulate the structure and dynamics of urban transportation systems, including aviation systems and ground transportation systems. More specificically, topological features, geometric properties, and dynamic evolution processes are explored and discussed in this thesis. To illustrate different construction mechanisms, as well as distinct evolving backgrounds of aviation systems and ground systems, China’s aviation network, U.S. airline network, Stockholm’s street network, Toronto’s street network, and Nanjing’s street network are respectively studied and compared.
Considering the existence of numerous studies, a clear and comprehensive literature review in this field is presented as the first step. Most studies on transport systems from the complex network perspective published within the last decade are reviewed and summarized. It is found that a majority of the studies focused on topological features of transportation systems, however geometric properties have not earned sufficient attention. On the other hand, since there is a long history of transportation systems and limited availability of related data, it is difficult for researchers to develop empirical evolution analyses on real transportation systems; instead, computer simulations based on mathematical reasoning are considered as the most widely used way to depict evolution features of transport systems. These findings lay the foundation of the author’s thesis research.
In this thesis, most network features, including both local and global measures, such as centrality, clustering, and efficiency, are respectively discussed in terms of aviation systems and urban street networks. Although transportation systems are affected by various factors, they do present a small-world structure and a
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double Pareto distribution regardless of sizes and socio-economic states. A hierarchical structure is also detected for aviation systems of China and the US. To study the geometric properties, the famous gravity model is introduced to detect the distance effect in aviation systems. It is found that a logarithmic hyperbolic model can better depict the distance decay effect than a power law model. In addition, the distribution patterns of the spatial distance of aviation systems and ground systems are explained. Spatial patterns of street networks in different urban development schemes are discussed and compared as well. Self- organized cities show more heterogeneity in both measures. In comparison, Toronto presents a single peak distribution for the cell area. The distribution of closeness centrality, which is regarded as a measure of accessibility, clearly depicts that the strictly planned city is better accessed from the overall perspective.
A static perspective is far from enough to understand and model transportation systems. The evolution process is an essential supplement to depict underlying features of complex systems. To further study network growth modeling, a model for the evolution of aviation systems is proposed. Both topological and geometric features are examined in the evolution process. The model suggests that the bigger the weight coefficient is, the slower the network efficiency decreases. The empirical analysis about the U.S. airline network during 1990– 2010 is also developed to validate this model.
The main contributions of this thesis can be summarized as follows. First, it provides a comprehensive review of complex network topology of transportation systems for researchers from different backgrounds. It not only lists the research results from various subjects, but it also bridges and compares the results from different viewpoints and angles, which is necessary to explore this interdisciplinary topic. Moreover, a spatial perspective constitutes the main focus of this thesis, trying to bridge the traditional geographic analysis and complex network theory. Lastly, empirical analyses are developed to explore the evolving process of transportation systems, providing a valuable baseline for theoretical simulations in this field. However, as mentioned above, the spatial structures are preliminarily explored in this work, and more efforts are expected in the near future. On the other hand, this work is generally driven by data analysis and computer simulation. Although these approaches are effective to analyze and model transportation systems, more real-world factors, such as urban development variables and spatial constraints, should be considered in further studies from a geographic perspective.
Keywords: Complex networks, transportation systems, network growth, Evolving networks, centrality, urban systems, topology, space syntax, spatial
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networks, distance effects, gravity laws, planarity, street network patterns, US airline system, China’s aviation system, network efficiency, cellular structure, hierarchy, network representations.
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Acknowledgment
It has been eight years since the first day I came to Sweden, how time flies! It is really difficult for me to summarize this special period with a few words, although which has to be done at this moment.
First and foremost, I would like to thank my PhD advisors, Prof. Yifang Ban and Prof. Bin Jiang. It has been an honor to be their student. They are supportive since the days I began my PhD study. I appreciate all their contributions of time, ideas, comments, and discussions to make my PhD experience amazing and unforgettable! I would specially acknowledge Prof. Yifang Ban for the funding and help during my studies, which helps me finish my thesis successfully. I am also indebted to Prof. Yanguang Chen, who guided me into the complex network field since my master and gave me the moral support which I needed to move on during the most difficult times.
My colleagues in the Division of Geoinformatics at KTH have helped and taught me a lot throughout these years. I would like to especially thank Dr. Hans Hauska for our valuable friendship; Dr. Takeshi Shirabe, Dr. Gyözö Gidófalvi and Ehsan Abshirini for their helpful suggestions and help!
I would like to greatly thank the China Scholarship Council (CSC) and the School of Architecture and Built Environment (ABE) at KTH for providing me the scholarship and stipend.
I also thank my friends (too many to list all names here) for their friendship and encouragement during the past years! I am so moved that they are always there for me! Especially those friends I met in Sweden, the time and memory shared with all of them is a vital asset of this particular stage!
My deepest thanks go to my parents for their unconditional love and care. I love them so much! They are the spiritual pillar of my life that makes me brave and fearless. In addition, the best outcome from these past years is finding my soul- mate and husband Shijing Xian, and giving birth to our dearest daughter Melinda. Shijing has faith in me even when I was irritable and depressed. The past five years have not been an easy ride, thanks to all of my family for sticking by my side!
Stockholm, November 2017 Jingyi Lin
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Contents
ABSTRACT ...... I ACKNOWLEDGMENT ...... IV CONTENTS ...... V LIST OF ABBREVIATIONS ...... VII LIST OF FIGURES ...... VIII LIST OF TABLES ...... X LIST OF PAPERS ...... XI 1. INTRODUCTION ...... 1 1.1 Background ...... 1 1.1.1 Complex network theory ...... 1 1.1.2 The development of transport geography...... 2 1.2 Research motivation and objectives ...... 3 1.2.1 Research motivation ...... 3 1.2.2 Research objectives ...... 5 1.3 Organization of the thesis ...... 6 2. NETWORK TOPOLOGY ...... 9 2.1 Network representation ...... 9 2.2 Basic measurements ...... 12 2.2.1 Degree ...... 12 2.2.2 Betweenness ...... 13 2.2.3 Closenesse ...... 13 2.2.4 Straightness ...... 13 2.2.5 Characteristic path length ...... 14 2.2.6 Clustering coefficient ...... 14 2.3 Correlation ...... 14 2.3.1 Assortativity ...... 14 2.3.2 Hierarchy ...... 15 2.4 Network efficiency ...... 15 2.5 Planar networks ...... 16 2.5.1 Definition ...... 16 2.5.2 Cellular structure ...... 16 2.6 Distribution patterns ...... 18 2.6.1 Power law ...... 18 2.6.2 Double Pareto distribution...... 18 3. MODELS AND DYNAMIC PROCESSES OF COMPLEX NETWORKS ...... 20 3.1 Basic models ...... 20
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3.1.1 Random model...... 20 3.1.2 Regular model ...... 20 3.1.3 Small-world model ...... 21 3.2 Network growth models ...... 22 3.2.1 Barabási and Albert (BA) model ...... 22 3.2.2 Barrat, Barthélemy and Vespignani (BBV) model ...... 23 3.2.3 Zhang, Fang, Zhou and Guan (ZFZG) model ...... 25 3.3 Spatial evolving models ...... 26 4. STUDY AREA AND METHODOLOGY ...... 30 4.1 Overview ...... 30 4.2 Study area and data acquisition ...... 31 4.2.1 Paper II & III: China’s and the US’s aviation systems ...... 31 4.2.2 Paper IV: Street systems of Stockholm, Nanjing and Toronto ...... 33 4.3 Network analysis ...... 35 4.3.1 Power law distribution ...... 35 4.3.2 Double Pareto distribution...... 36 4.4 Spatial distance effect ...... 37 4.5 Airline network evolution model ...... 39 4.6 Identifying critical nodes in a network ...... 41 5. RESULTS AND DISCUSSION ...... 42 5.1 Paper I: Reviewing studies on urban transportation networks ...... 42 5.2 Paper II: Analyzing topological and spatial structures of China’s aviation system ...... 44 5.3 Paper III: Modeling the evolving structure of US’s airline network ...... 47 5.4 Paper IV: Comparing the planar structures of three street networks...... 50 6. CONCLUSIONS AND FUTURE WORK ...... 55 6.1 Conclusions ...... 55 6.2 Limitations and Future research ...... 57 6.2.1 Limitations and challenges ...... 57 6.2.2 Future research ...... 57 REFERENCES ...... 59
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List of abbreviations
O/D - Origin/Destination GIS - Geographical information systems / science LBS - Location-based service CA - Cellular automata ABM - Agent-based modeling GDP - Gross Domestic Product ER - Erdős and Rényi model WS - Watts and Strogatz model BA - Barabási and Albert model BBV - Barrat, Barthélemy and Vespignani model WWW -World Wide Web ZFZG - Zhang, Fang, Zhou and Guan model CAS - China’s aviation system UAN - U.S.’s aviation network OLS - Ordinary least squares
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List of figures
Figure 1.1: The Seven Bridge of Königsberg problem. (This picture illustrates the real scenario of the town, and highlights the river and seven bridges. Source: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg) ...... 1 Figure 1.2: The organization of selected papers in this thesis ...... 8
Figure 2.1: A simple network with 7 nodes and 9 edges ...... 9 Figure 2.2: A directed network with 7 nodes and 9 edges ...... 9 Figure 2.3: The comparisons between the primal representation and the dual presentation of a simple street network in terms of different route definitions (Source: Porta et al., 2006a; 2006b) ...... 11 Figure 2.4: Two simplified public bus routes and the representations in b) L- space and c) P-space...... 12
Figure 3.1: The Erdős and Rényi model with NV =30 and p=0.1 (NE =82, C=0.073, L=3.11) ...... 20 Figure 3.2: The regular circle model with NV = 30 and
Figure 4.1: The basics of US airline network during 1990-2010 ...... 33 Figure 4.2: The basics of US airline network during 1990-2010 ...... 34 Figure 4.3: The distribution of (a)degree, (b)closeness and (c)betweenness of Nanjing, China ...... 35
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Figure 4.4: The comparison between distributions of a) the node degree of the BA model with 200000 nodes and b) the edge betweenness of the U.S. airline network in five years ...... 36 Figure 4.5: Degree distributions of CAN in 2009 and UAN in 2010 ...... 37 Figure 4.6: The distance distributions of a) China’s aviation network, b)U.S. airline network, c) Stockholm’s street network and d) Toronto’s street network...... 37 Figure 4.7: The linear relationships between traffic flow and si*sj for cities with in a) dij<500km, b) dij ϵ (900-1100km), c) dij ϵ (1800-2200km), and d) dij>2800km ...... 39 Figure 4.8: Models generated in terms of θ=1 and a) φ=0, b) φ=0.5, c) φ=1 and d) φ=2...... 40
Figure 5.1: The distributions of edge weight and node strength of CAS ...... 45 Figure 5.2: The distribution of weekly flight flows on spatial distances in CAS ...... 46 Figure 5.3: The evolution of distributions of node betweenness and edge betweenness of UAN during 1990-2010 ...... 48 Figure 5.4: Identifying the critical nodes in UAN by exploring the correlation between node degree and node betweenness ...... 49 Figure 5.5: Distributions of network efficiency centralities in terms of different values of φ=0 (black squares), 0.5 (red circles), 1 (green upward triangles), 2(blue downward triangles)...... 50 Figure 5.6: Distribution histograms of the cell area of urban street systems of Stockholm, Nanjing and Toronto...... 52 Figure 5.7: Cumulative probability distributions of closeness centrality of (a) Stockholm, (b) Nanjing and (c) Toronto...... 53 Figure 5.8: Betweenness distributions of the urban street systems of (a) Stockholm, (b) Nanjing and (c) Toronto...... 54
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List of tables
Table 2.1: The adjacent matrix for the basic model in Fig. 2.1 ...... 10 Table 2.2: The adjacent matrix for the directed model in Fig. 2.2 ...... 10 Table 4.1: Characteristics of nodes and links in Chinese Airline Network ...... 31 Table 4.2: The basics of Stockholm, Nanjing and Toronto in 2012 ...... 33 Table 5.1 Selected empirical analyses on transportation networks from a complex network perspective...... 42-43 Table 5.2: Regression results of gravitation model for different A(i) and B(j) measurements ...... 46 Table 5.3 The basics of urban street networks in Stockholm, Nanjing and Toronto ...... 50
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List of papers
I. Jingyi Lin and Yifang Ban, 2013. Complex Network Topology of Transportation systems. Transport Reviews, 33(6): 658-685. (Author contributions: JL: conceived the idea, collected and reviewed literatures, and wrote the papers. YB: edited the paper.)
II. Jingyi Lin, 2012, Network Analysis of China Aviation Systems, Statistical and Spatial Structure. Journal of Transport Geography, 22: 109-117.
III. Jingyi Lin and Yifang Ban, 2014. The evolving network structure of the US airline system during 1990-2010. Physica A, 410: 302-312. (Author contributions: JL: conceived the idea, performed the experiments, analyzed the data and wrote the papers. YB: helped to improve the conception and edited the paper)
IV. Jingyi Lin and Yifang Ban, 2017. Comparative analysis on topological structures of urban street networks. International Journal of Geo-Information, 6 (10), 295. (Author contributions: JL: conceived the idea, performed the experiments, analyzed the data and wrote the papers. YB: helped to improve the conception and analysis and edited the paper.)
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1. Introduction
1.1 Background
1.1.1 Complex network theory
In 1735, Leonard Euler presented his research to the St. Petersburg Academy and proved that the famous problem of the Seven Bridges of Königsberg has no solution (Euler, 1741), which means that it was not possible to walk through the seven bridges exactly one time (Fig. 1.1). This study laid the foundation for graph theory. Afterwards, graphs in one form or another arose in different fields of science, from Kekule's diagrams in chemistry to Kirchhoff's circuit laws in physics. However, it is worth mentioning that only in 1878 the term graph, which is the mathematical term for networks, was introduced by James J. Sylvester to describe such general structures that are visible everywhere (Sylvester, 1878). Since then, mathematicians have worked hard to study properties of general graphs, which are also defined as topologies. However, no standard definitions existed among multiple subjects until Frank Harary published the book Graph Theory in 1969, in which he broadened the reach of graph theory to include physics, psychology, sociology and anthropology by systematically explaining related definitions and measures (Harary, 1969).
Figure 1.1: The Seven Bridge of Königsberg problem. (This picture illustrates the real scenario of the town, and highlights the river and seven bridges. Source: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg)
Compared to graph theory, which possesses a long history (Diestel, 2005), the theory of complex networks is a young and active subject. It is mostly inspired by the empirical study of gigantic networks in the real world with the development of computer technologies (Watts and Strogatz, 1998; Barabási and Albert, 1999; Newman, 2003; Kim and Wilhelm, 2008; Zou et al., 2010). Real
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world graphs, including transport systems, ecosystems, human neural systems, metabolic networks, and the Internet present complex topological characteristics that deviate from random graphs or regular graphs (Faloutsos et al., 1999; Jeong el al., 2000; Bagler, 2008; Nagurney and Qiang, 2007; 2008). These graphs can be generalized as complex networks. Various subjects have tried to find ways to measure the complexity and underlying features of complex networks; therefore, this attention constitutes the prosperity of the complex network theory since the 1990s. Although complex network theory borrows the fundamental terminology and many definitions from graph theory, it introduced numerous measures and models to explore real-world systems, especially gigantic networks (Freeman, 1977; 1979; Watts and Strogatz, 1998; Barabási and Albert, 1999; Newman, 2003; Gao et al., 2016). More details about basic measures and models will be present in Chapter 2.
With the development of human knowledge and computer technologies, traditional borders between different academic subjects have become increasingly blurred. In other words, the interdisciplinary perspective is becoming popular and necessary. Complex network theory is just one example of this trend. Undoubtedly, network theory is an interdisciplinary field that combines ideas from various subjects (Newman, 2003), and it has benefited enormously from a wide range of viewpoints. The most famous example is the Sante Fe Institute in the U.S., which is dedicated to working toward trying to comprehend and explore complex systems that underlie many of the most profound problems facing science and society today (www.santafe.edu).
1.1.2 The development of transport geography
“The purpose of transportation is overcoming space, which is shaped by a variety of human and physical constraints such as distance, time, and administrative boundaries and so on.”
——Rodrigue et al., 2006
The basic purpose of transportation systems is to carry people, freight, and information from origins to destinations (Haggett and Chorley, 1969; Hiller and Hanson, 1984; Haggett, 2001; Ducruet and Beauguitte, 2014). Movements of people, goods, and information constitute the foundation of human societies, which are also the main research objectives of transport geography.
Although transport geography is an old subject, it has experienced prosperous development since the 1960s (Garrison, 1962; Rimmer, 1985; Black, 2003; Ausubel and Marchetti, 2001; Banister, 2002; Hoyle and Knowles, 1998; Hoyle and Smith, 1998). With the globalization of trade and emerging development of tele-communications, people started to realize the importance of locations. Air
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and rail transportations witnessed remarkable improvements at this stage, which commenced to provoke widespread discussions from researchers (O’Kelly, 1998; O’Connor, 1995; 2003; Shaw et al., 2009).
Transport geography emphasizes the importance of specific spatial features, such as nodes, locations, routes, and networks. As a result, network analysis becomes a traditional way to explore systems in this subject, and space syntax has become a popular technique in geospatial analysis as well (Hiller and Hanson, 1984). The former focuses on geometric features of transportation systems, while the latter emphasizes topological abstraction (Bafna, 2003; Batty, 2004). Accessibility and mobility and Origin/Destination (O/D) analysis are the most widely discussed (Hansen, 2009; Fuglsang and Hansen, 2011; Caschili and Montis, 2013; Djurhuus et al., 2014; 2015; Jonsson et al., 2014; Saberi et al., 2016). Many measures such as connectivity, cyclomatic number, diameter, and coefficient of α, β, γ are developed to quantify spatial characteristics of transportation systems (Rodrigue et al., 2009). With the innovation of computer technologies, Geography Information Science (GIS) provides an efficient way to represent and investigate transportation networks in a computer environment (Thill, 2000; Miller and Shaw, 2001; Shaw, 2002), and it further promotes transportation as an interdisciplinary topic.
1.2 Research motivation and objectives
1.2.1 Research motivation
As an essential part of infrastructure systems, the patterns and evolution processes of public transportation systems can, to a large extent, reflect the economic and social development of a city or a country. Especially with the globalization and informationization, efficiency and convenience are becoming more critical than ever, which drive urban planners and policy makers to pay more attention to improving public transportation systems. Under these circumstances, exploring the characteristics and mechanisms of transportation systems has become an active research topic in many subjects (Kansky, 1963; Leinbach, 1995; Black, 1996; 2003; Garrison and Souleyrette, 1996; Zhang, 1998; Garrison and Ward, 2000; Nagurney and Qiang, 2007; 2008; Börjesson et al., 2012; Cardillo et al., 2013; Dobruszkes and Graham, 2015; Mattsson and Jenelius, 2015; Mattsson et al., 2015; Aleta et al., 2017). Undoubtedly, much progress has been gained in various aspects, including performance evaluation, flow modeling, location-based service (LBS), and structure optimization (Iri, 1969; O’Kelly and Miller, 1994; Gutiérrez and Urbano, 1996; Gutiérrez et al., 1996; Jenelius et al., 2006; Cheng et al., 2013; Reggiani, 2013; Börjesson and Kristoffersson, 2014; Cats and Jenelius, 2014; 2015; Chow et al., 2014; Dong et al., 2014; Anbaroğlu et al., 2015; Jenelius and Mattsson, 2015; Jenelius and
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Koutsopoulos, 2015). It is worth mentioning that, along with the development of computer technologies, research methods in this field have also been evolving from qualitative descriptions to quantitative analyses to dynamic modeling (Watling, 1994; Barrat et al., 2004a; 2004b; Wong et al., 2006; Xie et al., 2010a; 2010b; Cheng et al., 2012; Rui and Ban, 2012; Fosgerau et al., 2013; Bolbol et al., 2014). As a consequence, many models emerged during the process, such as cellular automata (CA) and agent-based modeling (ABM); they are both well- known examples of studying human behaviors on transportation systems (Abdou et al., 2012; Hernàndez et al., 2002; Rossetti and Liu, 2005; Davidsson et al., 2005; Correia et al., 2013; Fuglsang et al., 2013; Manley et al., 2014; Monteiro et al., 2014).
As mentioned in the previous section, network analysis is a traditional method to study transportation systems (Haggett and Chorley, 1969; Taaffe et al., 1996). Modelling transportation systems as graphs is a very common and convenient practice, which also offers an efficient description of their topologies and their dynamic processes (Black, 2003; Rodrigue et al., 2009). However, it is worth mentioning that one important breakthrough in this field in the last decade has been the introduction of complex network theory, which provides a more novel and interdisciplinary viewpoint to analyze transportation systems.
Transportation systems are undoubtedly organized in a network pattern; therefore, they have attracted extensive attention from various disciplines since the introduction of complex network theory. Many real transportation systems have been studied from different perspectives (Guimerà and Amaral, 2004; 2005; Guimerà et al., 2005; Guida and Maria, 2007; Jiang, 2007; Majima et al., 2007; Montis et al., 2007; Guo and Cai, 2008; Ma and Timberlake, 2008; Jiang et al., 2014; Jiang and Okabe, 2014; Jiang and Ma, 2015; Jones, 2015; Dai et al., 2016; Song et al., 2016). Although much progress has been gained, many limitations still exist. At the early stage, most attempts to explore transportation systems from the complex network perspective do not take into account spatial distance (Guimerà et al., 2005; Liu and Zhou, 2007; Lu and Shi, 2007; Bagler, 2008), which should be an essential characteristic of transportation (Li et al., 2013). This can be attributed to the fact that much work came from physicists or social scientists, not geographers. This also constitutes the main motivation behind this thesis and related work. The specific motivations can be summarized as follows:
• With the development of complex network theory, transportation systems, as a typical case of networks, have attracted much attention from researchers in various subjects, such as physics, social sciences, geography and computer science. As a result, enormous work has been done analyzing and modeling transportation systems with a complex network approach during the last decade. It has been very difficult to
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capture the systematic schemes from both the theoretic and the empirical perspectives. • As an important part of transportation systems, aviation networks clearly possess similar geographical property as other ground systems. However, aviation systems are, to some extent, similar to social networks; it is unnecessary for them to construct physical airlines between two airports. In this sense, it is indispensable and interesting to explore the structure of aviation systems under the special construction mechanism. • Due to difficulties in getting time-series data about real transportation systems, many studies can only focus on the system scenarios during some fixed years. However, exploring the evolving dynamics of transportation systems is essential for urban planners and policy makers. • Urban street systems are the backbones of cities. There are various street structures in the real world in terms of different evolution processes, socio-economic environments, and geographical contexts of cities. Comparing the relationships among them is a challenging topic.
1.2.2 Research objectives
As discussed above, this research was initially motivated by the development of complex network theory. Therefore, the ambition of this dissertation is to explore and simulate the spatial structure of transportation systems from a complex network perspective. There are many ways of defining “spatial structure”, but all—in one way or another—express the fundamental idea that the information about locations is essential. This information includes topological, geometric, or geographic properties of objects. Topology deals with shapes and relative position of spatial features, which remains constant regardless of map distortion. In comparison, geometry is interested in measurements related to length, angle, and volume. It is worth mentioning that this study will cast more light on geometric properties of transportation systems, trying to bridge the gap between complex network theory and traditional urban studies. To this aim, the objectives of this work can be stated as follows:
• Summarize the theoretical and empirical progress in transportation system analysis and modeling from a complex network perspective, laying a solid foundation for the following research. • Explore the topological and geometic structures of urban transportation systems, including aviation systems and ground street networks, in terms of various complex network measures.
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• Analyze the evolving mechanism of transportation systems from a complex network perspective. To this aim, a growth model will be introduced to help understand the dynamic evolution process. • Investigate the distance effect in different transportation systems.
1.3 Organization of the thesis
This dissertation focuses on the analysis and modeling of urban transport systems from a complex network perspective. It is mainly based on the following papers. The relationship of the papers is presented in Fig. 1.2.
I. Jingyi Lin and Yifang Ban, 2013. Complex Network Topology of Transportation systems. Transport Reviews, 33(6): 658-685.
II. Jingyi Lin, 2012, Network Analysis of China Aviation Systems, Statistical and Spatial Structure. Journal of Transport Geography, 22: 109-117.
III. Jingyi Lin and Yifang Ban, 2014. The evolving network structure of the US airline system during 1990-2010. Physica A, 410: 302-312.
IV. Jingyi Lin and Yifang Ban, 2014. Comparative analysis on topological structures of urban street networks. International Journal of Geo-Information, 6 (10), 295.
Paper I is a comprehensive review on transportation networks from a complex network perspective, which covers the most related studies in this field during the last fifteen years. The complex transportation networks have earned much attention from various subjects, and the main outcomes can basically be divided into two directions: the topological properties and the spatial structures. Within each range, researchers have discussed the static measures and evolving mechanisms of transportation systems by utilizing empirical data and constructing theoretical models. This paper not only summarizes the existing work on complex transportation systems, but it also lays the foundation for the following papers.
Based on paper I, the following papers will develop a series of empirical analysis. To better compare characteristics of the air and ground transportation systems, paper II and paper III respectively discuss China’s and US’s aviation systems, and paper IV explores the urban street systems of three cities. In paper II, the authors analyze statistical structures of China’s aviation system in terms of complex network measures, including node and edge centralities and correlation analysis. Meanwhile, preliminary attempts, from a spatial
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perspective, are made on detecting distance effects using network measures in airline networks.
The static structure at a given time is far from enough to reflect the complicated growth mechanism of transportation systems. Therefore, the evolution process of US’s aviation system is discussed in paper III, by utilizing time-series data during 1990–2010. The evolving trends of various complex network measures are studied in this work. In contrast, a theoretic growth model including the distance parameter is proposed as well.
The structure of the aviation system is highly affected by the city system in a country; likewise, the structure of urban street networks is mainly shaped by the development of a city. In paper IV, the focus has been shifted to ground transportation systems. Three cities with distinct urban patterns are included to develop a comparative analysis in this work. Paper IV is indispensable to make the thesis complete and systematic.
This thesis is orhanized into six chapters. Chapter 1 provides a brief introduction of the background, the research motivation and objectives, and the structure of this thesis, therefore providing an overall scheme of the thesis. Chapter 2 summarizes the basics of complex network theory, including the origin, development and progresses during the last decade in this field. This part is essential since it lays a solid theoretical foundation for consecutive sections. Chapter 3 shifts the focus to transportation systems. Various models are discussed, from the most simple and static models to dynamic evolving models. To better capture spatial features of transportation systems, the distance factor is introduced in this section. Chapter 4 presents the methodologies used in this dissertation. Since the dissertation is based on four papers, this section summarizes the related study areas and data processing in these papers. Besides, this section also explains the interrelationships between the papers. Chapter 5 discusses the results of each paper and highlights the contributions. Chapter 6 concludes the thesis and indicates our research directions, as well as challenges in the future based on the outcomes of this dissertation.
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Figure 1.2: The organization of the papers in this thesis
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2. Network topology
Section 2 will introduce the basic definitions, representations, as well as measures used in this thesis. The basic definitions and some measures originated from graph theory (Diestel, 2005), but some others are proposed in terms of the substantial non-trivial features of complex networks (Freeman, 1977; 1979).
2.1 Network representation
According to graph theory, a graph G = (V, E) is composed of a pair of vertices (nodes) V and a set of edges (links) E, in which we assume that the number of V and E respectively equals to n and m. Fig. 2.2 illustrates a graph with n=7 and m=9. Here V(G) = {v1, v2, v3, v4, v5, v6, v7}, and E(G) = {e1, e2, e3, e4, e5, e6, e7, e8, e9}. An edge ek = (vi, vj) connects the node vi and the node vj. For example, the edge e1 connects the nodes v1 and v2.
Figure 2.1: A simple network with 7 nodes and 9 edges
Fig. 2.1 is a basic network model. To depict complex scenarios of the real world, a directed model was proposed (Fig. 2.2). In this model, an edge ek = (vi, vj) has a direction, starting from node vi and ending at node vj. Unlike in Fig. 2.1, the edge e1 in this model is from v2 to v1. It is worth mentioning that most transport systems are directed networks.
Figure 2.2: A directed network with 7 nodes and 9 edges
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In a computational environment, a network model needs to be transformed into a computer recognizable pattern for further analysis. To this end, the adjacency matrix is introduced. Generally, if there is an edge between node vi and vj, then the corresponding value aij in the adjacent value equals to 1, otherwise it is 0 (Tab. 2.1). For a basic model, note that the value of both directions equal to 1 once there is an edge between two nodes. Comparatively, for a directed model, the direction can be easily identified in the adjacent matrix (Tab. 2.2).
Table 2.1: The adjacent matrix for the basic model in Fig. 2.1 Node v1 v2 v3 v4 v5 v6 v7 v1 0 1 0 0 1 0 1 v2 1 0 1 0 0 1 0 v3 0 1 0 1 0 0 0 v4 0 0 1 0 1 0 0 v5 1 0 0 1 0 1 0 v6 0 1 0 0 1 0 1 v7 1 0 0 0 0 1 0
Table 2.2: The adjacent matrix for the directed model in Fig. 2.2 Node v1 v2 v3 v4 v5 v6 v7 v1 0 0 0 0 1 0 0 v2 1 0 1 0 0 1 0 v3 0 0 0 1 0 0 0 v4 0 0 0 0 0 0 0 v5 0 0 0 1 0 0 0 v6 0 0 0 0 1 0 0 v7 1 0 0 0 0 1 0
In the real world, a node or an edge usually possesses a value; then the model becomes a weighted network. For example, for an aviation system, the traffic flow can be counted as the weight of the link (Lin, 2012).
What is described above is for the general situation, however, many specific features should be taken into account when analyzing different systems. As for empirical analysis on real world systems, the basic and core step involves the network representation. In some sense, how to present the system accurately and informatively is the first problem that every researcher should solve.
In transportation systems, both the roads (railways, airlines, etc.) and the intersections (railway stations, airports, etc.) are important. Because there are different analysis ends, two opposite representation approaches have been
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proposed to describe them (e.g., the primal representation and the dual representation) (Porta et al., 2006a; 2006b).
The primal approach is an intuitional way to illustrate a system, by which the system is projected as the real scenario. That is to say, road segments are regarded as edges, while intersections are nodes (Fig. 2.3). This method is widely used in many studies on urban street networks, highway systems, and aviation systems (Chi et al., 2003; Li and Cai, 2004; Scellato et al., 2006; Liu and Zhou, 2007; Xu et al., 2007a; Bagler, 2008; Jung et al., 2008; Hu and Zhu, 2009; Jiang, 2009; Porta et al., 2009; Rocha, 2009; Kaluza et al., 2010; Paleari et al., 2010; Zhang et al., 2010; Sapre and Parekh, 2011; Wang et al., 2011; Zanin and Lillo, 2013; Wang et al., 2014; Chatterjee, 2015; Rui et al., 2015; Du et al., 2016; Hong et al., 2016). On the contrary, the dual approach is taking road segments that are regarded as nodes, and there is called an edge between two nodes if they intersect with each other (Anez et al., 1996; Batty, 2004; Porta et al., 2006b; Jiang 2007; Hu et al., 2008; 2009; Zeng et al., 2009; Barthelemy, 2011). This method originates from the space syntax (Hiller and Hanson, 1984), and it is effective to reveal underlying topologies of real systems since it emphasizes functional relationships between routes. In short, the primal representation contains the real geometric features of a transportation system; however, the dual representation highlights the topological features, therefore, decreasing the complexity of the whole system (Fig. 2.3).
Figure 2.3: The comparisons between the primal representation and the dual presentation of a simple street network in terms of different route definitions (Source: Porta et al., 2006a; 2006b)
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As an essential part of urban infrastructure, public transportation systems have earned wide attention from different perspectives (Latora and Marchiori, 2002; Sen et al., 2003; Seaton and Hackett, 2004; Sienkiewicz and Holyst, 2005; Chen et al., 2007; Li and Cai, 2007; Ferber et al., 2007; 2009; Xu et al., 2007b; Wang et al., 2008; Derrible and Kennedy, 2009; 2010a; 2010b; 2012; Lee et al., 2008; Guo and Cai, 2008; Liu et al., 2009; Wang et al., 2009; Wang et al., 2009; Soh et al., 2010; Roth et al., 2012, Wang et al., 2014; Xu et al., 2015; Regt et al., 2017). Undoubtedly, a public transportation system is comprehensive, which can be composed of subway, bus, metro, and railway systems. The most significant feature of such systems is that they are all designed in terms of the route-stop (station) pattern. Accordingly, two representation approaches have been proposed respectively in the L-space and the P-space. As Fig. 2.4 shows, in the L-space, routes and stops (stations) are respectively regarded as links and nodes. In this situation, only two consecutive stops (stations) can be connected. In the P-space, however, there is a link between two stops (stations) as long as they are accessible no matter how many stops (stations) are between them.
a) b) c) Figure 2.4: Two simplified public bus routes and the representations in b) L- space and c) P-space.
2.2 Basic measurements
2.2.1 Degree
Degree is the most straightforward centrality measure in network analysis, although many other ones have been proposed. Degree calculates the number of edges from/to the nodes. Therefore, the degree of node vi can be defined as: kai ij (2-1) vVj
In which ki represents the degree of node vi, aij indicates the entry value in the adjacent matrix of node vi and vj, and V is the set of nodes in the network. Likewise, the average degree of a network measures the accessibility of the whole network, which is written as:
k k Ci/2 N V N E N V (2-2) vVi
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Here NV and NE respectively are the number of the nodes and edges in the network.
2.2.2 Betweenness
Researchers in social studies had commenced the exploration of the importance of the bridging of nodes in a network since last century (Freeman, 1977; 1979). It is also called betweenness in complex networks (Barthélemy, 2004; Crucitti et al., 2006; Rui and Ban, 2014), which is written as:
1 nv() C B jgi (2-3) i NNn1(2) VVjg vvvijg
Here NV is the number of nodes in a network, nig(vi) is the number of shortest paths between nodes vj and vg that passes the node vi, and nig is the total number of shortest paths between nodes vj and vg. It is worth mentioning that it can be extended to the edge of betweenness in terms of the same definition (Scellato et al., 2006; Barthélemy and Flammini, 2008). Both are important centrality measures and will be discussed in this thesis.
2.2.3 Closenesse
The closeness centrality measures the importance of a node by calculating its distance to all other ones in the network. It is written as:
C NV 1 Ci (2–4) dij vvji
Here NV is the number of nodes in a network, and dij indicates the shortest path length between node vi and vj. Notice that the distance could be the Euclidean distance, the social distance, or the topological distance.
2.2.4 Straightness
Straightness centrality introduced the spatial distance in addition to the topological distance in a network. It measures the efficiency by calculating the extent that the shortest path between node vi and vj deviates from their Euclidean distance in the real world.
Eucl S 1 dij Ci (2–5) NdV1 vvji ij
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Eucl In which NV is the number of nodes in a network, dij indicates the Euclidean distance between node vi and vj, while dij is their shortest path length in the network.
2.2.5 Characteristic path length
As indicated above, dij represents the shortest path length between vi and vj. In this sense, the characteristic path length of a network can be derived as the average of the shortest path length between all its node pairs.
1 Ld= ij (2–6) NNVV(1) vvij
2.2.6 Clustering coefficient
In addition, there is another important global measure in network theory, e.g., the clustering coefficient. As the name indicates, it quantifies the level of clustering of a network. Suppose that node vi has ki neighbors, and they have mi links between each other, then the clustering coefficient equals to:
2mi Ci (2–7) vVi kkii(1)
Where ki is the degree of node vi, then ki (ki - 1)/2 gives the maximum edges between ki neighbors. For a network that contains NE nodes, its clustering coefficient can be calculated as the average of values of all nodes:
112ki CCi (2–8) NNkkVViivii Vv V (1)
Note that this definition only considers the immediate neighbors of a node; however, the n-clustering coefficient can be derived likewise by considering the n-neighborhood (Jiang, 2004).
2.3 Correlation
Correlation analysis is finding relationships between two measurements. In complex network theory, multiple measures have been proposed to describe a system. In some sense, exploring correlations between them is even more important than analyzing distributions of the measures themselves.
2.3.1 Assortativity
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Assortativity measures the vertex correlation, which further reflects the local architecture in a network (Newman, 2003). It calculates the average degree of a node’s immediate neighbors (Maslov and Sneppen, 2002):
1 kk (2–9) nnvj, i ki vVij ()
In which ki is the degree of node vi, then V(i) includes all nearest neighbors of node vi. In a network, if a node with a high degree value intends to be connected with other high-degree nodes, then assortativity is detected, otherwise it is regarded as disassortativity.
2.3.2 Hierarchy
The hierarchy in a network can be detected by exploring the correlation between the node degree and the clustering coefficient (Rabasz and Barabási, 2003):
1 C(k)= C ∼ k -a ' å i (2–10) N k =k ki =k i
’ Where N ki=k represents the number of nodes whose degree equals to k, then ci is the clustering coefficient of node vi. In a complex system, a hierarchy structure often indicates the coexistence of the scale-free property and a high clustering coefficient. Explicitly speaking, small communities of nodes are assembled into increasingly large communities in a hierarchical way.
2.4 Network efficiency
Network efficiency measures the performance of the information exchange of a network, which can be affected by many factors. Latora and Marchiori (2001) propose a clear way to represent this quantity as:
11 E= (2–11) NNdEEij(1) ij
Here NE is the number of nodes in a network, and dij is the distance between nodes vi and vj. Despite the global efficiency, this measure can also be extended to a local one by taking the average of efficiency values of sub-graphs (Rui and Ban, 2012). Based on the network efficiency definition, the information centrality is proposed as follows (Crucitti et al., 2006; Porta et al., 2006a; 2006b; Gastner and Newman, 2006a; Latora and Marchiori, 2007):
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EGEG[]['] C I = (2–12) i EG[]
In this example, G represents the original graph with the network efficiency E[G]. G’ represents the subgraph of G in which the node i and related links have been removed, and E[G’] calculates the resulting efficiency. In this sense, the information centrality measures the importance of a node by quantifying the relative difference of network efficiency when it is inactivated from a graph.
2.5 Planar networks
2.5.1 Definition
Most ground systems are planar. In graph theory, a planar graph is a graph that can be embedded in the plane. In other words, a graph is planar only if any two edges meet each other at a vertex (Clark and Holton, 1991; Diestel, 2005; Cardillo et al., 2006; Wilson, 2010; Chan et al., 2011; Mozes, 2013). As far as transportation systems are concerned, street systems in cities are a case in point, although there are a few bridges and tunnels in the real world (Xie and Levinson, 2007; 2009; 2011). A planar graph can be described in a mathematical manner:
Nf = Ne – Nv –2 (2–13)
In this explanation, Nf, Ne, and Nv respectively represent the number of faces, edges, and nodes in a graph. The face is also referred to as a cell in the remainder of this section.
2.5.2 Cellular structure
The alpha index is a measure of connectivity, evaluating the number of cells in a graph compared to the maximum number of faces (Garrison and Marble, 1962; Kansky, 1963). It is calculated as follows:
N NN2 = f ev (2–14) NNmax 25v
Here Nf, Ne, and Nv respectively represent the number of faces, edges, and nodes in a graph. The alpha index always ranges from 0 to 1, and the higher the alpha index, the more a graph is connected. In some studies, the alpha index is also called meshedness coefficient (Buhl et al., 2006), which quantifies the tree structure of a graph.
Calculating the average number of links per node in a graph can generate the beta index:
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N = e (2–15) Nv
A larger value of beta indicates a higher connectivity of a graph. In addition, the gamma index compares the actual number of edges with the number of all possible edges in a graph (Rodrigue et al., 2009), which is written as:
N = e (2– 3 ( 2)Nv 16)
In this explanation, Ne and Nv respectively represent the number of edges and nodes in a graph. Based on the definition, the value of gamma lies between 0 and 1, and 1 means a completely connected network.
Network structure shapes traffic flows on it. To further depict the cellular pattern of a graph, Xie and Levinson (2007; 2009; 2011) proposed the ringness, webness, treeness, and circuitness to measure the topological structure of a network. The treeness Φtree can be defined as:
Ltree tree (2–17) Ltotal
Here Ltree is the length of links in the part of the tree structure, while Ltotal is the total length of all links in a graph g. Then the circuitness Φcir can be defined as:
cirtree1 (2–18)
Note that these two measures both range from 0 to 1. A high treeness indicates a tree or branching structure of a graph, while a high circuitness means a circuit network.
The Shannon entropy (1948) can be introduced to measure the heterogeneity in a network:
n H iilog2 () (2–19) i1
In which n indicates the number of components in the system, ρi is the probabality of component i. The entropy value of a homogenous system equals to 0 and a higher value indicates a greater heterogeneity. For a transport system, the roads can be divided into different subsets based on different properties including traffic volume, administrative levels, and functional types (Ben-Naim et al., 2004; Xie and Levinson, 2007; 2009).
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There are also some attempts to study cell properties, including cell degree, cell area, and form factor. The cell degree measures the number of adjacent cells of a cell (Lammer et al., 2006), and the form factor can be calculated as the ratio of the cell area and the area of the circumscribed circle that is covered by the cell:
SSff4 g 2 (2–20) Slcir max
In this example, Sf is the cell area, and lmax indicates the diameter of the circumscribed circle. In this sense, the form factor measures the compactness or roundness of a cell. The minimum value 0 and the maximum value 1 respectively represent infinitely narrow and perfectly circular cells.
2.6 Distribution patterns
2.6.1 Power law
The scaling property or power law distribution has been recognized as the most typical characteristic for complex phenomena and has been validated by many empirical analyses (Newman, 2003; Kalapala et al., 2006; Barthelemy, 2011; Lin and Ban, 2013). For example, population size distribution of cities, intensities of earthquakes, and wealth distributions all follow the power law. This law indicates the heterogeneity of a system. Let c represent the quantity whose distribution in which we are interested; the power-law distribution can be expressed as:
PCcc() (2–21)
In this example, p(c) is the probability of quantity c, and α is a constant parameter of the distribution known as the exponent or scaling parameter, which typically lies in the range 2<α<3.
2.6.2 Double Pareto distribution
The double Pareto Law is a distribution between the lognormal distribution and the power law distribution. Explicitly speaking, a double Pareto distribution can match the body of a lognormal distribution and the tail of a power law distribution (Mitzenmacher, 2004). It exhibits power law behavior at both the upper tail and the lower tail regions, which can be written as (Reed and Jorgensen, 2004):
-1 c, c c P() C c -2 (2–22) c, c c
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In this example, C represents a centrality measure. cη is the transition value, based on the distribution that can be divided into two parts. They both follow a scaling pattern with different scaling exponents δ1 and δ2. Note that it is a cumulative probability distribution, and the original distribution can be derived as:
-11 P() C c c, c c PC() c -12 (2–23) c, c c
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3. Models and dynamic processes of complex networks
3.1 Basic models
3.1.1 Random model
Modeling is an efficient and straightforward way to depict the dynamics of the real world in both natural and social sciences. During the last decades, a large number of models of networks have been proposed in various subjects, from simple random models to complicated evolving models. In this section, frequently used models of the complex network theory will be discussed. The most famous random model was proposed by Erdős and Rényi (1959), which is called Erdős and Rényi (ER) model. It can be generated by randomly connecting two nodes from the initial N nodes in terms of a probability p. The ER model presents a small characteristic path length and a low clustering coefficient. Fig.3.1 gives an example based on 30 nodes and the probability of 0.1. The modeling process generates 82 links. In addition, the clustering coefficient and the average path length respectively are equal to 0.073 and 3.11.
Figure 3.1: The Erdős and Rényi model with NV =30 and p=0.1 (NE =82, C=0.073, L=3.11)
3.1.2 Regular model
The regular model can be generated by giving the number of nodes and the average degree per node. For example, if the number of nodes is set to n and the average degree per nodes to k, then the number of edges in an undirected model equals (n*k)/2. According to this definition, the level of clustering of the network is high; however, the value of the characteristic path length is large. The regular model presents the opposite properties compared to the ER model. To better compare them, a regular ring with NE=30 and
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Fig. 3.2. The network consists of 60 links. The clustering coefficient and the average path length respectively are equal to 0.5 and 4.14.
Figure 3.2: The regular circle model with NV = 30 and
3.1.3 Small-world model
The above-mentioned models present two extreme scenarios, and neither of them is suitable to describe the real world. In most real networks, it is possible to reach a node from another one by going through a number of edges that is small compared to the total number of nodes in the system. To better represent the real world with a large clustering coefficient and a low average path length from a global perspective, researchers have worked for a long time to improve and construct another model. An important breakthrough was made by Watts and Strogatz (1998). They proposed the small-world model, which is also known as the Watts and Strogatz (WS) model. The WS model is generated by rewiring any links in a regular network based on the probability p. P can range from 0 to 1, corresponding to a regular network to a random one. Consequently, two arbitrary nodes are likely connected by only a small number of intermediate links in a small-world network, which indicates a small characteristic path length and a large clustering coefficient. In Fig. 3.3, a small-world model is generated based on the regular ring model in Fig. 3.2 and a rewiring probability p=0.3. In contrast, its characteristic path length is 2.717, which is smaller than the regular model (Fig. 3.2). However, its clustering coefficient equals to 0.322, which is much bigger than the random model (Fig. 3.1).
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Figure 3.3: A small-world model with NV =30, p=0.30 (NE=60, C=0.322, L=2.717)
3.2 Network growth models
3.2.1 Barabási and Albert (BA) model
In addition to the efforts made on static topological models, much progress has been gained in understanding the dynamic evolution process. The network growth model is much more complicated, since an evolution perspective is introduced. Barabási and Albert (1999) suggested that many real networks exhibit preferential connectivity during their evolution process. That is to say, a node that possesses a higher connectivity has a higher probability to be connected with a new node in a network. In one word, the big one will get bigger. As discussed in section 2.2.1, the node degree is the most straightforward way to measure the connectivity; therefore, the attaching probability that a new vertex will be connected with node vi can be caculated as:
k t1 i (3–1) ni k t1 j j
t-1 In this example, ki indicates the degree of node vi at time t-1. It is assumed that the original network only possesses one node, and a new node with c new links is added per step. Then the whole process can be divided into two parts:
One new vertex is added per time step with c links, thus the number of nodes and edges in the network equal to:
Ntt N1 11 t (3–2) vv
Ntt N1 c ct (3–3) ee
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Preferential attachment: the new vertex connects with c existing nodes
according to the probability ni , then the total degree of the network at time t is:
t k ci t 2 i (3–4)
Figure 3.4: The growth mechanism of the BA model with c = 2. A new node nt is connected to vertices i and i’ at time t.
Fig. 3.4 explicitly explains the mechanism with c = 2. Note that the degree distribution of a BA model always follows a scale-free pattern, thus it is also called a scale free model. Together with the small-world model, it is acknowledged as one of the two most significant models that mark the birth of complex network theory.
3.2.2 Barrat, Barthélemy and Vespignani (BBV) model
The BBV model is a model introducing the weight-driven mechanisms, which plays an important role in real-world systems (Yook et al., 2001). The contribution of the BBV model lies in the proposal of weight reinforcement instead of a static weight mechanism. The basic idea of the BBV model is that every new edge would introduce a new weight to its existing neighborhood edges, and the evolutions of edges and related vertices could be captured under such a mechanism (Barrat et al., 2004a, 2004b, 2004c; 2005; 2008). In a scientific collaboration network, a new collaboration would not affect the previous collaboration. However, in most real systems, such as an aviation network and World Wide Web (WWW), a new edge leads to a redistribution of the previous weights of the links. For simplicity, the BBV model assumes that the new edge only affects the vertex it connects and the vertex’s direct neighborhood links.
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As defined above, the node strength for node i is the sum of all edge weight of its links, which can be stated as: swiij jvi () , (3–5)
In this example, Vi is the set of neighbors of vertex i. In a weight-driven growth mechanism, the probability of a vertex being chosen at time t is based on its node strength at time t-1:
st1 i ni st1 j j . (3–6)
This means that new vertices are more likely to connect to the vertices with the larger strength, instead of the bigger degree in the BA model. The weight of new links is uniform during the growth process, which is set to wc. We can now explain the dynamics of weights in terms of two kinds of vertices:
For the vertex i to which a new edge attaches at time t, the node strength is
sswtt1 iii c . (3–7)
For the weight of previous edges connecting to vertex i:
wt wwtt1 ij ijiji st i (3–8)
Here the coefficient δi could be a constant or a variable. Obviously, in the long run, the larger δi would generate a larger Δsi.
In this process, a new node and c new edges are added per time step, and c is a constant. Fig. 3.5 depicts the above mechanism with c = 2. If we assume the network grows from one node in the original stage, then:
The total degree of the network at time t:
t ki 2 ct i (3–9)
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Figure 3.5: The growth mechanism of the BBV model with c = 2. A new node nt is connected to the vertices i and i’at time t, and the weight of the two new edges equals to wc (Source: Barret et al., 2004b).
Based on this framework, Barrat et al. (2004a, 2004b) explored the evolution and distribution of strength, degree, and weight. The author will not illustrate the detailed analytic process in this section. In summary, it is worth noting that the parameter δi plays a key role in the model. It controls the microscopic weight dynamics and the collective distribution in the network. These quantities all present a power-law distribution, with the exponent being highly dependent on the parameter δi. Obviously, for δi =0, the BBV model is equivalent to the BA model.
3.2.3 Zhang, Fang, Zhou and Guan (ZFZG) model
In contrast, with the m-edge addition per time in BBV model, Zhang et al. (2008) believe that an accelerating growth mechanism could better depict complex systems in the real world. To this end, they introduced another parameter θ called the accelerating exponent to control the growth dynamics. In this sense, the new edges produced by a new added node at time t can be written as a function tθ.
It is worth mentioning that the ZFZG model also conforms to the weight growth distribution in Fig. 3.5, although the total degree in the network would be different due to the different numbers of new edges tθ:
2t1+ ktt dt2 i i 1+ (3–10)
In summary, this model is governed by two parameters θ and δi, which can significantly affect the network structure. With the change of θ from 0 to 1, the distributions of strength, weight, and degree are subject to the transition from a power-law to an exponential pattern. Furthermore, if we let θ=0 and δi =0, then
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it becomes the basic BA model. In addition, for θ=0 it is reduced to a special case of the BBV model with c=1.
3.3 Spatial evolving models
In the previous sections, a number of theoretical models have been discussed to capture the dynamics of real-world networks. However, it is worth noting that geographical factors are absent in those models. For some networks, e.g., social networks and biologic networks, ignoring geographical factors may be reasonable since they are not embedded in Euclidean space. However, distance plays an intrinsic role for a special class of networks (e.g., spatial networks) where the Euclidean coordinates of nodes and the physical lengths of edges are meaningful (Sienkiewicz and Holyst, 2005; Wong et al., 2006; Erath et al., 2009, Ferber et al., 2009; Barthélemy, 2011). Many researchers have realized this and made much progress on including the spatial distance in topological growth models (Manna and Sen, 2002; Jost and Joy, 2002; Xulvi-Brunet and Sokolov, 2002; Barthélemy, 2003; Yook et al., 2003; Gastner and Newman, 2006b; Xie et al., 2007a, 2007b; Rui et al., 2014). In this section, only two of them are selected to illustrate the simulation frames.
It is widely accepted that preferential attachment is the probable explanation for the scale-free property in many real networks. For spatial networks, a distance preference function is further added to the attachment probability function as:
k t1 i fd() (3–11) ninik t1 j j
Here fd()ni is a general distance preference function, and different definitions can lead to distinct model features. For example, Jost and Joy (2002) believed that the shortest distances are always selected when a new node selects its connections. However, the simple linear function cannot generate reasonable results in most cases. Therefore, Xulvi-Brunet and Sololov (2002) proposed that the distance preference should follow a power-law function as follows: f() d d (3–12) ni In this example, d is the distance from the new node to other candidate nodes, and ɑ is a continuously varying parameter. In Fig. 3.6, distinct results are obtained with ɑ varying from -∞ to +∞, indicating that the node is connecting from the closest to the farthest node in the network (Manna and Sen, 2002). Note that this model becomes the BA model when ɑ =0.
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Figure 3.6: The simulation results based on different values of ɑ (-∞, +∞) (Source: Manna and Sen; 2011)
Xie et al. (2007b) made some changes by adding a scaling exponent to the degree at time step t instead of the distance; therefore, the attachment function is written as:
2 ni (3-13) ni kt () i In this example, the node n is newly added, and i is an existing node in the 2 network. ni represents the geographical distance between nodes n and i. Ki(t) is the degree of node i at time t, and ɑ is a free parameter. In this model, only one node n is added, and one node i is chosen to connect with it at one time, by minimizing the quantity ni . Fig. 3.7 illustrates the evolution of the model as ɑ grows. Clearly, when ɑ=2 almost all new nodes are connected to the most central one. To summarize, the distance effect is decreasing as the parameter ɑ increases.
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Figure 3.7: Network growth with different parameters: (a) α = 0, (b) α = 0.5, (c) α = 1 and (d) α = 2. (Source: Xie et al., 2007b)
For real systems, traffic optimization is also a critical factor to be considered (Ferrer i Cancho and Sole, 2003; Gastner and Newman, 2006b; Aldous, 2008; Barthélemy and Flammini, 2008; Brede, 2010). Barthélemy and Flammini (2006) find the tree connecting N nodes in a plane, which minimizes the following cost function:
bd (3-14) e ee
In this example, be is the betweenness centrality of an edge, and de is the Euclidean distance of the edge. In this case, the parameters u and v control the relative importance of spatial distance against node centrality. Figure 3.8 gives some examples of different values of them. When (u,v) = (0,1), the cost function d equals to (0,1) e e . In this case, it means the new node always connects to the nearest node to minimize the total length of the network (Fig 3.8(d)). In b comparison, when (u,v) = (1,0), the cost function becomes (0,1) e e , indicating that all nodes are connected to the most centralized one (Fig. 3.8(d)).
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Figure 3.8: Different spanning trees after global optimization. (a) Minimum spanning tree for (μ,ν) = (0,1). (b) Optimal traffic tree for (μ,ν) = (1/2,1/2). (c) Minimum Euclidean distance tree for (μ,ν) = (1,1). (d) Optimal betweenness centrality tree for (μ,ν) = (1,0), which is also the shortest path tree (Source: Barthélemy and Flammini, 2006).
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4. Study area and methodology
4.1 Overview
Understanding urban transportation systems is a complex task. It usually covers various perspectives, from a global level to an individual level; from geometric properties to topological structures; and from the spatial expansion to the temporal evolution. It is necessary to investigate and provide comprehensive measures to better capture and model real transport systems.
In this chapter, the author will introduce the proposed models and analyses in detail. Since this thesis is composed of four articles—which involve different subjects, like geography, statistics and computer science—the author is obliged to summarize the methodology in terms of the common cognitive process, instead of introducing them one by one. In this sense, this chapter is divided into four parts as follows:
(1) The first step to analyze a transportation system in a computer environment is to extract and represent it; therefore, the first part is defining study areas and representing them in section 4.2. Paper I introduces comprehensive empirical analyses and summarizes various representations in terms of different features of transportation systems. Based on this, China’s and US’s aviation networks are respectively discussed in paper II and paper III. Then the focus shifts to ground transport systems. Urban street systems of Stockholm, Nanjing, and Toronto are explicitly compared in paper IV. Note that, although there are sometimes many representations for a system, the author selects the one which could retain the most geographic characteristics.
(2) From a complex network perspective, exploring various centrality measures is an indispensable part to discuss both the global and the local network structures. As a result, numerous different centrality measures have been proposed (Freeman, 1977; 1979; Boccaletti et al., 2006; Chan et al., 2011) and are summarized in paper I. Paper II, paper III, and paper IV respectively discuss centrality distributions and correlations between them for aviation systems and urban street networks. This part focuses on statistical analyses and lays a necessary foundation for further analyses.
(3) In the real world, transportation systems have been undergoing a long history of development; therefore, an evolution perspective is necessary in this thesis. The US’s aviation network is explored in paper III over a 20-year period. To better capture the dynamics underlying the evolution process, a network model is proposed with nonlinear centrality growth and neighboring connections.
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Besides investigating the preferential attachment mechanism, the model validates the empirical studies as well.
(4) As for transportation systems, aviation systems and urban street networks are embedded in space and thus affected by various geographic factors. To depict the dependence of flight flows on centrality and distance, the gravitational model is introduced in paper III. Moreover, paper III investigates the relationships between spatial distance and centrality measures. The authors propose a spatial evolving model to study the network growth mechanism. Lastly, more light is shed on planar properties of urban street networks in paper IV.
4.2 Study area and data acquisition
4.2.1 Paper II & III: China’s and the US’s aviation systems
The aim of this thesis is to analyze and model transport systems, which are composed of aviation, metro, bus, railway, and urban street networks. As paper I summarizes, these transportation networks have a long history of working together to constitute an efficient and comprehensive system for our human society. In the first two articles, aviation systems are selected as the study objects. It is worth mentioning that the aviation system is, to some extent, similar to social networks, since it is unnecessary to construct physical connections in such networks. However, aviation is definitely constrained by geographical factors considering that all airports are embedded in real spaces and have fixed coordinates. In this sense, exploring the features of airline systems is of significant importance for transportation network analysis.
In paper II, China’s aviation network (CAN) is constructed by extracting airport cities as nodes and air routes links. During the construction process, there are always two principles to obey. First, all cities that have their own airports, no matter how many they possess, are included in this network. That is to say, if there are two airports in one city—for example, Beijing has the International Capital Airport and Nanyuan Airport—they will be combined as one node in this case. In this study, there are 166 civil airports in the mainland of China in 2009 and, after the combination, only 140 nodes are constructed. Naturally, related traffic flow and air routes are also combined in this analysis.
Second, all detailed domestic flight data is collected from different airlines’ websites. If there are at least 20 regular flights between two cities in 2009, then a link is set up between corresponding nodes. In order to avoid random perturbations, most seasonal flights or temporary flights are deleted from this study. To better understand the interactions between cities, the weekly flight
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numbers are further adopted as link weight so that we can obtain a weighted but undirected network.
After establishing the basic network structure, some socio-economic features are assigned to nodes and links in the network (Fig. 4.1). Data about Gross Domestic Product (GDP) and population of cities can be obtained from the Chinese City Statistical Yearbook 2009 (NBSC, 2009a) and China Statistical Yearbook for Regional Economy 2009 (NBSC, 2009b). The lengths of links, e.g., the spatial distance between city pairs, are calculated in ArcGIS based on coordinates of cities.
Table 4.1. Characteristics of nodes and links in Chinese Airline Network
Airport Cities Air Routes Coordinates Spatial Length (Distance)
Features GDP The Number of Flights per Week
Population
Likewise, the U.S.’s airline networks (UAN) were constructed from 1990 to 2010 (Fig. 4.2). All preliminary data in this study come from Bureau of Transportation Statistics, and the author executed further data integration, screening, and standardization.
Note that there is only one difference between the construction processes of UAN and CAN. For the UAN, a link is set up only for city pairs that have at least 100 flights between them per year. Compared with the CAN, the U.S. possesses much more developed airline systems. In 1990, the UAN is composed of 304 airport cities and 12,626 links. These numbers have increased in the duration of the study period. In 2009, the number of airport cities reached 891, and the number of links in the network reached 17,910 (Fig. 4.2a). The passenger turnover also exploded from 0.41 billion in 1990 to 0.63 billion in 2010 (Fig. 4.2b).
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(a) (b) Figure 4.1: The basics of US airline network during 1990-2010
4.2.2 Paper IV: Street systems of Stockholm, Nanjing and Toronto
Both paper I and paper II discuss the aviation system from different perspectives; in paper IV, the focus is shifted to the ground transportation system, i.e. urban street networks. The urban street network shapes the backbone of a city, which can be best reflected in the shape and the form of a city (Masucci et al., 2009). It is acknowledged that urban street networks can be affected by various factors, from historical, economical, geographical, to social factors. To better compare interactions between street patterns with these potential factors, three street systems with totally distinct characteristics were selected in this study. They are Stockholm in Sweden, Nanjing in China, and Toronto in Canada (Tab 4.1).
From an urban planning perspective, Toronto is a famous well-planned city, whose development has followed a strict plan. In comparison, Nanjing is an old city that possesses a long and prosperous history. It means that its development may be more affected by historical than planning factors. In some sense, Stockholm and Nanjing can be regarded as self-organized cities, while Toronto is strictly a planned one. Which kind is more efficient? This is a controversial topic that has invoked extensive discussions in urban studies and also constitutes the motivation of this article.
As discussed in section 2.1, there are two ways to represent planar networks, e.g., the primal approach and the dual way (Fig. 2.3). In this study, the primal representation is selected to reflect the real geometry of the three systems (Fig. 4.3). In other words, every node denotes an origin, an end, or an intersection in these systems, and it possesses geographical coordinates. In this case, one road (in the administrative sense) may be broken into many segments. Each segment is represented by a link in the system, and a length value is assigned correspondingly.
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To obtain more precise networks, these preliminary GIS files are carefully edited to avoid duplicate areas and lines, broken lines, and isolated points. In the resulting networks, Toronto presents the biggest size with 26,093 nodes and 37,123 segments. In comparison, Stockholm possesses the most concise structure with 2,612 nodes and 3,603 segments. Nanjing lies in the middle with 6,121 nodes and 9,275 segments.
(a) (b)
(c) Figure 4.2: Street networks of (a) Stockholm, (b) Nanjing and (c) Toronto.
Table 4.2. The basics of Stockholm, Nanjing and Toronto in 2012
City NO. NO. Population Area Population GDP Mean of of density ($bil) income($) nodes edges (km2) (Per/km2) Stockholm 2612 3603 881,235 188 4687 34.5 48780 Nanjing 6121 9275 6,376,500 3098 7573 59.5 5906 Toronto 26093 37123 2,791,140 630 4430 143.8 42736
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4.3 Network analysis
4.3.1 Power law distribution
Centrality is a measure that has been extensively discussed for complex networks. As introduced in section 2.2, there are various centrality measures. From a local perspective, node degree, betweenness, closeness, and straightness measure different features of nodes or edges. From a global perspective, average path length, clustering coefficient, and network efficiency depict underlying characteristics of the whole system. These measures are respectively calculated in this thesis.
In Fig. 4.4, the node degree, closeness, and betweenness values of the city of Nanjing are plotted. Considering the practical efficiency of real-world street networks, normally there are a maximum of three roads intersecting with each other at one point. That is to say, the maximum of the node degree value in this network can be six, which is also the case in this research. In contrast, the minimum value should be one for an end or a starting point of a road. The degree possesses a relatively even distribution from an overall perspective (Fig. 4.4(a)).
In contrast, the node closeness and the betweenness show more heterogeneity for urban street systems (Fig. 4.4). These two indices both calculate the node centrality from a global perspective; therefore, they theoretically present very low values near the edge of the system (Prota et al., 2006a; Crucitti et al., 2006), and maps are an excellent approach to capture this edge effect.
(a) (b) (c) Figure 4.3: The distribution of (a)degree, (b)closeness and (c)betweenness of Nanjing, China
As a universal property for complex systems, the power law distribution has been widely discussed in centrality analyses of transportation networks. Although the mathematic formula is straightforward, detecting power laws in empirical analyses is complicated due to the difficulties of estimating the large
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fluctuations that occur in the tail of the distribution and the ranges over which a power-law phenomenon holds. The topic has invoked much discussion over last decades (Adler et al., 1998; Malevergne et al., 2005; Clauset et al., 2009). In this thesis, the least-squares fitting is used to analyze the data (Fig. 4.5). In the meantime, the goodness-of-fit between the data and the power law is calculated to judge whether the fitting hypothesis can be accepted.
(a) (b) Figure 4.4: The comparison between distributions of a) the node degree of the BA model with 200000 nodes and b) the edge betweenness of the U.S. airline network in five years
4.3.2 Double Pareto distribution
A perfect scaling pattern only occurs in a mathematical sense. For most phenomena observed in nature or in the social world, some other heavy-tailed distributions, including exponential distribution, stretched exponential distribution, log-normal distribution, or a double Pareto Law are normally detected (Reed, 2003; Mitzenmacher, 2004; Reed and Jorgensen, 2004; Al- Athari, 2011; Toda, 2012; Nadarajah et al., 2013). There have been many studies comparing these different distributions (Giesen et al., 2010; Jiang and Jia, 2011). In this study, the double Pareto Law, also acknowledged as the two- regime power law, is detected for many centralities. On a log-log plot, the power law distribution presents a straight line, while the double Pareto distribution consists of two straight-line segments that meet at a transition point (Fig. 4.5). In practice, the most difficult part of detecting a double Pareto distribution is deciding the transition value of the centrality measure. A best transition value should balance the goodness-of-fit of the data and the power law for both regimes. It is woth mentioning that not all centrality distributions could be fitted by double Pareto distribution, although they present similar two-regime patterns.
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(a)CAN (b)UAN Figure 4.5: Degree distributions of CAN in 2009 and UAN in 2010
4.4 Spatial distance effect
The spatial perspective has acknowledged as an important branch of network theory and revoked widespread interest from various subjects (Kurant and Thiran, 2006; Lammer et al., 2006; Lambiotte et al., 2008; Zhang et al., 2009; Barthelemy, 2011; Levinson, 2012). A statistic distance histogram is a straightforward way to depict the spatial interaction in a real-world system. In this thesis, distance distribution patterns are captured for both aviation systems and ground transportation systems (Fig. 4.7).
Frequency
Cumulative% Frequency Frequency Frequency Cumulative%
Dij Dij
(a) (b)
Frequency Frequency
Frequency Frequency Cumulative% Cumulative%
Dij Dij (c) (d) Figure 4.6: The distance distributions of a) China’s aviation network, b)U.S. airline network, c) Stockholm’s street network and d) Toronto’s street network.
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To better explore the relationship between spatial distance and socio-economic factors, a well-known model from geography is introduced, e.g., the gravitational model (Curry, 1972; Viitala, 1977; Anderson, 1979; Ullman, 1980; Batty and Sikdar, 1982; Fotheringham and O’Kelly, 1989; Evenett and Keller, 2002). This model is extended from the famous Newton’s Law of Universal Gravitation. In spatial economics, the model is used to depict the spatial interaction between two places (Eaton J. and Kortum, 2002). In mathematics, the spatial interaction U(i, j) between the origin i and destination j can be described as:
UijAiBjfd(,)()()() ij (4–1)
In this example, A(i) and B(j) respectively represent a measurement for the origin i and destination j. Then f(dij) is the most complicated part and also the one that needs further research. It is also called the distance dependent function (Fischer and Griffith, 2008). In empirical analyses, U(i, j), A(i), and B(j) can be different measures in terms of specific research objectives.
First, the trend of the function can be detected by transforming the equation 4.4 as follows:
Uij(,) fd() AiBj()() ij (4–2)
According to this equation, if U(i, j) and A(i)*B(j) follow a linear relationship, then the slope should be the distance effect value. In paper II, the traffic flow between two airports is considered to measure the interaction between two cities, while the node strength of airports represents the measurements A(i) and B(j). In this sense, the correlation between u(i, j) and A(i)*B(j) can be depicted in Fig. 4.8. In the figure, four subplots respectively describe the correlation trends in terms of different distance ranges. Based on the preliminary image about the fitable range and the model of the distance effect can be detected by ordinary least squares (OLS) analysis. It is worth mentioning that the distance effect that is calculated here can be used as the baseline for the further evolution modelling of aviation systems.
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Figure 4.7: The linear relationships between traffic flow and si*sj for cities with in a) dij<500km, b) dij ϵ (900-1100km), c) dij ϵ (1800-2200km), and d) dij>2800km
4.5 Airline network evolution model
Many network evolution models have been discussed in section 3.3. In paper III, we try to explain the evolution process of the US airline system from different perspectives. For comparison, a theoretical model is proposed. As mentioned above, airline systems are similar to social networks, since it is needless to construct physical links between airports. It means the systems are less constrained by geographical factors. On the other hand, all airports are embedded in spaces, and it is inefficient and costly to operate long routes from a business perspective. This means the probability for an airport to connect with another one decreases with the distance between each other. This architecture works in the real world. A new airport would firstly connect with the nearest regional hub, then with other centralized neighbors. As a result, the airline evolution model can be set up in three steps.
There is two nodes (airports) and one link (airline) between them in the original state, and the node degree of each airport equals to 1. One airport i is added in this system per time step. The new airport is connected with an existing airport in terms of the attachment probability function as follows,
kd/ i j j ij (4-3)
Here kj is the degree of node j and dij measures the spatial distance between node i and j. φ is the weight coefficient of the degree centrality, whileθis the distance
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decay coefficient. The attachment probability function is set up based on the BA model and the spatial distance effect. Note that whenθ=0, this model becomes the BA model.
Recalculate the degree distribution after the attachment at time step t.
Based on this probability function, a new added airport would link with an existing node with the large degree and short spatial distance. When φ = 0, only distance is taken into account. The node degree becomes progressively important with the increase of φ. Fig. 4.9 describes four models based onθ=1 and different values of φ = 0, 0.5, 1, 2. Correspondingly, centralities also present distinct distributions.
(a) φ=0 (b) φ=0.5
(c) φ=1 (d) φ=2
Figure 4.8: Models generated in terms of θ=1 and a) φ=0, b) φ=0.5, c) φ=1 and d) φ=2.
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4.6 Identifying critical nodes in a network
In a network, there are always some important nodes, which play a critical role in the whole network. Detecting such nodes can impose significant impacts on network structure optimization, traffic monitoring, and emergency response.
In this thesis, exploring the correlation between node degree and node betweenness is used to find critical nodes in transportation systems. As discussed in section 2.2, node degree measures the number of immediate neighbors of a node, while node betweenness indicates the level of bridgeness of a node. Generally speaking, there is a positive correlation between the two measures. It means that nodes with a bigger degree value are supposed to present a higher betweenness centrality. However, this is not the case for critical nodes. A critical node has an extremely high betweenness compared with its degree values. This method is used to identify important transfer hubs of the aviation systems in paper III.
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5. Results and discussion
This chapter summarizes the main results of the four papers and discusses the contributions to the knowledge about urban complex network and related research fields. The relationships between the papers are illustrated in Fig. 1.1.
From a complex network perspective, both topological features and geometric characteristics are studied in these four papers. They complement each other with various focuses, and they progressively work together to achieve the research objectives.
5.1 Paper I: Reviewing studies on urban transportation networks
This paper presents a broad overview of the topological network structure of transportation systems from a complex network perspective. The objective of this paper is to lay the necessary foundations for the whole thesis from the theoretical and the empirical perspective. For the theoretical part, network representations and topological measures are summarized at a variety of spatial scales. For the empirical part, most research from the last decade is discussed in terms of railway networks, public transport networks, aviation networks, maritime networks, and urban street networks.
Different transportation systems have distinct ways of representation in terms of their topological characteristics. In comparison with planar networks, non-planar systems—for example, social networks, aviation systems, and maritime networks—are much easier to generalize. It is worth mentioning that the degrees in non-planar networks could be much bigger than those in planar networks due to the topological difference between them. This statement is also verified in papers II, III, and IV. Different representation for ground transportation systems can lead to distinct analysis results. To be specific, a dual approach depicts the functional structure while a primal approach keeps the geographical characteristics of a planar network. In paper IV, the primal approach is used to generalize urban street networks in Stockholm, Nanjing, and Toronto to reflect their spatial features.
There has been a huge amount of measures since the emergence of complex network theory. It is impossible and unnecessary to list all of them. In paper I, those measures that are most widely used in studying transportation systems are selected and classified into node centrality, local clustering, global measures, and cell properties. This classification makes it easy for readers to locate necessary measures according to specific aims.
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These theoretical reviews are followed by empirical observations. The authors compare various studies on different kinds of transportation systems, and they disclose the underlying relationships between them. It is interesting that some common features exist across distinct systems regardless of function, size, or representation approaches. For example, the small-world property is widely detected in empirical analyses. Especially for aviation systems, the average path length lies between 2.3 for national systems to 4.4 for worldwide systems. In comparison, urban public transportation systems present more complicated situations. Clearly, the generalization space significantly affects the studies. The value of average path length is much bigger in a L-space than the one in a P- space (Tab. 5.1).
To sum up, this paper can be referred to as an introductory article for transport geographers who are interested in complex network theory, as well as researchers with interdisciplinary backgrounds.
Table 5.1: Selected empirical analyses on transportation networks from a complex network perspective. (N denotes the number of nodes;
Name N
Indian railway network (Sen 587 － 0.69 2.16 Exponential P et al, 2003) China railway network (Li & 3915 － 0.835 3.5 Scale-free P Cai, 2007) China railway network 3110 － 0.2798 11.18 Shifted power L (Wang et al., 2008) law Polish railway network 1533 － 0.1681 19 Unclear L (Kurant & Thiran, 2006) － 0.6829 2.3 Exponential P
Swiss railway network 1613 － 0.0949 16.3 Unclear L (Kurant & Thiran, 2006) － 0.9095 3.6 Exponential P
22 Polish public transport 152-2811 2.48-3.08 0.032-0.161 6.83- Power-law L network (Sienkiewicz & 21.52 Holyst, 2005) 32.94-90.93 0.682-0.847 1.71- Exponential P 2.90 3 Chinese public transport 192-495 2.26-3.06 0.01-0.16 10.41- No universal L networks (Lu & Shi, 2007) 11.38 pattern 23.59-45.54 0.73-0.89 2.00- No universal P 2.18 pattern 4 Chinese bus transport 827-4374 44.46-84.01 0.73-0.78 2.53- Exponential P networks (Chen et al., 2007) 2.89
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3 Chinese bus-transport 1150-3938 2.88-3.22 0.09-0.21 7.13- Power-law L networks (Xu et al., 2007a) 12.56 41.06-94.19 0.73-0.78 2.54- Exponential P 2.66 14 urban public transport 1544-46244 － 0.01-0.13 7.08- Heavy-tail L networks (Ferber et al., 85.84 2007) － 0.69-0.95 2.26- Heavy-tail P 4.79 US flight network (Chi et al., 215 － 0.618 2.403 Two-regime － 2003) power-law World-wide airport network 3883 － 0.62 4.4 Truncated － (Guimera &d Amaral, 2004; power-law Indian2005) airport network 79 － 0.733 2.067 Scale-free － (Bagler, 2008) Europe airport network 467 － 0.38 2.80 Two-regime － (Paleari et al., 2010) power-law China airport network 144 － 0.49 2.34 Two-regime － (Paleari et al., 2010) power-law China’s ship-transport 162 3.11 0.54 5.86 Scale-free L network (Xu et al., 2007b) 162 8.27 0.83 3.87 Truncated P power-law Worldwide maritime 878 9.04 0.40 3.60 Truncated L transportation network (Hu power-law & Zhu, 2009) 878 28.44 0.71 2.66 Exponential P
Global cargo-ship network 951 76.4 0.49 2.5 Right-skewed L (Kaluza et al, 2010) Bulk dry carriers network 616 44.6 0.43 2.57 Right-skewed L (Kaluza et al, 2010)
5.2 Paper II: Analyzing topological and spatial structures of China’s aviation system
Aviation systems, as an even more important part in the current information society, plays an indispensable role in urban infrastructure systems, especially in long-distance commuting. Paper II investigates the topological structure and the geometric mechanisms of China’s aviation system.
In conjunction with the findings in paper I, China’s aviation network presents a small-world property with a clustering coefficient of 0.737 and an average path length of 2.108. A total of 78 percent of the city pairs in the network can reach each other by at most one transfer, while the maximum topological distance of the whole network equals 4, existing between some cities in Xinjiang Province and Tongren City of Guizhou Province. Similarly, China’s aviation system presents an evolving scale-free property as most study cases in paper I, which is attributed to the absence of dominant hubs in the system. Besides, as a weighted network in which the weekly flight frequency is introduced to measure the
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traffic volume between city pairs, the edge weight and node strength of CAS present a similar double Pareto pattern (Fig. 5.1).
Figure 5.1: The distributions of edge weight and node strength of CAS
Rabasz and Barabasi (2003) stated that distance-driven networks do not present the hierarchical structure; however, paper II describes the hierarchy of CAS by exploring the scaling relationship between the node centrality ki and the clustering coefficient ci. It can be explained by the industrial development in the last decade and the hub-and-spoke structure of CAS. Local airports are connected together to enhance their regional hubs and then weaved into the higher level of the system.
Besides, paper II also makes an effort to explore geometric features of CAS. Fig. 5.2 indicates that the flight flows are not evenly distributed in terms of distance. Instead, 80% of the traffic flows concentrate on distances ranging from 500km to 2000km, which can be regarded as medium-distance trips in CAS. It is easy to understand since the high-speed rail system has been taking an increasing number of market shares for short-haul trips. In the meantime, passengers prefer to take a transfer rather than a direct flight for long-distance trips considering the convenience and economy.
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Figure 5.2: The distribution of weekly flight flows on spatial distances in CAS
Fig. 5.2 conveys an intuitive impression on the distance decay effect in CAS. To detect the underlying mechanism, a gravitational model is introduced to detect the detailed mechanism from a statistical perspective. It is generally written as:
UijAiBjfd(,)()()() (5–1) ij It is well known that cities are a complex combination of various elements. To take different urban factors into account, three scenarios are considered in this work:
• Strength-based measurements: taking the weekly flight frequency between city i and city j as U(i, j), while the node strength of city i and city j as A(i) and B(j). • Population-based measurements: taking the weekly flight frequency between city i and city j as U(i, j), while the population of city i and city j as A(i) and B(j). • GDP-based measurements: taking the weekly flight frequency between city i and city j as U(i, j), while the GDP value of city i and city j as A(i) and B(j).
Obviously, these three scenarios respectively cast light on the node centrality, the demographic, and the socio-economic features. The statistical results based on the OLS regression approach are listed in Tab. 5.2. Three scenarios yield similar results. The strength-based and GDP-based measurements yield similar results for both the suitable distance range and the decay function. As mentioned above, short-haul trips in China’s domestic market are dominated by high-speed rail systems; thus, no clear spatial effect is observed for trips shorter than 500km. For middle- and long-distance trips, a scaling decay function is calculated for both scenarios with similar decay coefficients of -0.697 and -0.702. Compared
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with these two scenarios, the modeling for the population-based one is more complicated. It is found that a logarithmic hyperbolic model can better depict the distance decay effect than a power law model.
Table 5.2: Regression results of gravitation model for different A(i) and B(j) measurements Parameter Strength-based(si*sj) Population-based(pi*pj) GDP-based(gi*gj) Suitable distance(km) >474 >145 >577 1/d f(dij) ad* ae* Decay coefficient -0.697 — -0.92 -0.702 R2 0.956 0.988 0.892 0.911
The main contributions of paper II can be concluded as follows:
• This paper provides a novel perspective to explore the topological structure of China’s aviation system. Besides, the widely-adopted hub- and-spoke structure can also be explained from a complex network perspective. • This work describes the spatial distance effects of CAS. Intense competition from high-speed rail systems imposes significant effects on the spatial pattern of CAS as well. For medium- and long-distance trips, the distance decay effect is detected.
5.3 Paper III: Modeling the evolving structure of US’s airline network
Paper II only discusses CAS at a specific point in time to further validate the features and to better capture the dynamic properties of aviation systems. Paper III explores the evolving structure of US’s airline network (UAN) considering the availability of a completely historic dataset.
UAN is extracted by examining airport cities as nodes and flight routes as edges based on a 20-year range. It is worth pointing out that US’s airline industry underwent a comprehensive restructuring in 2001. The number of airports doubled from 403 in 2001 to 848 in 2002. In the meantime, many short- and middle-length routes were constructed to decrease operation cost and optimize network efficiency. Nevertheless, UAN keeps a stable structure from the topological perspective. The small-world property is detected throughout the study period. The distributions of centrality measures, including in-degree kin, out-degree kout, node strength si and node betweenness Bi, consistently present a double Pareto pattern as well, except that the edge betweenness Be follows a scaling distribution (Fig. 5.3).
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(a) (b) Figure 5.3: The evolution of distributions of node betweenness and edge betweenness of UAN during 1990-2010
On the other hand, although the distribution patterns were not affected by the industry restructuring, the average betweenness
This paper also investigates the hierarchical architecture of UAN. Unlike China’s aviation system, the hierarchy only exists for high-degree cities in UAN. This difference can be attributed to the more developed structure of UAN, especially with the emergence of low-degree airports after the industry restructuring in 2001.
Paper III introduces the method explained in section 4.6 to locate critical hubs in the system. Fig. 5.4 illustrates the correlation between the out-degree kout and the node betweenness Bi during 1990-2010. Obviously, there are some nodes that possess extremely high values of Bi compared with kout. Seattle is a case in point. This city is not prominent in connectivity, but it essentially acts as a regional hub in UAN even in the global aviation system.
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Anchorage
Seattle Anchorage Seattle
Honolulu Fairbanks Elko
Honolulu
Anchorage Anchorage
Fairbanks Seattle Fairbanks Seattle
Figure 5.4: Identifying the critical nodes in UAN by exploring the correlation between node degree and node betweenness
In paper II, the distance effect is captured in aviation systems, and a scaling model is detected as the distance decay function. On this basis, this paper further proposes a model to simulate the evolution process of aviation networks. The model is explained in section 4.5, and the attachment probability function can be written as:
ijjij kd/ (5–2)
Note that when equals to 0, the model becomes the BA model. With the increase of , the distance effect in constructing the network is becoming increasingly important. However, it is found that no significant difference is detected among corresponding models whenθranges from 0.5 to 1.5. In this case, the decay coefficient is set up as 1 in this paper for simplicity’s sake.
In Fig. 5.4, the authors calculate the network efficiency in terms of different values of φ based on this model. Note that in this study, the network efficiency is measured based on the shortest path length between node pairs instead of actual distances or flows on paths. It is because airline passengers care about the number of transfers rather than the spatial flight distance in the real world. Fig. 5.5 shows that the network efficiency decreases as the network evolves. The bigger the weight coefficient, the slower the network efficiency decreases. Moreover, it is found that the scenario generated in terms of φ=0.5 can best reflect the real situation. When the number of nodes in this scenario grows to
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900, the network efficiency decreases to 0.31. In comparison, the network efficiency of UAN equals 0.33 in 2009 and 2010, while the number of airports in the system stays around 891 over the same period.
Figure 5.5: Distributions of network efficiency centralities in terms of different values of φ=0 (black squares), 0.5 (red circles), 1 (green upward triangles), 2(blue downward triangles).
This paper also discusses the relationship between network efficiency and two distance measures, the characteristic path length L and average flight distance
In conclusion, the contribution of paper III lies in the evolution and dynamic perspective, which can be specifically explained as follows:
• In complex network theory, most studies on evolving structure of transportation system are limited on numerical modeling and simulation due to the availability of empirical data. This work makes a valuable step forward by exploring the dynamic process of US’s aviation system during 1990-2010. • Based on the findings of paper II, this paper introduces a distance-based model to simulate the evolution of aviation networks. It is helpful to explain many empirical observations.
5.4 Paper IV: Comparing the planar structures of three street networks
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Paper II and paper III discuss aviation systems from different perspectives. This paper shifts the focus from aviation systems to ground transportation systems. Urban street networks in Stockholm, Sweden; Nanjing, China; and Toronto, Canada are studied and compared. Note that the three cities are located on different continents and present distinct geographic contexts and evolving history. In brief, Stockholm is a small-sized and self-organized city, Nanjing is a middle-sized and self-organized city, and Toronto is a large-sized and planned city. It is expected that a comparative analysis from a complex network perspective can provide informative meaning.
This paper investigates basic planar features of the three networks (Tab. 5.3). Due to the absence of triangles in systems, the small-world property is not detected for all three networks. Although the number of nodes and edges of Nanjing are less than Toronto, the former presents the biggest number of total street length, clustering coefficient, α,β, andγ.This phenomenon conflicts with the conclusion that these coefficients increase with the city size (Levinson, 2012). It can be attributed to the self-organized mechanism of Nanjing, which has resulted in many long route segments in the city. The impacts of development mechanisms on urban patterns are also acquired in terms of distributions of segment length and cell area (Fig. 5.6). Self-organized cities show more heterogeneity in both measures. In comparison, Toronto presents a single peak distribution for the cell area.
Table 5.3: The basics of urban street networks in Stockholm, Nanjing and Toronto Name N E Ltotal cc Lgeo α β γ Ρ (km) (km/km2) Stockholm 2612 3603 971.3 0.052 40 0.190 1.379 0.460 5.17 Nanjing 6121 9275 6219.9 0.070 50.2 0.258 1.515 0.505 7.39 Toronto 26093 37123 5650 0.041 69.6 0.211 1.423 0.474 8.97 (Note: NV and NE respectively indicate the number of nodes and edges of the network, Ltotal is the total length of the streets; cc represents clustering coefficient of the network; Lgeo is the characteristic length path of the network and P calculates the road network density.)
350 120%
300 100% 250 80% 200 60% 150
Frequency Frequency 40% 100 Cumulative% 50 20%
0 0% 10000 210000 410000 610000 810000 1010000 Cell area (m2) Stockholm
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800 120% 700 100% 600 80% 500 400 60%
300 Frequency
Frequency 40% 200 Cumulative% 20% 100 0 0% 10000 210000 410000 610000 810000 1010000 Cell area (m2) Nanjing
9000 120% 8000 100% 7000 6000 80% 5000 60% 4000
Frequency 3000 Frequency 40% 2000 Cumulative% 20% 1000 0 0% 10000 210000 410000 610000 810000 1010000 Cell area (m2) Toronto Figure 5.6: Distribution histograms of the cell area of urban street systems of Stockholm, Nanjing, and Toronto.
The paper investigates node centralities in a primal space. The closeness centrality is discussed (Fig. 5.7) to explore the accessibility of three systems. It clearly depicts that the strictly planned city is more accessible from an overall perspective, although they all present a heavy-tailed distribution instead of a standard power law. This distribution shows a two-regime pattern, which can be divided in terms of a threshold value. From Stockholm to Toronto, the threshold value is decreasing while the size is increasing. This indicates that Stockholm and Nanjing have some dominant streets with the highest closeness, while Toronto possesses a more even distribution. When a city size increases, the average closeness decreases as well.
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Figure 5.7: Cumulative probability distributions of closeness centrality of (a) Stockholm, (b) Nanjing and (c) Toronto.
Besides, betweenness centrality depicts an interesting difference among street networks. Although the measure can successfully pick out the most important routes in the three networks, Toronto does present a more even distribution pattern compared to the other two cities. It means that the street network of Toronto is more robust regarding network safety, presenting a high tolerance for deliberate attacks, and a higher overall efficiency in traffic distribution.
(a) (b)
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(c) Figure 5.8: Betweenness distributions of the urban street systems of (a) Stockholm, (b) Nanjing and (c) Toronto.
The contributions of paper IV lie in many aspects:
• As a supplement to paper II and paper III, this paper explores ground transportation systems to make the whole research more systematic and complete. • This paper explores urban street networks from a complex network perspective. In contrast to a traditional geographical analysis of planar networks, this paper not only discusses spatial distance distributions of networks, but it also depicts complex network features. • This paper compares urban street networks and explains differences between various measures in terms of the urban size and urban form, providing quantitative measures that the strictly planned city is more efficient regarding accessibility and connectivity. The results have meaningful implications for urban planners and policy makers.
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6. Conclusions and future work
6.1 Conclusions
As the backbone of a city or country, transportation networks are an essential part of the infrastructure system; thus, they have attracted much attention from various subjects. Complex network theory provides a novel perspective for researchers to explore topological structures of transportation systems that are organized as a typical network pattern. This thesis focuses on analyzing and modeling transportation systems from a complex network perspective, including topological structures, geometric features, and dynamic evolution. The four papers in this thesis endeavor to discuss these topics step by step. They work together to make this thesis one of complete and valuable research.
The objectives of this thesis proposed in section 1.2.2 have respectively been addressed.
(1) This thesis reviews the current state of research on transportation systems from a complex network perspective. Paper I explicitly explains that a suitable representation approach is necessary for generalizing systems in a computer environment. For non-planar networks, an intuitive way is enough to generalize their topologies. Urban public transportation, such as metro, railway, and bus systems, should be generalized in terms of specific research objectives. For urban street networks, primal and dual representations are widely used to reflect topological and geometric features of transportation systems.
Various measures in terms of spatial scales, including node centralities, regional clustering, and global efficiency, are discussed and compared. They are followed by a summary of empirical analyses in this field during the last decade in terms of different types of transportation systems. This work is useful for researchers in geography who are interested in or would like to devote themselves to complex network theory. It can also be used as an introductory reference for novices with other backgrounds.
(2) Exploring topological features is of the utmost importance in understanding transportation systems. Many different measures are proposed and discussed to investigate the heterogeneity of networks. Although transportation systems are affected by various factors, they do have many more similar properties than one would expect. First, CAS and UAN both present the small-world structure regardless of size and socio-economic status, which is referred to as an efficient organization pattern in the real world. Second, although these centralities— including node degree, node straight, node betweenness, edge betweenness, node closeness—depict distinct features of a node or an edge in a network, their
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distributions share a similar pattern, e.g., the scaling relationship. In empirical analyses, a double Pareto distribution is detected instead of a perfect power law for most transportation systems. It is worth mentioning that a hierarchical structure is also detected for CAS and UAN. It overturns the conclusion that a distance-driven system does not present a hierarchy (Rabasz and Barabási, 2003).
(3) Following the topological structure, geometric features are an indispensable frontier in complex network theory. The distribution of route lengths for aviation systems and urban street networks are studied in this thesis. Most airline routes are between 500–1500km in aviation systems, and more short routes should be constructed with the system developing out of economy and efficiency considerations. In contrast, the distribution of lengths of urban street segments are more related to their development paths. A strictly planned city usually presents a more homogeneous pattern of spatial features than a self-organized one. In a sense, the former is believed to be more robust and efficient from the structure perspective.
Furthermore, the gravitational model is introduced to measure the spatial interactions in China’s aviation system. For medium- and long- distance trips, the distance decay effect can be captured by a general scaling relationship. That is to say, the number of traffic flow decreases while the spatial distance increases. Therefore, it is believed that the advantage in medium- and long- distance travels should be maintained and enhanced by optimizing the structure of networks. In contrast, short-distance travels are mostly initiated by special reasons, and they face increasing competition from ground transportation systems.
(4) The evolution of transportation systems is studied from both the theoretic and the modeling perspectives. It is proved that the basic scheme of an aviation network is established in the early stage of development and maintained during the whole evolution. The distance effects become increasingly important with the system evolving. The model proposed in paper III reflects the real scenario of the evolution of aviation systems. It provides a valuable baseline for network optimization and planning. Both the simulation and the empirical analysis show that—along with the evolution of aviation systems—the importance of some critical nodes is enhanced, although the average betweenness and the overall efficiency decreases. To sum up, this new understanding will have a broad impact on transportation optimization and ultimately on the development of cities and regions.
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6.2 Limitations and Future research
6.2.1 Limitations and challenges
The work in this thesis is far from complete, and there are ample opportunities to improve with further research. In paper II and paper III, only domestic flights are considered in the network. The absence of international data may underestimate some regional hubs and thus impose negative effects of depicting geometric and topological properties of the whole network.
Admittedly, only weekly flight numbers are not enough to explore spatial interactions on aviation networks. Considering the development of big data technologies, more and more online registration data becomes available now. Multi-source data should be introduced to better capture the spatial structure and human dynamics on the network.
There are many potential improvements on the network evolution model. More socio-economic features should be considered in the attachment probability function. At this time, the model only captures the evolution process of aviation systems; how to improve the model for a planar network is still a problem deserving more efforts.
6.2.2 Future research
Considering the complex network theory is a continuously developing subject, there are many aspects that are absolutely worthy of more attention.
First, there is a growing concern with human dynamics on transportation systems. As a relatively new subject, many issues in this field remain to be resolved. One fundamental question questions how human behave on transportation systems by asking, how do these mobility patterns emerge? Besides, some findings consistently manifest that the scaling property of human movement is highly affected by geographic structures of transport systems instead of human interests on a collective level (Jiang, 2009; Jiang et al., 2009). However, other researchers argued that it is human intentions lead to the heavy- tail nature (Rhee et al., 2011). As to the author’s knowledge, this is still an open question.
Although much progress has been made on modeling and optimizing transport systems with technology development, how to include geographic factors in a theoretical model remains a challenging topic. Transportation systems are
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embedded in space and interact with geographical contexts all the time. In this sense, geographers can contribute more to this subject than expected.
To sum up, most studies in this field are still in the theoretical stage. Much work lies ahead to explore potential applications, such as: how to use the simulation to optimize the real traffic systems; how to reduce congestion; and how to improve the emergency response mechanism in cities. These practical topics require researchers from different subjects to collaborate with each other to generate more data and technologies. An interdisciplinary perspective can be expected in this field.
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