Small-world properties of the Indian railway network

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.

Structural network characteristics affect epidemic severity and prediction in social contact networks

Understanding and mitigating epidemic spread in complex networks requires the measurement of structural network properties associated with epidemic risk. Classic measures of epidemic thresholds like the basic reproduction number (R0) have been adapted to account for the structure of social contact networks but still may be unable to capture epidemic potential relative to more recent measures based on spectral graph properties. Here, we explore the ability of R0 and the spectral radius of the social contact network to estimate epidemic susceptibility. To do so, we simulate epidemics on a series of constructed (small world, scale-free, and random networks) and a collection of over 700 empirical biological social contact networks. Further, we explore how other network properties are related to these two epidemic estimators (R0 and spectral radius) and mean infection prevalence in simulated epidemics. Overall, we find that network properties strongly influence epidemic dynamics and the subsequent utility of R0 and spectral radius as indicators of epidemic risk.

An approach for accessibility assessment and vulnerability analysis of national multimodal transport systems

A series of ambitious accessibility-oriented policies have been launched in the world, which center around improving accessibility via the national multimodal transport systems including road, airline, and railway. The national multimodal transport accessibility assessment is one of the "basic" analyses for the design and implementation of these policies, whereas existing national-scale accessibility studies either focus on single-mode or two-mode transport or ignore the schedule-dependent nature of railway and airline. This article models the integrated road, railway, airline, and walking transport as a four-layer network with the consideration of their interdependencies. An algorithm is then developed to accurately assess the travel time-based accessibility on the four-layer network with the consideration of the daily schedule of trains and flights. The proposed approach is applied to map accessibility to 363 cities in mainland China and analyze the optimal travel modes. In addition, this article investigates the travel time-based vulnerability of the national multimodal transport system in mainland China under the extreme storm recently occurred in Zhengzhou (July 2021). The findings in this work provide insightful suggestions for transport planners to design the national multimodal transport systems and for stakeholders to schedule travels.

Analysis of Weather Factors on Aircraft Cancellation using a Multilayer Complex Network

Airlines provide one of the most popular and important transportation services for passengers. While the importance of the airline industry is rising, flight cancellations are also increasing due to abnormal weather factors, such as rainfall and wind speed. Although previous studies on cancellations due to weather factors considered both aircraft and weather factors concurrently, the complex network studies only treated the aircraft factor with a single-layer network. Therefore, the aim of this study was to apply a multilayer complex network (MCN) method that incorporated three different factors, namely, aircraft, rainfall, and wind speed, to investigate aircraft cancellations at 14 airports in the Republic of Korea. The results showed that rainfall had a greater impact on aircraft cancellations compared with wind speed. To find out the most important node in the cancellation, we applied centrality analysis based on information entropy. According to the centrality analysis, Jeju Airport was identified as the most influential node since it has a high demand for aircraft. Also, we showed that characteristics and factors of aircraft cancellation should be appropriately defined by links in the MCN. Furthermore, we verified the applicability of the MCN method in the fields of aviation and meteorology. It is expected that the suggested methodology in this study can help to understand aircraft cancellation due to weather factors.

Network spreading among areas: A dynamical complex network modeling approach

With the outbreak of COVID-19, great loss and damage were brought to human society, making the study of epidemic spreading become a significant topic nowadays. To analyze the spread of infectious diseases among different areas, e.g., communities, cities, or countries, we construct a network, based on the epidemic model and the network coupling, whose nodes denote areas, and edges represent population migrations between two areas. Each node follows its dynamic, which describes an epidemic spreading among individuals in an area, and the node also interacts with other nodes, which indicates the spreading among different areas. By giving mathematical proof, we deduce that our model has a stable solution despite the network structure. We propose the peak infected ratio (PIR) as a property of infectious diseases in a certain area, which is not independent of the network structure. We find that increasing the population mobility or the disease infectiousness both cause higher peak infected population all over different by simulation. Furthermore, we apply our model to real-world data on COVID-19 and after properly adjusting the parameters of our model, the distribution of the peak infection ratio in different areas can be well fitted.

Absorbing phase transition in a unidirectionally coupled layered network

We study the contact process on layered networks in which each layer is unidirectionally coupled to the next layer. Each layer has elements sitting on (i) an Erdös-Réyni network, and (ii) a d-dimensional lattice. The top layer is not connected to any layer and undergoes an absorbing transition in the directed percolation class for the corresponding topology. The critical infection probability p_{c} for the transition is the same for all layers. For an Erdös-Réyni network the order parameter decays as t^{-δ_{l}} at p_{c} for the lth layer with δ_{l}∼2^{1-l}. This can be explained with a hierarchy of differential equations in the mean-field approximation. The dynamic exponent z=0.5 for all layers and ν_{∥}→2 for larger l. For a d-dimensional lattice, we observe a stretched exponential decay of the order parameter for all but the top layer at p_{c}.

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.

Multiplex network disintegration strategy inference based on deep network representation learning

Multiplex networks have attracted more and more attention because they can model the coupling of network nodes between layers more accurately. The interaction of nodes between layers makes the attack effect on multiplex networks not simply a linear superposition of the attack effect on single-layer networks, and the disintegration of multiplex networks has become a research hotspot and difficult. Traditional multiplex network disintegration methods generally adopt approximate and heuristic strategies. However, these two methods have a number of drawbacks and fail to meet our requirements in terms of effectiveness and timeliness. In this paper, we develop a novel deep learning framework, called MINER (Multiplex network disintegration strategy Inference based on deep NEtwork Representation learning), which transforms the disintegration strategy inference of multiplex networks into the encoding and decoding process based on deep network representation learning. In the encoding process, the attention mechanism encodes the coupling relationship of corresponding nodes between layers, and reinforcement learning is adopted to evaluate the disintegration action in the decoding process. Experiments indicate that the trained MINER model can be directly transferred and applied to the disintegration of multiplex networks with different scales. We extend it to scenarios that consider node attack cost constraints and also achieve excellent performance. This framework provides a new way to understand and employ multiplex networks.

A Novel Trajectory Based Prediction Method for Urban Subway Design

In recent years, with the development of various types of public transportation, they are also more and more closely connected. Among them, subway transportation has become the first choice of major cities. However, the planning of subway stations is very difficult and there are many factors to consider. Besides, few methods for selecting optimal station locations take other public transport in to consideration. In order to study the relationship between different types of public transportation, the authors collected and analyzed the travel data of subway passengers and the passenger trajectory data of taxis. In this paper, a method based on LeaderRank and Gaussian Mixed Model (GMM) is proposed to conduct subway station locations selection. In this method, the author builds a subway-passenger traffic zone weighted network and a station location prediction model. First, we evaluate the nodes in the network, then use the GPS track data of taxis to predict the location of new stations in future subway construction, and analyze and discuss the land use characteristics in the prediction area. Taking the design of the Beijing subway line as an example, the suitability of this method is illustrated.

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.

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.

A growth model for water distribution networks with loops

Water distribution networks (WDNs) expand their service areas over time. These growth dynamics are poorly understood. One facet of WDNs is that they have loops in general, and closing loops may be a functionally important process for enhancing their robustness and efficiency. We propose a growth model for WDNs that generates networks with loops and is applicable to networks with multiple water sources. We apply the proposed model to four empirical WDNs to show that it produces networks whose structure is similar to that of the empirical WDNs. The comparison between the empirical and modelled WDNs suggests that the empirical WDNs may realize a reasonable balance between cost, efficiency and robustness in terms of the network structure. We also study the design of pipe diameters based on a biological positive feedback mechanism. Specifically, we apply a model inspired by Physarum polycephalum to find moderate positive correlations between the empirical and modelled pipe diameters. The difference between the empirical and modelled pipe diameters suggests that we may be able to improve the performance of WDNs by following organizing principles of biological flow networks.

Measuring the effect of distance on the network topology of the Global Container Shipping Network

This paper examines how spatial distance affects network topology on empirical data concerning the Global Container Shipping Network (GCSN). The GCSN decomposes into 32 multiplex layers, defined at several spatial levels, by successively removing connections of smaller distances. This multilayer decomposition approach allows studying the topological properties of each layer as a function of distance. The analysis provides insights into the hierarchical structure and (importing and exporting) trade functionality of the GCSN, hub connectivity, several topological aspects, and the distinct role of China in the network's structure. It also shows that bidirectional links decrease with distance, highlighting the importance of asymmetric functionality in carriers' operations. It further configures six novel clusters of ports concerning their spatial coverage. Finally, it reveals three levels of geographical scale in the structure of GCSN (where the network topology significantly changes): the neighborhood (local connectivity); the scale of international connectivity (mesoscale or middle connectivity); and the intercontinental market (large scale connectivity). The overall approach provides a methodological framework for analyzing network topology as a function of distance, highlights the spatial dimension in complex and multilayer networks, and provides insights into the spatial structure of the GCSN, which is the most important market of the global maritime economy.

Analyzing Influencing Factors of Transfer Passenger Flow of Urban Rail Transit: A New Approach Based on Nested Logit Model Considering Transfer Choices

With the continuous improvement of the operation line network of urban rail transit, analyzing influencing factors of transfer passenger flow of urban rail transit is critical to improve the transfer demand analysis of urban rail transit. Using data collected from questionnaires, transfer passenger flow surveys and smart cards, this study proposes an approach base on nested logit passenger flow assignment model considering transfer choice behaviours of passengers. The transfer passenger flow at seven transfer stations in Nanjing is obtained. Subsequently, this study investigates the potential influencing factors of transfer passenger flow, including the node degree, geographic location (located in the city center, urban fringe, suburbs or suburban fringe), economic location (distance from the city center) and transportation locations (if it is close to a transportation hub or in combination with the hub) of rail transit transfer stations. The results indicate that a positive correlation between the transfer passenger flow and the node degrees of transfer stations. However, the relationship between transfer passenger flow and the economic, geographic, and transportation locations of transfer stations is not clear. The finding have reference value for the network design of rail transit transfer stations and transfer facilities, and provide reference for the analysis of passenger flow under network operation.

Same Stats, Different Graphs: Exploring the Space of Graphs in Terms of Graph Properties

Data analysts commonly utilize statistics to summarize large datasets. While it is often sufficient to explore only the summary statistics of a dataset (e.g., min/mean/max), Anscombe's Quartet demonstrates how such statistics can be misleading. We consider a similar problem in the context of graph mining. To study the relationships between different graph properties, we examine low-order non-isomorphic graphs and provide a simple visual analytics system to explore correlations across multiple graph properties. However, for larger graphs, studying the entire space quickly becomes intractable. We use different random graph generation methods to further look into the distribution of graph properties for higher order graphs and investigate the impact of various sampling methodologies. We also describe a method for generating many graphs that are identical over a number of graph properties and statistics yet are clearly different and identifiably distinct.

Reliability Optimization of a Railway Network

With the increase of the railway operating mileage, the railway network is becoming more and more complicated. We expect to build more railway lines to offer the possibility to offer more high quality service for the passengers, while the investment is often limited. Therefore, it is very important to decide the pairs of cities to add new railway lines under the condition of limited construction investment in order to optimize the railway line network to maximize the reliability of the railway network to deal with the railway passenger transport task under emergency conditions. In this paper, we firstly define the reliability of the railway networks based on probability theory by analyzing three minor cases. Then we construct a reliability optimization model for the railway network to solve the problem, expecting to enhance the railway network with the limited investment. The goal is to make an optimal decision when choosing where to add new railway lines to maximize the reliability of the whole railway network, taking the construction investment as the main constraint, which is turned to the building mileage limit. A computing case is presented based on the railway network of Shandong Province, China. The computing results prove the effectiveness of the model and the efficiency of the algorithm. The approach presented in this paper can provide a reference for the railway investors and builders to make an optimal decision.

Importance rankings of nodes in the China Railway Express network under the Belt and Road Initiative

China Railway Express (CR express) refers to the regular container trains transporting between China and European countries under the Belt and Road Initiative (BRI). This paper aims to conduct the systematic and in-depth research on the importance ranking of logistics nodes across the complex CR express network from China's national plan of the BRI perspective, with consideration of the connectivity between the 21st-Century Maritime Silk Road (MSR) and the Silk Road Economic Belt (SREB). In doing so, first, it sets up the complex network in the CR express transport. Second, based on the restraint coefficients in the theory of structural hole such as network scale, efficiency, grading and clustering co-efficiency, this paper applies the Multiple Attribute Decision Making (MADM) model in association with algorithm development in calculating the importance of the CR express nodes including both inland nodes and seaport nodes. The paper has three-fold contributions. In theory, it confirms the accuracy and practicability of the structural hole theory in the importance ranking of nodes in the complex network. In practice, it identifies important logistics nodes of CR express network across the BRI. Further, the findings in the paper contribute to optimising the structure of CR express transport and improving its network stability.

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.

Scaling in the recovery of urban transportation systems from massive events

Public transportation is a fundamental infrastructure for life in cities. Although its capacity is prepared for daily demand, congestion may rise when huge crowds gather in demonstrations, concerts or sport events. In this work, we study the robustness of public transportation networks by means of a stylized model mimicking individual mobility through the system. We find scaling relations in the delay suffered by both event participants and other citizens doing their usual traveling in the background. The delay is a function of the number of participants and the event location. The model is solved analytically in lattices proving the existence of scaling relations and the connection of their exponents to the local dimension. Thereafter, extensive and systematic simulations in eight worldwide cities reveal that a newly proposed measure of local dimension explains the exponents found in the network recovery. Our methodology allows to dynamically probe the local dimensionality of a transportation network and identify the most vulnerable spots in cities for the celebration of massive events.

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 Flow Network of Chinese Scientists and Its Driving Mechanisms Based on the Spatial Development Path of CAS and CAE Academicians

Scientists are a core element of regional technological innovation and economic growth. Academicians of the Chinese Academy of Sciences (CAS) and Chinese Academy of Engineering (CAE) are outstanding representatives of Chinese scientists. The purpose of this paper is to reveal the characteristics of Chinese scientists’ space migration, depict the structural topology of their flow network and to analyze the driving mechanisms behind the characteristics of their space migration. We mainly used spatial statistics, complex network analysis, and a modified location quotient. Our main conclusions are as follows: (1) The spatial distribution of Chinese scientists at different development stages tends to converge. The Chinese scientists’ birthplaces are highly concentrated in the eastern coastal areas and the Yangtze River basin. Education locations are highly consistent, with clusters of global high-level educational resources at home and abroad, and their workplaces are mainly distributed in economically developed cities. (2) In terms of network characteristics, the spatial heterogeneity of the growth of Chinese scientists is obvious. Only a few cities, such as Beijing, Shanghai, and Nanjing, have higher degrees of node centrality, strength centrality, and betweenness centrality, so their networks’ hierarchical structures are pyramid-shaped and their core-periphery structures are prominent. By identifying the roles of the cities as network nodes, we found that only Beijing plays accomplished and stable roles, whereas other cities play a single role. (3) Regarding driving mechanisms, Chinese scientists’ spatial migration has been influenced by national policy, regional economic level, education level, and individual traits at national, regional, and personal scales. Therefore, the combined effects of various forces have driven the evolution of the flow network of Chinese scientists.

Relaxation dynamics of semiflexible treelike small-world polymer networks

We study the relaxation dynamics of the polymer networks that are constructed based on a degree distribution specific to small-world networks. The employed building algorithm generates polymers with a large variety of architectures, thus allowing for a detailed study of the structural transition from a pure linear chain to dendritic polymer networks. This is done by varying a single parameter p, which measures the randomness in the degree of the network's nodes. The dynamics is investigated in the framework of the generalized Gaussian structures model by monitoring the influence of the parameter p and of the stiffness parameter q on the behavior of the relaxation quantities: averaged monomer displacement, storage modulus, and loss modulus. The structure properties of the constructed polymers are described by the mean-square radius of gyration. In the absence of stiffness, in the intermediate frequencies domain of the dynamical quantities we encounter different behaviours, such as a dendritic behavior followed by a linear one for very small values of p or a single well-marked dendritic behavior for higher values of p. The stiffness parameter q influences drastically the relaxation dynamics of these polymer networks and in general no evident scaling regions were encountered. However, for some values of the parameter set (p,q), such as (0.8,0.4), an extremely short constant slope region, less than one order of magnitude, was found.

General scores for accessibility and inequality measures in urban areas

In the last decades, the acceleration of urban growth has led to an unprecedented level of urban interactions and interdependence. This situation calls for a significant effort among the scientific community to come up with engaging and meaningful visualizations and accessible scenario simulation engines. The present paper gives a contribution in this direction by providing general methods to evaluate accessibility in cities based on public transportation data. Through the notion of isochrones, the accessibility quantities proposed measure the performance of transport systems at connecting places and people in urban systems. Then we introduce scores ranking cities according to their overall accessibility. We highlight significant inequalities in the distribution of these measures across the population, which are found to be strikingly similar across various urban environments. Our results are released through the interactive platform: www.citychrone.org, aimed at providing the community at large with a useful tool for awareness and decision-making.

Local Attachment Explains Small World-like Properties of Fibroblastic Reticular Cell Networks in Lymph Nodes

Fibroblastic reticular cells (FRCs) form a cellular network that serves as the structural backbone of lymph nodes and facilitates lymphocyte migration. In mice, this FRC network has been found to have small-world properties. Using a model based on geographical preferential attachment, we simulated the formation of a variety of cellular networks and show that similar small-world properties robustly emerge under such natural conditions. By estimating the parameters of this model, we generated FRC network representations with realistic topological properties. We found that the topological properties change markedly when the network is expanded from a thin slice to a three-dimensional cube. Typical small-world properties were found to persist as network size was increased. The simulated networks were very similar to two-dimensional and three-dimensional lattice networks. According to the used metrics, these lattice networks also have small-world properties, indicating that lattice likeness is sufficient to become classified as a small-world network. Our results explain why FRC networks have small-world properties and provide a framework for simulating realistic FRC networks.

Sustainable Development of Urban Rail Transit Networks: A Vulnerability Perspective

Urban rail transit (URT) systems are critical to modern public transportation services. Unfortunately, disruptions in URT systems can lead to dysfunction and threaten sustainable development. This study analyses URT network sustainability from a vulnerability perspective. Two network attack scenarios, including random attacks and intentional attacks, are designed to assess different kinds of disruptions to URT networks. Under random attacks, nodes are randomly removed from the network. In contrast, under intentional attacks, key nodes are identified and removed based on topological metrics and passenger flow volume. Then, URT network vulnerability is evaluated by quantifying the changes in network efficiency and structural integrity under the network attacks from a spatio-temporal point of view. The real-world case of the Shanghai URT system from 1993 to 2020 is used to illustrate the vulnerability in the evolution of the URT system. The results indicate that the URT network is increasingly fault-tolerant and structurally robust over time. The URT network is more vulnerable to intentional attacks than to random failures. Additionally, there are significant spatial differences in the vulnerability of Shanghai URT network. Stations in the central activity zone (CAZ) are more fault-tolerant and robust than stations located outside of the CAZ. Furthermore, stations with large centrality and greater passenger flow volumes and lines with many key nodes and greater passenger flow volumes, are vulnerable to disruptions in the URT networks. This study provides a new index to comprehensively quantify node centrality; it also fills a research gap by analysing the vulnerability of URT networks based on both longitudinal and spatial patterns. Finally, this paper highlights significant practical implications for the sustainable development of URT networks, as well as the sustainable development of public transportation services.

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.

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.

The Stochastic Transport Dynamics of a Conserved Quantity on a Complex Network

The stochastic dynamics of conserved quantities is an emergent phenomena observed in many complex systems, ranging from social and to biological networks. Using an extension of the Ehrenfest urn model on a complex network, over which a conserved quantity is transported in a random fashion, we study the dynamics of many elementary packets transported through the network by means of a master equation approach and compare with the mean field approximation and stochastic simulations. By use of the mean field theory, it is possible to compute an approximation to the ensemble average evolution of the number of packets in each node which, in the thermodynamic limit, agrees quite well with the results of the master equation. However, the master equation gives a more complete description of the stochastic system and provides a probabilistic view of the occupation number at each node. Of particular relevance is the standard deviation of the occupation number at each node, which is not uniform for a complex network. We analyze and compare different network topologies (small world, scale free, Erdos-Renyi, among others). Given the computational complexity of directly evaluating the asymptotic, or equilibrium, occupation number probability distribution, we propose a scaling relation with the number of packets in the network, that allows to construct the asymptotic probability distributions from the network with one packet. The approximation, which relies on the same matrix found in the mean field approach, becomes increasingly more accurate for a large number of packets.

Exploring the Hierarchical Structure of China’s Railway Network from 2008 to 2017

The analysis of transport networks is an important component of urban and regional development and planning. Based on the four main stages of China’s railway development from 2008 to 2017, this paper analyzes the hierarchical and spatial heterogeneity distribution of train flows. We found a high degree of spatial matching with the distribution of China’s main railway corridors. Then, using a classical community detection algorithm, this paper attempts to describe the functional structure and regional effects of China’s railway network. We also explore the impacts of construction policies and changes to train operations on the spatial organizing pattern and evolution of network hierarchies. The results of this empirical study reveal a clear pattern of independent communities, which in turn indicates the existence of a hierarchical structure in China’s railway network. The decreases in both the number of communities and average distance between community centers indicate that the newer high-speed rail services have shortened the connections between cities. In addition, the detected communities are inconsistent with China’s actual administrative divisions in terms of quantity and boundaries. The spatial spillover and segmentation effects cause the railway network in different regions to be self-contained. Finally, the detected communities in each stage can be divided into the categories of monocentric structure, dual-nuclei structure, and polycentric structure according to the number of extracted hubs. The polycentric structure is the dominant mode, which shows that the railway network has significant spatial dependence and a diversified spatial organization mode. This study has great significance for policymakers seeking to guide the future construction of high-speed rail lines and optimize national or regional railway networks.

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.

Flow rate of transport network controls uniform metabolite supply to tissue

Life and functioning of higher organisms depends on the continuous supply of metabolites to tissues and organs. What are the requirements on the transport network pervading a tissue to provide a uniform supply of nutrients, minerals or hormones? To theoretically answer this question, we present an analytical scaling argument and numerical simulations on how flow dynamics and network architecture control active spread and uniform supply of metabolites by studying the example of xylem vessels in plants. We identify the fluid inflow rate as the key factor for uniform supply. While at low inflow rates metabolites are already exhausted close to flow inlets, too high inflow flushes metabolites through the network and deprives tissue close to inlets of supply. In between these two regimes, there exists an optimal inflow rate that yields a uniform supply of metabolites. We determine this optimal inflow analytically in quantitative agreement with numerical results. Optimizing network architecture by reducing the supply variance over all network tubes, we identify patterns of tube dilation or contraction that compensate sub-optimal supply for the case of too low or too high inflow rate.

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.

Optimal paths on the road network as directed polymers

We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. We find that, to a good approximation, the optimal paths can be described as directed polymers in a disordered medium, which belong to the Kardar-Parisi-Zhang universality class of interface roughening. Comparing the scaling behavior of our data with simulations of directed polymers and previous theoretical results, we are able to point out the few characteristics of the road network that are relevant to the large-scale statistics of optimal paths. Indeed, we show that the local structure is akin to a disordered environment with a power-law distribution which become less important at large scales where long-ranged correlations in the network control the scaling behavior of the optimal paths.

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.

Circuity analyses of HSR network and high-speed train paths in China

Circuity, defined as the ratio of the shortest network distance to the Euclidean distance between one origin-destination (O-D) pair, can be adopted as a helpful evaluation method of indirect degrees of train paths. In this paper, the maximum circuity of the paths of operated trains is set to be the threshold value of the circuity of high-speed train paths. For the shortest paths of any node pairs, if their circuity is not higher than the threshold value, the paths can be regarded as the reasonable paths. With the consideration of a certain relative or absolute error, we cluster the reasonable paths on the basis of their inclusion relationship and the center path of each class represents a passenger transit corridor. We take the high-speed rail (HSR) network in China at the end of 2014 as an example, and obtain 51 passenger transit corridors, which are alternative sets of train paths. Furthermore, we analyze the circuity distribution of paths of all node pairs in the network. We find that the high circuity of train paths can be decreased with the construction of a high-speed railway line, which indicates that the structure of the HSR network in China tends to be more complete and the HSR network can make the Chinese railway network more efficient.

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.

The Structure of Interurban Traffic: A Weighted Network Analysis

We study the structure of the network representing the interurban commuting traffic of the Sardinia region, Italy, which amounts to 375 municipalities and 1600 000 inhabitants. We use a weighted network representation in which vertices correspond to towns and the edges correspond to the actual commuting flows among those towns. We characterize quantitatively both the topological and weighted properties of the resulting network. Interestingly, the statistical properties of the commuting traffic exhibit complex features and nontrivial relations with the underlying topology. We characterize quantitatively the traffic backbone among large cities and we give evidence for a very high heterogeneity of the commuter flows around large cities. We also discuss the interplay between the topological and dynamical properties of the network as well as their relation with sociodemographic variables such as population and monthly income. This analysis may be useful at various stages in environmental planning and provides analytical tools for a wide spectrum of applications ranging from impact evaluation to decision making and planning support.

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.

Physics of transportation: Towards optimal capacity using the multilayer network framework

Because of the critical role of transportation in modern times, one of the most successful application areas of statistical physics of complex networks is the study of traffic dynamics. However, the vast majority of works treat transportation networks as an isolated system, which is inconsistent with the fact that many complex networks are interrelated in a nontrivial way. To mimic a realistic scenario, we use the framework of multilayer networks to construct a two-layered traffic model, whereby the upper layer provides higher transport speed than the lower layer. Moreover, passengers are guided to travel along the path of minimal travelling time and with the additional cost they can transfer from one layer to another to avoid congestion and/or reach the final destination faster. By means of numerical simulations, we show that a degree distribution-based strategy, although facilitating the cooperation between both layers, can be further improved by enhancing the critical generating rate of passengers using a particle swarm optimisation (PSO) algorithm. If initialised with the prior knowledge from the degree distribution-based strategy, the PSO algorithm converges considerably faster. Our work exemplifies how statistical physics of complex networks can positively affect daily life.

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.

Phase transitions in Pareto optimal complex networks

The organization of interactions in complex systems can be described by networks connecting different units. These graphs are useful representations of the local and global complexity of the underlying systems. The origin of their topological structure can be diverse, resulting from different mechanisms including multiplicative processes and optimization. In spatial networks or in graphs where cost constraints are at work, as it occurs in a plethora of situations from power grids to the wiring of neurons in the brain, optimization plays an important part in shaping their organization. In this paper we study network designs resulting from a Pareto optimization process, where different simultaneous constraints are the targets of selection. We analyze three variations on a problem, finding phase transitions of different kinds. Distinct phases are associated with different arrangements of the connections, but the need of drastic topological changes does not determine the presence or the nature of the phase transitions encountered. Instead, the functions under optimization do play a determinant role. This reinforces the view that phase transitions do not arise from intrinsic properties of a system alone, but from the interplay of that system with its external constraints.

Generalization of the Ehrenfest urn model to a complex network

The Ehrenfest urn model is extended to a complex directed network, over which a conserved quantity is transported in a random fashion. The evolution of the conserved number of packets in each urn, or node of the network, is illustrated by means of a stochastic simulation. Using mean-field theory we were able to compute an approximation to the ensemble-average evolution of the number of packets in each node which, in the thermodynamic limit, agrees quite well with the results of the stochastic simulation. Using this analytic approximation we are able to find the asymptotic dynamical state of the system and the time scale to approach the equilibrium state, for different networks. The study is extended to large scale-free and small-world networks, in which the relevance of the connectivity distribution and the topology of the network for the distribution of time scales of the system is apparent. This analysis may contribute to the understanding of the transport properties in real networks subject to a perturbation, e.g., the asymptotic state and the time scale required to approach it.

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.

Spatially embedded growing small-world networks

Networks in nature are often formed within a spatial domain in a dynamical manner, gaining links and nodes as they develop over time. Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded. In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes. We find that such growth processes naturally result in networks with small-world features, including a short characteristic path length and nonzero clustering. We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space. The results do depend strongly on dimension, and higher-dimensional spaces result in shorter path lengths but less clustering.

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.

Random and targeted interventions for epidemic control in metapopulation models

In general, different countries and communities respond to epidemics in accordance with their own control plans and protocols. However, owing to global human migration and mobility, strategic planning for epidemic control measures through the collaboration of relevant public health administrations is gaining importance for mitigating and containing large-scale epidemics. Here, we present a framework to evaluate the effectiveness of random (non-strategic) and targeted (strategic) epidemic interventions for spatially separated patches in metapopulation models. For a random intervention, we analytically derive the critical fraction of patches that receive epidemic interventions, above which epidemics are successfully contained. The analysis shows that the heterogeneity of patch connectivity makes it difficult to contain epidemics under the random intervention. We demonstrate that, particularly in such heterogeneously connected networks, targeted interventions are considerably effective compared to the random intervention. Our framework is useful for identifying the target areas where epidemic control measures should be focused.

Mean field approximation for biased diffusion on Japanese inter-firm trading network

By analysing the financial data of firms across Japan, a nonlinear power law with an exponent of 1.3 was observed between the number of business partners (i.e. the degree of the inter-firm trading network) and sales. In a previous study using numerical simulations, we found that this scaling can be explained by both the money-transport model, where a firm (i.e. customer) distributes money to its out-edges (suppliers) in proportion to the in-degree of destinations, and by the correlations among the Japanese inter-firm trading network. However, in this previous study, we could not specifically identify what types of structure properties (or correlations) of the network determine the 1.3 exponent. In the present study, we more clearly elucidate the relationship between this nonlinear scaling and the network structure by applying mean-field approximation of the diffusion in a complex network to this money-transport model. Using theoretical analysis, we obtained the mean-field solution of the model and found that, in the case of the Japanese firms, the scaling exponent of 1.3 can be determined from the power law of the average degree of the nearest neighbours of the network with an exponent of −0.7.