Statistical analysis of 22 public transport networks in Poland

An Event-Link Network Model Based on Representation in P-Space

The L-space and P-space are two essential representations for studying complex networks that contain different clusters. Existing network models can successfully generate networks in L-space, but generating networks in P-space poses significant challenges. In this study, we present an empirical analysis of the distribution of the number of a line's nodes and the properties of the networks generated by these data in P-space. To gain insights into the operational mechanisms of the network of these data, we propose an event-link model that incorporates new nodes and links in P-space based on actual data characteristics using real data from marine and public transportation networks. The entire network consists of a series of events that consist of many nodes, and all nodes in an event are connected in the P-space. We conduct simulation experiments to explore the model's topological features under different parameter conditions, demonstrating that the simulation outcomes are consistent with the theoretical analysis of the model. This model exhibits small-world characteristics, scale-free behavior, and a high clustering coefficient. The event-link model, with its adjustable parameters, effectively generates networks with stable structures that closely resemble the statistical characteristics of real-world networks that share similar growth mechanisms. Moreover, the network's growth and evolution can be flexibly adjusted by modifying the model parameters.

Robustness evaluation of bus-subway composite network considering accessibility

The purpose of evaluating the robustness of the bus-subway composite network is to assess its ability to maintain normal operation under interference or failure. Most existing research focuses on the network's structure, often overlooking critical factors such as spatial information and site accessibility. To address this, ArcGIS and network mapping services are employed to enhance the network coupling method and construct a composite network model. Three types of site importance indices are proposed, direct reachability, inter-site reachability, and off-site accessibility. After simulating various deliberate and random attacks, we analyze changes in robustness indicators, including the maximum connectivity subgraph rate, network efficiency, and network reachability. The CRITIC weighting method is employed to assess the impact of various attack modes on network robustness. The study, focusing on the bus-subway composite network in the core areas of Dalian City, shows that: sites with high off-site accessibility, have the most significant impact on overall network performance. Network reachability reflects the accessibility level of the public transportation system. Network efficiency is most influenced by critical sites. It holds significant application value for the optimization and emergency management of the bus-subway composite network.

gtfs2net: Extraction of General Transit Feed Specification Data Sets to Abstract Networks and Their Analysis

Mass transportation networks of cities or regions are interesting and important to be studied to get a picture of the properties of a somehow better topology and system of transportation. One way to do this lies on the basis of spatial information of stations and routes. As we show however interesting findings can be gained also if one studies the abstract network topologies of these systems. To get these abstract types of networks, we have developed a tool that can extract a network of connected stops from General Transit Feed Specification feeds. As we found during the development, service providers do not follow the specification in coherent ways, so as a kind of postprocessing we have introduced virtual stations to the abstract networks that gather close stops together. We analyze the effect of these new stations on the abstract map as well.

Assessing the dynamic resilience of Urban Rail Transit Networks during their evolution using a ridership-weighted network

The assessment of the resilience of Urban Rail Transit Networks (URTNs) and the analysis of their evolutionary characteristics during network growth can help in the design of efficient, safe, and sustainable networks. However, there have been few studies regarding the change of resilience in long-term network development. As for the existing resilience studies, they rarely consider the entire cycle of accident occurrence and repair; furthermore, they ignore the changes in network transportation performance during emergencies. Moreover, the measurement metrics of the important nodes have not been comprehensively considered. Therefore, to remedy these deficiencies, this paper proposes a URTN dynamic resilience assessment model that integrates the entire cycle of incident occurrence and repair, and introduces the network transport effectiveness index E(Gw) to quantitatively assess the network resilience. In addition, a weighted comprehensive identification method of the important nodes (the WH method) is proposed. The application considers the Xi'an urban rail transit network (XURTN) during 2011-2021. The obtained results identify the resilience evolutionary characteristics during network growth. And longer peripheral lines negatively affect the resilience of XURTN during both the attack and the repair processes. The central city network improves the damage index Rdam and the recovery index Rrec by up to 123.46% and 11.65%, respectively, over the overall network. In addition, the WH method can comprehensively and accurately identify the important nodes in the network and their evolutionary characteristics. Compared to the single-factor and topological strategies, the Rdam is 1.17%~178.89% smaller and the Rrec is 1.68%~84.81% larger under the WH strategy. Therefore, this method improves the accuracy of the important node identification. Overall, the insights from this study can provide practical and scientific references for the synergistic development of URTN and urban space, the enhancement of network resilience, and the protection and restoration of important nodes.

Relation between the degree and betweenness centrality distribution in complex networks

The centrality measures, like betweenness b and degree k in complex networks remain fundamental quantities helping to classify them. It is realized from Barthelemy's paper [Eur. Phys. J. B 38, 163 (2004)10.1140/epjb/e2004-00111-4] that the maximal b-k exponent for the scale-free (SF) networks is η_{max}=2, belonging to SF trees, based on which one concludes δ≥γ+1/2, where γ and δ are the scaling exponents for the distribution functions of the degree and the betweenness centralities, respectively. This conjecture was violated for some special models and systems. Here we present a systematic study on this problem for visibility graphs of correlated time series, and show evidence that this conjecture fails in some correlation strengths. We consider the visibility graph of three models: two-dimensional Bak-Tang-Weisenfeld (BTW) sandpile model, one-dimensional (1D) fractional Brownian motion (FBM), and 1D Levy walks, the two latter cases are controlled by the Hurst exponent H and the step index α, respectively. In particular, for the BTW model and FBM with H≲0.5, η is greater than 2, and also δ<γ+1/2 for the BTW model, while the Barthelemy's conjecture remains valid for the Levy process. We assert that the failure of the Barthelemy's conjecture is due to large fluctuations in the scaling b-k relation resulting in the violation of hyperscaling relation η=γ-1/δ-1 and emergent anomalous behavior for the BTW model and FBM. Universal distribution function of generalized degree is found for these models which have the same scaling behavior as the Barabasi-Albert network.

The Evolution of China's Railway Network (CRN) 1999-2019: Urbanization Impact and Regional Connectivity

With the rapid development of China's economy and society, China's railway transportation system has been dramatically improved in terms of its scale and operational efficiency. To uncover the underlying relationship between urbanization and railway network structure, this paper examines the evolution of China's railway transportation system from 1999 to 2019 by applying complex network theory. The results show that China's railway network (CRN) has become more connected, more "small-world" and more heterogeneous since the beginning of the twenty-first century. Based on the train flow and train travel distance, the evolutionary course of CRN is found to undergo two apparent stages, with a turning point in 2007. By calculating the regional railway connection index (RRCI), it is revealed that the planned core cities in different regions act as bridges connecting the regions to the rest of the whole network.

Research on the Destruction Resistance of Giant Urban Rail Transit Network from the Perspective of Vulnerability

Giant urban rail transit (GURT) systems have been formed in many metropolises and play a critical role in addressing serious traffic congestion. Unfortunately, as a dynamic and complex system, the vulnerability of GURT networks under various failure scenarios will be more prominent as the network expansion continues. Thus, it is imperative to explore the complex structural characteristics of the network and improve the ability to deal with the disturbance of emergencies. In this study, the destruction resistance of GURT networks with scale growth is illustrated from a vulnerability perspective. Specifically, taking Shanghai rail transit (SHRT) system as an example, the network topology model is constructed using the Space L method, and the network structure characteristics are analyzed based on the complex network theory. In addition, five attack strategies are developed to represent random and targeted attacks during the simulation of network failure, and two metrics are determined to evaluate the network vulnerability. Some meaningful results have been obtained: (i) The Shanghai rail transit planning network (SHRTPN) has increased the network efficiency by more than 10% over the Shanghai rail transit operating network (SHRTON) and has effectively enhanced the network destruction resistance. (ii) The SHRT network is a small-world network and shows significant vulnerability under the targeted attacks. The failure of only 3% high betweenness stations in SHRTON can lead to a 66.2% decrease in the network efficiency and a 75.8% decrease in the largest connected component (LCC) ratio. (iii) Attacking stations will cause more severe network failures than attacking edges, and it is necessary to focus on preventing catastrophic network failure caused by the critical station’s failure breaking the threshold. Finally, the strategies for improving the destruction resistance of GURT networks are proposed. The findings of this research can provide an essential reference for the rational planning, safety protection, and sustainable construction of GURT systems.

Research on Optimization of Urban Public Transport Network Based on Complex Network Theory

The urban public transportation system is an important part of urban transportation, and the rationality of public transportation routes layout plays a vital role in the transportation of the city. Improving the efficiency of public transportation can have a positive impact on the operation of the public transportation system. This paper uses complex network theory and the symmetry of the up and down bus routes and stations to establish an urban public transit network model and calculates the probability of passengers choosing different routes in the public transit network according to passenger travel impedance. Based on passenger travel impedance, travel path probability and passenger travel demand, the links are weighed, and the network efficiency calculation method is improved. Finally, the public transit network optimization model was established with network efficiency as the objective function and solved by the ant colony algorithm. In order to verify the effectiveness of the model and the solution method, this paper selects areas in Nanguan District of Changchun City for example analysis. The result shows that the efficiency of the optimized network is 8.5% higher than that of the original network, which proves the feasibility of the optimized model and solution method.

MLCOR Model for Suppressing the Cascade of Edge Failures in Complex Network

As a decisive parameter of network robustness and network economy, the capacity of network edges can directly affect the operation stability and the construction cost of the network. This paper proposes a multilevel load–capacity optimal relationship (MLCOR) model that can substantially improve the network economy on the premise of network safety. The model is verified in artificially created networks including free-scale networks, small-world networks, and in the real network structure of the Shanghai Metro network as well. By numerical simulation, it is revealed that under the premise of ensuring the stability of the network from the destruction caused by initial internal or external damage on edge, the MLCOR model can effectively reduce the cost of the entire network compared to the other two linear load–capacity models regardless of what extent of the destruction that the network edges suffer initially. It is also proved that there exists an optimal tunable parameter and the corresponding optimal network cost for any BA and NW network topology, which can provide the reference for setting reasonable capacities for network edges in a real network at the stage of network planning and construction, promoting security and stability of network operation.

The Evolvement of Rail Transit Network Structure and Impact on Travel Characteristics: A Case Study of Wuhan

The expansion of the rail transit network has a positive impact on travel characteristics under spatial and temporal constraints by changing accessibility. However, few empirical studies have examined the longitudinal evolution of the impact of accessibility and travel characteristics. In this paper, a model of the Wuhan rail transit network is constructed and the evolution of the spatial pattern of accessibility over different periods is analyzed. The correlation of accessibility with rail transit travel characteristics is studied longitudinally to provide theoretical support for rail transit construction and traffic demand management. The study shows that: (1) Wuhan’s rail transit network has evolved from a tree to a ring, improving the operational efficiency. (2) The accessibility of Wuhan’s rail transit network has evolved into a circular structure, showing a decreasing trend away from the city center. (3) The change of accessibility greatly affects travel characteristics. The higher the accessibility, the higher the traffic volume, and the lower the travel frequency, the more residents travel during peak hours, and the shorter the travel distance. These findings are useful for gaining insight into public transportation demand in large cities, and thus for developing reasonable transportation demand management policies.

Structural Analysis in Transit System Using Network Theory Case of Guadalajara, Mexico

Structural analysis in a transit network is a key aspect used to evaluate in a planning process. In this sense, the use of network science was applied in this work to generate a framework of the main structural features of a transport network. In this case, an alternative transport network in Guadalajara, Mexico was taken as an example. The network properties selected were grade of accessibility, spatial friction, and vulnerability. In the case of the grade of accessibility, this propriety makes reference to the efficiency of the travel time that the network gives due to its structural features. The spatial friction measures how direct in terms of distance the trips that the network provides are, and the vulnerability relates to the ease with which the network can comprise its performance by affectations to their nodes or links. In this sense, this work presents a detailed methodology and a set of open-source tools that can be used to measure these key structural elements for decision making.

Link prediction of the world container shipping network: A network structure perspective

Although the world container shipping network (WCSN) has gradually been shaped with ever-increasing complexity in link evolution over the last decades, its evolving mechanism remains to be unveiled. This motivates us to explore the evolutionary pattern of the WCSN, which can be achieved by advancing the existing link prediction models. Using the k-shell decomposition method, the network hierarchy can be decomposed and evaluated by four indices which are KS-Salton, KS-AA, KS-RA, and KS-LRW. The results show that the network hierarchy depends largely on trade patterns and demonstrates certain geographic characteristics. Meanwhile, the KS-LRW index performs best and, therefore, is further simulated for the future WCSN by predicting its top 1677 potential edges, which significantly enhances the overall network connectivity and efficiency. These findings create profound implications for shipping companies to strategically reduce the trail cost for new lines by analyzing the network data.

Modelling virus spreading in ride-pooling networks

Urban mobility needs alternative sustainable travel modes to keep our pandemic cities in motion. Ride-pooling, where a single vehicle is shared by more than one traveller, is not only appealing for mobility platforms and their travellers, but also for promoting the sustainability of urban mobility systems. Yet, the potential of ride-pooling rides to serve as a safe and effective alternative given the personal and public health risks considerations associated with the COVID-19 pandemic is hitherto unknown. To answer this, we combine epidemiological and behavioural shareability models to examine spreading among ride-pooling travellers, with an application for Amsterdam. Findings are at first sight devastating, with only few initially infected travellers needed to spread the virus to hundreds of ride-pooling users. Without intervention, ride-pooling system may substantially contribute to virus spreading. Notwithstanding, we identify an effective control measure allowing to halt the spreading before the outbreaks (at 50 instead of 800 infections) without sacrificing the efficiency achieved by pooling. Fixed matches among co-travellers disconnect the otherwise dense contact network, encapsulating the virus in small communities and preventing the outbreaks.

Modular gateway-ness connectivity and structural core organization in maritime network science

Around 80% of global trade by volume is transported by sea, and thus the maritime transportation system is fundamental to the world economy. To better exploit new international shipping routes, we need to understand the current ones and their complex systems association with international trade. We investigate the structure of the global liner shipping network (GLSN), finding it is an economic small-world network with a trade-off between high transportation efficiency and low wiring cost. To enhance understanding of this trade-off, we examine the modular segregation of the GLSN; we study provincial-, connector-hub ports and propose the definition of gateway-hub ports, using three respective structural measures. The gateway-hub structural-core organization seems a salient property of the GLSN, which proves importantly associated to network integration and function in realizing the cargo transportation of international trade. This finding offers new insights into the GLSN’s structural organization complexity and its relevance to international trade.

It is crucial to understand the evolving structure of global liner shipping system. Here the authors unveiled the architecture of a recent global liner shipping network (GLSN) and show that the structure of global liner shipping system has evolved to be self-organized with a trade-off between high transportation efficiency and low wiring cost and ports’ gateway-ness is most highly associated with ports’ economic performance.

Road Landscape Morphology of Valley City Blocks under the Concept of “Open Block”—Taking Lanzhou City as an Example

The unique valley geographical environment and the congestion-prone road landscape make valley city traffic jammed easily. In this paper, under the background of “open blocks”, two open patterns, which correspond to two different road landscapes ("ideal grid opening" and "open under realistic conditions"), are proposed. Taking Lanzhou city as an example, six basic statistical characteristics are used to compare and analyze the changes of road network topology in blocks to find out which open pattern is more suitable for valley cities. The results show that the pattern "open under realistic conditions" has a significant effect on the improvement of network performance and capacity. Specifically, breaking the "large blocks" and developing the small-scale blocks help to alleviate the traffic pressure. Besides, the opening of blocks located along river valley has a more positive effect on improving road network performance than the blocks sited in the inner area of cities.

Exploring the Characteristics of an Intra-Urban Bus Service Network: A Case Study of Shenzhen, China

The urban bus service system is one of the most important components of a public transport system. Thus, exploring the spatial configuration of the urban bus service system promotes an understanding of the quality of bus services. Such an understanding is of great importance to urban transport planning and policy making. In this study, we investigated the spatial characteristics of an urban bus service system by using the complex network approach. First, a three-step workflow was developed to collect a bus operating dataset from a public website. Then, we utilized the P-space method to represent the bus service network by connecting all bus stop pairs along each bus line. With the constructed bus network, a set of network analysis indicators were calculated to quantify the role of nodes in the network. A case study of Shenzhen, China was implemented to understand the statistical properties and spatial characteristics of the urban bus network configuration. The empirical findings can provide insights into the statistical laws and distinct convenient areas in a bus service network, and consequently aid in optimizing the allocation of bus stops and routes.

The Evolution and Growth Patterns of the Road Network in a Medium-Sized Developing City: A Historical Investigation of Changchun, China, from 1912 to 2017

Understanding the evolution and growth patterns of urban road networks helps to design an efficient and sustainable transport network. The paper proposed a general study framework and analytical workflow based on network theory that could be applied to almost any city to analyze the temporal evolution of road networks. The main tasks follow three steps: vector road network drawing, topology graph generation, and measure classification. Considering data availability and the limitations of existing studies, we took Changchun, China, a middle-sized developing city that is seldom reported in existing studies, as the study area. The research results of Changchun (1912–2017) show the road networks sprawled and densified over time, and the evolution patterns depend on the historical periods and urban planning modes. The evolution of network scales exhibits significant correlation; the population in the city is well correlated with the total road length and car ownership. Each network index also presents specific rules. All road networks are small-world networks, and the arterial roads have been consistent over time; however, the core area changes within the adjacent range but is generally far from the old city. More importantly, we found the correlation between structure and function of the urban road networks in terms of the temporal evolution. However, the temporal evolution pattern shows the correlation varies over time or planning modes, which had not been reported

Urban Transit Network Properties Evaluation and Optimization Based on Complex Network Theory

Urban public transportation contributes greatly to sustainable urban development. An urban public transportation network is a complex system. It is meaningful for theory and practice to analyze the topological structure of an urban public transportation network and explore the spatial structure of an urban transportation network so as to mitigate and prevent traffic congestion and achieve sustainability. By examining the Xi’an bus network, the degree distribution, average path length, aggregation coefficient, and betweenness centrality of the bus station network were computed using models in complex network theory. The results show that the node degrees of the Xi’an bus network are unevenly distributed and present a polarization diagram with long average path length and high aggregation. A model based on betweenness and its solution method was developed to improve the public transportation network’s sustainability and discuss the possibility of optimizing the sustainability by network analyzing methods.

Hierarchical structure in the world's largest high-speed rail network

Presently, China has the largest high-speed rail (HSR) system in the world. However, our understanding of the network structure of the world's largest HSR system remains largely incomplete due to the limited data available. In this study, a publicly available data source, namely, information from a ticketing website, was used to collect an exhaustive dataset on the stations and routes within the Chinese HSR system. The dataset included all 704 HSR stations that had been built as of June, 2016. A classical set of frequently used metrics based on complex network theory were analyzed, including degree centrality, betweenness centrality, and closeness centrality. The frequency distributions of all three metrics demonstrated highly consistent bimodal-like patterns, suggesting that the Chinese HSR network consists of two distinct regimes. The results indicate that the Chinese HSR system has a hierarchical structure, rather than a scale-free structure as has been commonly observed. To the best of our knowledge, such a network structure has not been found in other railway systems, or in transportation systems in general. Follow-up studies are needed to reveal the formation mechanisms of this hierarchical network structure.

Cascading Failures and Vulnerability Evolution in Bus⁻Metro Complex Bilayer Networks under Rainstorm Weather Conditions

In recent years, the frequent occurrence of rainstorms has seriously affected urban–public transport systems. In this study, we examined the impact of rainstorms on the vulnerability of urban–public transport systems consisting of both ground bus and metro systems, which was abstracted into an undirected weighted Bus–Metro complex bilayer network (Bus–Metro CBN) and the passenger volume was regarded as its weight. Through the changes in the node scale, network efficiency, and passenger volume in the maximal connected component of the Bus–Metro CBN, we constructed a vulnerability operator to quantitatively calculate the vulnerability of the Bus–Metro CBN. Then, the flow-based couple map lattices (CMLs) model was proposed to simulate cascading failure scenarios of the Bus–Metro CBN under rainstorm conditions, in which the rainstorm is introduced through a perturbation variable. The simulation results show that under the condition of passenger flow overload, the network may have a two-stage cascading failure process. The impact analysis shows that there is a rainstorm intensity threshold that causes the Bus–Metro CBN to collapse. Meanwhile, we obtained the optimal node and edge capacity through capacity analysis. In addition, our analysis implies that the vulnerability of the Bus–Metro CBN network in most scenarios is mainly caused by the degradation of network structure rather than the loss of passenger flow. The network coupling strength analysis results show that the node coupling strength has greater potential to reduce the vulnerability than edge coupling strength. This indicates that traffic managers should prioritize controlling the mutual influence between bus stops (or metro stations) to reduce the vulnerability of the Bus–Metro CBN more effectively.

Measuring health of highway network configuration against dynamic Origin-Destination demand network using weighted complex network analysis

Ideal configuration or layout of highways should resemble the actual demands for the roads represented by Origin-Destination (OD) information. It would be beneficial if existing highways can be evaluated for their configurational fitness against the current demands, and newly planned highways can carefully be designed in terms of their layouts and topologies that would reflect the demands. Analysis techniques used for complex networks in the matured field of network theory can be applied for the highway layout health monitoring against the current OD information. This paper proposes a methodology of measuring the fitness of existing highways by comparing their structural configuration against conceptual OD networks using well-established techniques in network theory for complex networks. In the first phase, this paper conducts an empirical analysis and finds that both structural highway network and OD network follow the "power law" distribution as they are weighted by capacity and traffic volume respectively. It is also found that the power law coefficient of the OD network dynamically changes throughout the day and week. In the second phase, a noble methodology of weighting and measuring the health, of structural highway networks against OD networks by means of comparing their power law coefficients is proposed. It is found that the proposed method is effective at detecting deviations from ideal structural configurations associated with actual demands.

Exploring Railway Network Dynamics in China from 2008 to 2017

China’s high speed rail (HSR) network has been rapidly constructed and developed during the past 10 years. However, few studies have reported the spatiotemporal changes of railway network structures and how those structures have been affected by the operation of high speed rail systems in different periods. This paper analyzes the evolving network characteristics of China’s railway network during each of the four main stages of HSR development over a 10-year period. These four stages include Stage 1, when no HSR was in place prior to August 2008; Stage 2, when several HSR lines were put into operation between August 2008, and July 2011; Stage 3, when the network skeleton of most main HSR lines was put into place. This covered the period until January 2013. Finally, Stage 4 covers the deep intensification of several new HSR lines and the rapid development of intercity-HSR railway lines between January 2013, and July 2017. This paper presents a detailed analysis of the timetable-based statistical properties of China’s railway network, as well as the spatiotemporal patterns of the more than 2700 stations that have been affected by the opening of HSR lines and the corresponding policy changes. Generally, we find that the distribution of both degrees and strengths are characterized by scale-free patterns. In addition, the decreasing average path length and increasing network clustering coefficient indicate that the small world characteristic is more significant in the evolution of China’s railway network. Correlations between different network indices are explored, in order to further investigate the dynamics of China’s railway system. Overall, our study offers a new approach for assessing the growth and evolution of a real railway network based on train timetables. Our study can also be referenced by policymakers looking to adjust HSR operations and plan future HSR routes.

A collection of public transport network data sets for 25 cities

Various public transport (PT) agencies publish their route and timetable information with the General Transit Feed Specification (GTFS) as the standard open format. Timetable data are commonly used for PT passenger routing. They can also be used for studying the structure and organization of PT networks, as well as the accessibility and the level of service these networks provide. However, using raw GTFS data is challenging as researchers need to understand the details of the GTFS data format, make sure that the data contain all relevant modes of public transport, and have no errors. To lower the barrier for using GTFS data in research, we publish a curated collection of 25 cities' public transport networks in multiple easy-to-use formats including network edge lists, temporal network event lists, SQLite databases, GeoJSON files, and the GTFS data format. This collection promotes the study of how PT is organized across the globe, and also provides a testbed for developing tools for PT network analysis and PT routing algorithms.

Pendular behavior of public transport networks

In this paper, we propose a methodology that bears close resemblance to the Fourier analysis of the first harmonic to study networks subjected to pendular behavior. In this context, pendular behavior is characterized by the phenomenon of people's dislocation from their homes to work in the morning and people's dislocation in the opposite direction in the afternoon. Pendular behavior is a relevant phenomenon that takes place in public transport networks because it may reduce the overall efficiency of the system as a result of the asymmetric utilization of the system in different directions. We apply this methodology to the bus transport system of Brasília, which is a city that has commercial and residential activities in distinct boroughs. We show that this methodology can be used to characterize the pendular behavior of this system, identifying the most critical nodes and times of the day when this system is in more severe demanded.

A network-based framework for assessing infrastructure resilience: a case study of the London metro system

Modern society is increasingly dependent on the stability of a complex system of interdependent infrastructure sectors. It is imperative to build resilience of large-scale infrastructures like metro systems for addressing the threat of natural disasters and man-made attacks in urban areas. Analysis is needed to ensure that these systems are capable of withstanding and containing unexpected perturbations, and develop heuristic strategies for guiding the design of more resilient networks in the future. We present a comprehensive, multi-pronged framework that analyses information on network topology, spatial organization and passenger flow to understand the resilience of the London metro system. Topology of the London metro system is not fault tolerant in terms of maintaining connectivity at the periphery of the network since it does not exhibit small-world properties. The passenger strength distribution follows a power law, suggesting that while the London metro system is robust to random failures, it is vulnerable to disruptions on a few critical stations. The analysis further identifies particular sources of structural and functional vulnerabilities that need to be mitigated for improving the resilience of the London metro network. The insights from our framework provide useful strategies to build resilience for both existing and upcoming metro systems.

Comparative Analysis on Topological Structures of Urban Street Networks

Street systems are the backbone of cities. With global urbanization and economic development, street systems have undergone significant development along with the growth of cities. In this paper, the authors select three cities with varying sizes, histories, locations, and growth dynamics: Stockholm, Toronto, and Nanjing. We analyze topological structures of their public street systems based on GIS and complex network theory. Considering the planarity of street systems, we first calculate various topological measures, including α, β, and γ indices, and density. This is followed by comparing three centrality measures, i.e., degree, betweenness, and closeness in complex network theory. In this part, we investigate these characteristics of nodes and edges in a primal representation, and discuss their relations with urban growth mechanisms.

Effectiveness of rapid rail transit system in Beijing

The effectiveness of rapid rail transit system is analyzed using tools of complex network for the first time. We evaluated the effectiveness of the system in Beijing quantitatively from different perspectives, including descriptive statistics analysis, bridging property, centrality property, ability of connecting different part of the system and ability of disease spreading. The results showed that the public transport of Beijing does benefit from the rapid rail transit lines, and the benefit of different regions from RRTS is gradually decreased from the north to the south. The paper concluded with some policy suggestions regarding how to promote the system. This study offered significant insight that can help understand the public transportation better. The methodology can be easily applied to analyze other urban public systems, such as electricity grid, water system, to develop more livable cities.

Analysing human mobility patterns of hiking activities through complex network theory

The exploitation of high volume of geolocalized data from social sport tracking applications of outdoor activities can be useful for natural resource planning and to understand the human mobility patterns during leisure activities. This geolocalized data represents the selection of hike activities according to subjective and objective factors such as personal goals, personal abilities, trail conditions or weather conditions. In our approach, human mobility patterns are analysed from trajectories which are generated by hikers. We propose the generation of the trail network identifying special points in the overlap of trajectories. Trail crossings and trailheads define our network and shape topological features. We analyse the trail network of Balearic Islands, as a case of study, using complex weighted network theory. The analysis is divided into the four seasons of the year to observe the impact of weather conditions on the network topology. The number of visited places does not decrease despite the large difference in the number of samples of the two seasons with larger and lower activity. It is in summer season where it is produced the most significant variation in the frequency and localization of activities from inland regions to coastal areas. Finally, we compare our model with other related studies where the network possesses a different purpose. One finding of our approach is the detection of regions with relevant importance where landscape interventions can be applied in function of the communities.

Statistical Analysis of Bus Networks in India

In this paper, we model the bus networks of six major Indian cities as graphs in L-space, and evaluate their various statistical properties. While airline and railway networks have been extensively studied, a comprehensive study on the structure and growth of bus networks is lacking. In India, where bus transport plays an important role in day-to-day commutation, it is of significant interest to analyze its topological structure and answer basic questions on its evolution, growth, robustness and resiliency. Although the common feature of small-world property is observed, our analysis reveals a wide spectrum of network topologies arising due to significant variation in the degree-distribution patterns in the networks. We also observe that these networks although, robust and resilient to random attacks are particularly degree-sensitive. Unlike real-world networks, such as Internet, WWW and airline, that are virtual, bus networks are physically constrained. Our findings therefore, throw light on the evolution of such geographically and constrained networks that will help us in designing more efficient bus networks in the future.

User-based representation of time-resolved multimodal public transportation networks

Multimodal transportation systems, with several coexisting services like bus, tram and metro, can be represented as time-resolved multilayer networks where the different transportation modes connecting the same set of nodes are associated with distinct network layers. Their quantitative description became possible recently due to openly accessible datasets describing the geo-localized transportation dynamics of large urban areas. Advancements call for novel analytics, which combines earlier established methods and exploits the inherent complexity of the data. Here, we provide a novel user-based representation of public transportation systems, which combines representations, accounting for the presence of multiple lines and reducing the effect of spatial embeddedness, while considering the total travel time, its variability across the schedule, and taking into account the number of transfers necessary. After the adjustment of earlier techniques to the novel representation framework, we analyse the public transportation systems of several French municipal areas and identify hidden patterns of privileged connections. Furthermore, we study their efficiency as compared to the commuting flow. The proposed representation could help to enhance resilience of local transportation systems to provide better design policies for future developments.

Complex Network Theory Applied to the Growth of Kuala Lumpur's Public Urban Rail Transit Network

Recently, the number of studies involving complex network applications in transportation has increased steadily as scholars from various fields analyze traffic networks. Nonetheless, research on rail network growth is relatively rare. This research examines the evolution of the Public Urban Rail Transit Networks of Kuala Lumpur (PURTNoKL) based on complex network theory and covers both the topological structure of the rail system and future trends in network growth. In addition, network performance when facing different attack strategies is also assessed. Three topological network characteristics are considered: connections, clustering and centrality. In PURTNoKL, we found that the total number of nodes and edges exhibit a linear relationship and that the average degree stays within the interval [2.0488, 2.6774] with heavy-tailed distributions. The evolutionary process shows that the cumulative probability distribution (CPD) of degree and the average shortest path length show good fit with exponential distribution and normal distribution, respectively. Moreover, PURTNoKL exhibits clear cluster characteristics; most of the nodes have a 2-core value, and the CPDs of the centrality’s closeness and betweenness follow a normal distribution function and an exponential distribution, respectively. Finally, we discuss four different types of network growth styles and the line extension process, which reveal that the rail network’s growth is likely based on the nodes with the biggest lengths of the shortest path and that network protection should emphasize those nodes with the largest degrees and the highest betweenness values. This research may enhance the networkability of the rail system and better shape the future growth of public rail networks.

The multilayer temporal network of public transport in Great Britain

Despite the widespread availability of information concerning public transport coming from different sources, it is extremely hard to have a complete picture, in particular at a national scale. Here, we integrate timetable data obtained from the United Kingdom open-data program together with timetables of domestic flights, and obtain a comprehensive snapshot of the temporal characteristics of the whole UK public transport system for a week in October 2010. In order to focus on multi-modal aspects of the system, we use a coarse graining procedure and define explicitly the coupling between different transport modes such as connections at airports, ferry docks, rail, metro, coach and bus stations. The resulting weighted, directed, temporal and multilayer network is provided in simple, commonly used formats, ensuring easy access and the possibility of a straightforward use of old or specifically developed methods on this new and extensive dataset.

Anatomy and efficiency of urban multimodal mobility

The growth of transportation networks and their increasing interconnections, although positive, has the downside effect of an increasing complexity which make them difficult to use, to assess, and limits their efficiency. On average in the UK, 23% of travel time is lost in connections for trips with more than one mode, and the lack of synchronization decreases very slowly with population size. This lack of synchronization between modes induces differences between the theoretical quickest trip and the 'time-respecting' path, which takes into account waiting times at interconnection nodes. We analyse here the statistics of these paths on the multilayer, temporal network of the entire, multimodal british public transportation system. We propose a statistical decomposition - the 'anatomy' - of trips in urban areas, in terms of riding, waiting and walking times, and which shows how the temporal structure of trips varies with distance and allows us to compare different cities. Weaknesses in systems can be either insufficient transportation speed or service frequency, but the key parameter controlling their global efficiency is the total number of stop events per hour for all modes. This analysis suggests the need for better optimization strategies, adapted to short, long unimodal or multimodal trips.

Scaling in transportation networks

Subway systems span most large cities, and railway networks most countries in the world. These networks are fundamental in the development of countries and their cities, and it is therefore crucial to understand their formation and evolution. However, if the topological properties of these networks are fairly well understood, how they relate to population and socio-economical properties remains an open question. We propose here a general coarse-grained approach, based on a cost-benefit analysis that accounts for the scaling properties of the main quantities characterizing these systems (the number of stations, the total length, and the ridership) with the substrate's population, area and wealth. More precisely, we show that the length, number of stations and ridership of subways and rail networks can be estimated knowing the area, population and wealth of the underlying region. These predictions are in good agreement with data gathered for about subway systems and more than railway networks in the world. We also show that train networks and subway systems can be described within the same framework, but with a fundamental difference: while the interstation distance seems to be constant and determined by the typical walking distance for subways, the interstation distance for railways scales with the number of stations.

Network centrality of metro systems

Whilst being hailed as the remedy to the world's ills, cities will need to adapt in the 21(st) century. In particular, the role of public transport is likely to increase significantly, and new methods and technics to better plan transit systems are in dire need. This paper examines one fundamental aspect of transit: network centrality. By applying the notion of betweenness centrality to 28 worldwide metro systems, the main goal of this paper is to study the emergence of global trends in the evolution of centrality with network size and examine several individual systems in more detail. Betweenness was notably found to consistently become more evenly distributed with size (i.e. no "winner takes all") unlike other complex network properties. Two distinct regimes were also observed that are representative of their structure. Moreover, the share of betweenness was found to decrease in a power law with size (with exponent 1 for the average node), but the share of most central nodes decreases much slower than least central nodes (0.87 vs. 2.48). Finally the betweenness of individual stations in several systems were examined, which can be useful to locate stations where passengers can be redistributed to relieve pressure from overcrowded stations. Overall, this study offers significant insights that can help planners in their task to design the systems of tomorrow, and similar undertakings can easily be imagined to other urban infrastructure systems (e.g., electricity grid, water/wastewater system, etc.) to develop more sustainable cities.

A long-time limit for world subway networks

We study the temporal evolution of the structure of the world's largest subway networks in an exploratory manner. We show that, remarkably, all these networks converge to a shape that shares similar generic features despite their geographical and economic differences. This limiting shape is made of a core with branches radiating from it. For most of these networks, the average degree of a node (station) within the core has a value of order 2.5 and the proportion of k = 2 nodes in the core is larger than 60 per cent. The number of branches scales roughly as the square root of the number of stations, the current proportion of branches represents about half of the total number of stations, and the average diameter of branches is about twice the average radial extension of the core. Spatial measures such as the number of stations at a given distance to the barycentre display a first regime which grows as r(2) followed by another regime with different exponents, and eventually saturates. These results--difficult to interpret in the framework of fractal geometry--confirm and yield a natural explanation in the geometric picture of this core and their branches: the first regime corresponds to a uniform core, while the second regime is controlled by the interstation spacing on branches. The apparent convergence towards a unique network shape in the temporal limit suggests the existence of dominant, universal mechanisms governing the evolution of these structures.

Negative edges and soft thresholding in complex network analysis of resting state functional connectivity data

Complex network analyses of functional connectivity have consistently revealed non-random (modular, small-world, scale-free-like) behavior of hard-thresholded networks constructed from the right-tail of the similarity histogram. In the present study we determined network properties resulting from edges constrained to specific ranges across the full correlation histogram, in particular the left (negative-most) tail, and their dependence on the confound signal removal strategy employed. In the absence of global signal correction, left-tail networks comprised predominantly long range connections associated with weak correlations and were characterized by substantially reduced modularity and clustering, negative assortativity and γ<1 Deconvolution of specific confound signals (white matter, CSF and motion) resulted in the most robust within-subject reproducibility of global network parameters (ICCs~0.5). Global signal removal altered the network topology in the left tail, with the clustering coefficient and assortativity converging to zero. Networks constructed from the absolute value of the correlation coefficient were thus compromised following global signal removal since the different right-tail and left-tail topologies were mixed. These findings informed the construction of soft-thresholded networks, replacing the hard thresholding or binarization operation with a continuous mapping of all correlation values to edge weights, suppressing rather than removing weaker connections and avoiding issues related to network fragmentation. A power law adjacency function with β=12 yielded modular networks whose parameters agreed well with corresponding hard-thresholded values, that were reproducible in repeated sessions across many months and evidenced small-world-like and scale-free-like properties.

Evolution of Chinese airport network

With the rapid development of the economy and the accelerated globalization process, the aviation industry plays a more and more critical role in today's world, in both developed and developing countries. As the infrastructure of aviation industry, the airport network is one of the most important indicators of economic growth. In this paper, we investigate the evolution of the Chinese airport network (CAN) via complex network theory. It is found that although the topology of CAN has remained steady during the past few years, there are many dynamic switchings inside the network, which have changed the relative importance of airports and airlines. Moreover, we investigate the evolution of traffic flow (passengers and cargoes) on CAN. It is found that the traffic continues to grow in an exponential form and has evident seasonal fluctuations. We also found that cargo traffic and passenger traffic are positively related but the correlations are quite different for different kinds of cities.

Scaling of human behavior during portal browsing

We investigated flows of visitors migrating between different portal subpages. Two various portals were studied as weighted networks where nodes are portal subpages and edge weights are numbers of user transitions. Such networks differ from networks of portal subpages connected by hyperlinks prepared by portal designers. Distributions of link weights, node strengths, and times spent by visitors at one subpage follow power laws over several decades for data collected during two different days and for weekly data. The distribution of numbers P(z) of unique subpages visited during one session is exponential and there is a square-root dependence between the total number of transitions n during a single visit and the average z . A model of portal surfing is developed where the browsing process corresponds to a self-attracting walk on the weighted network with a short memory. Results of numerical simulation are in agreement with weekly and daily portal data, and our analytical approach fits empirical data in the center part of scaling regime.

Dynamics in scheduled networks

When studying real or virtual systems through complex networks theories, usually time restrictions are neglected, and a static structure is defined to characterize which node is connected to which other. However, this approach is oversimplified, as real networks are indeed dynamically modified by external mechanisms. In order to bridge the gap, in this work we present a scheduled network formalism, which takes into account such dynamical modifications by including generic time restrictions in the structure of an extended adjacency matrix. We present some of its properties and apply this formalism to the specific case of the air transportation network in order to analyze its efficiency. Real data are used at this point. We finally discuss on the applicability of this formalism to other complex systems.

Synchronization and spatiotemporal patterns in coupled phase oscillators on a weighted planar network

To reveal the relation between network structures found in two-dimensional biological systems, such as protoplasmic tube networks in the plasmodium of true slime mold, and spatiotemporal oscillation patterns emerged on the networks, we constructed coupled phase oscillators on weighted planar networks and investigated their dynamics. Results showed that the distribution of edge weights in the networks strongly affects (i) the propensity for global synchronization and (ii) emerging ratios of oscillation patterns, such as traveling and concentric waves, even if the total weight is fixed. In-phase locking, traveling wave, and concentric wave patterns were, respectively, observed most frequently in uniformly weighted, center weighted treelike, and periphery weighted ring-shaped networks. Controlling the global spatiotemporal patterns with the weight distribution given by the local weighting (coupling) rules might be useful in biological network systems including the plasmodial networks and neural networks in the brain.

Log-periodic oscillations due to discrete effects in complex networks

We show how discretization affects two major characteristics in complex networks: internode distances (measured as the shortest number of edges between network sites) and average path length, and as a result there are log-periodic oscillations of the above quantities. The effect occurs both in numerical network models as well as in such real systems as coauthorship, language, food, and public transport networks. Analytical description of these oscillations fits well numerical simulations. We consider a simple case of the network optimization problem, arguing that discrete effects can lead to a nontrivial solution.

Empirical analysis of the ship-transport network of China

Structural properties of the ship-transport network of China (STNC) are studied in the light of recent investigations of complex networks. STNC is composed of a set of routes and ports located along the sea or river. Network properties including the degree distribution, degree correlations, clustering, shortest path length, centrality, and betweenness are studied in different definitions of network topology. It is found that geographical constraint plays an important role in the network topology of STNC. We also study the traffic flow of STNC based on the weighted network representation, and demonstrate the weight distribution can be described by power-law or exponential function depending on the assumed definition of network topology. Other features related to STNC are also investigated.

Geographical networks evolving with an optimal policy

In this article we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a node, having randomly assigned coordinates in a 1x1 square, is added and connected to a previously existing node i, which minimizes the quantity ri2/kialpha, where ri is the geographical distance, ki the degree, and alpha a free parameter. The degree distribution obeys a power-law form when alpha=1, and an exponential form when alpha=0. When alpha is in the interval (0, 1), the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge length will sharply increase when alpha exceeds the critical value alphac=1, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.

Extraction and analysis of traffic and topologies of transportation networks

The knowledge of real-life traffic patterns is crucial for a good understanding and analysis of transportation systems. These data are quite rare. In this paper we propose an algorithm for extracting both the real physical topology and the network of traffic flows from timetables of public mass transportation systems. We apply this algorithm to timetables of three large transportation networks. This enables us to make a systematic comparison between three different approaches to construct a graph representation of a transportation network; the resulting graphs are fundamentally different. We also find that the real-life traffic pattern is very heterogenous, in both space and traffic flow intensities, which makes it very difficult to approximate the node load with a number of topological estimators.

Layered complex networks

Many complex networks are only a part of larger systems, where a number of coexisting topologies interact and depend on each other. We introduce a layered model to facilitate the description and analysis of such systems. As an example of its application, we study the load distribution in three transportation systems, where the lower layer is the physical infrastructure and the upper layer represents the traffic flows. This layered view allows us to capture the fundamental differences between the real load and commonly used load estimators, which explains why these estimators fail to approximate the real load.