|
| Author and Study time | Case city | PTN type / Whether multilayered systems are considered | Modeling method | Network node size1 | Complexity indicator | Major contribution or conclusion |
|
Latora et al.[34]
2002 | Boston | Subway, Bus / Yes | Space L | 124 | Average shortest path length, Clustering coefficient, Network efficiency, Network cost | Only subway network is not a closed system, but the extended PTN (subway plus bus) is a generic closed transportation system that can exhibit the small-world behavior. As a result, the diffusion of small-world networks can be interpreted as the need to create networks that are both globally and locally efficient. |
Seaton et al.[35]
2004 | Boston, Vienna | Subway / No | Space P | 124, 76 | Degree, Average shortest path length, Clustering coefficient | Two subway networks are investigated through comparing the properties of networks having different architecture with each other and with those of the appropriate random bipartite model, which extends the application of bipartite analysis from social affiliation network to technological network. |
Wu et al.[36]
2004 | Beijing | Bus / No | Space L | 441 | Growth characteristic, Preferential attachment characteristic, Clustering characteristic, Degree and cumulative degree distribution | The cumulative degree distribution follows a power-law, indicating a scale-free topological structure. This characteristic makes network can resist random failures successfully, if the hub nodes are controlled well. |
Ferber et al.[37]
2005 | Berlin, Dusseldorf, Paris | Bus, Tram, and Subway / Yes | Space L | 2952, 1615, 4003 | Degree and degree distribution, Direct neighbourhood size of a given station and its integrated distribution, Segment distribution | Through fitting the distributions of various complexity indicators (following Zipf law distribution), all PTNs are examined to have the scale-free structure. Additionally, the possibility of new scaling laws that govern intrinsic properties of common subsets of stations serviced by many routes, is discussed. |
Sienkiewicz et al.[38]
2005 | 22 cities in Poland | Bus, Tram / Yes | Space L, Space P | 152-2811 | Degree and degree distribution, Degree-degree correlations, Betweenness, Average shortest path length, Clustering coefficient, Network assortativity coefficient | The degree distributions of all PTNs follow a power-law or an exponential function, the path length distributions are given by asymmetric and unimodal functions, and all PTNs exhibit the small-world behavior and are hierarchically organized. |
Zhao et al.[39]
2005 | Beijing | Bus / No | Modified Space L, Space P, Modified Space R | 4177 (L,P);
616 (R) 2 | Degree and degree distribution, Strength and strength distribution, Average shortest path length | With Modified Space L representation method, the edge weight is the number of routes passing through this edge. With Modified Space R representation method, the node weight is the number of same stations between two routes.
Some routes have a transit function, some stations have a central function, and node weight distributions differ significantly from other weighted networks. |
Ferber et al.[40]
2007 | Berlin, Paris and other twelve cities | Bus, Electric trolleybus, Ferry, Subway, Tram, Urban train / Yes | Modified Space L, Space L, Space P, Space R | 1544-46244 | Degree and degree distribution, Average shortest path length, Clustering coefficient, “Harness” distribution | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
The proposed “Harnessing” effect is a first example in this study area. Additionally, an evolutionary model of PTN based on effectively interacting self-avoiding walks that reproduces the key features, is subsequently proposed and simulated. |
Xu et al.[41]
2007 | Beijing, Shanghai, Nanjing | Bus / No | Space L, Space P, Modified Space R | 3938, 2063, 1150 (L,P);
516, 501, 174 (R) | Degree and degree distribution, Distribution for the number of routes that service each station, Strength and strength distribution, Clustering coefficient | With Modified Space R representation method, the node weight is the number of same stations between two routes.
With Space L representation method, degree distribution and distribution for the number of routes that service each station both obey power-law, while the cumulative degree distribution under P space follows an exponential distribution. Small-world behavior is observed in both topologies, but it is much more distinct in the space P, as well as the hierarchical structure of network. Furthermore, with modified Space R representation method, the BTN is regarded as a weighted network with routes mapped to nodes and number of common stations to weights between routes, so that a heavy tailed power-law for the weight distribution and a linear dependence between the strength and degree are observed. |
Chen et al.[42]
2007 | Nanjing, Hangzhou, Beijing, Shanghai | Bus / No | Space P | 827-4374 | Degree and cumulative degree distribution, Average shortest path length, Clustering coefficient, Cumulative distribution of the number of routes a station joins, Cumulative distribution of the number of stations in a route | The cumulative degree distribution follows an exponential function, the cumulative distribution of the number of routes a station joins follows a asymmetric and unimodal function, and the cumulative distribution of the number of stations in a route follows an exponential function. |
Lu et al.[43]
2007 | Dalian, Langfang, Jining | Bus / No | Modified Space L, Space P, Modified Space R | 495, 192, 449 (L,P);
93, 16, 31 (R) | Degree and degree distribution, Strength and strength distribution, Average shortest path length, Clustering coefficient | With Modified Space L representation method, the edge weight is the number of routes passing through this edge. With Modified Space R representation method, the node weight is the number of same stations between two routes.
Network topological parameters have significantly effects on the accessibility, convenience, and terrorist security capability of BTN, which is considered as a new path to solve traffic system problem. |
Qin et al.[44]
2008 | Shanghai | Bus / No | Modified Space L | 1997 | Strength and strength distribution, Strength-degree correlations, Weighted clustering coefficient, Degree-degree correlations | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
There is a superlinear relationship between the station saturated transit capacity and the connection density between stations. |
Yan et al.[45]
2008 | Shijiazhuang | Bus / No | Space P | 500-1033 | Mean number of stations per route, Degree and cumulative degree distribution, Average shortest path length, Clustering coefficient | Through the empirical analysis on the evolution of BTN topological characteristics from 1996 to 2008, it can get the conclusion that the BTN is analyzed as a small-world, accelerated growing network, and a hybrid between a scale-free and a random network. |
Sun et al.[46]
2009 | Qingdao | Bus / No | Modified Space R | 129 3 | Degree and degree distribution, Connected edge weight distribution, Relationship between length and degree of routes, Average shortest path length, Clustering coefficient | Modified Space R representation method, the node weight is the number of same stations between two routes.
Bus transit route network has the small-world characteristic, and the length of route is positively correlated with its degree. However, the change of degree is significantly lower than that of length, so the effect of increasing length to increase transfer times is not obvious. |
Ferber et al.[47]
2009 | Berlin, Paris and other twelve cities | Bus, Electric trolleybus, Ferry, Subway, Tram, Urban train / Yes | Modified Space L, Space L, Space P, Space R | 1494-44629 | Degree and cumulative degree distribution, Clustering coefficient, Network assortativity coefficient, Average shortest path length, Betweenness centrality, Cumulative “Harness” distribution | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
This study covers BTN, tram network, and subway network, and it can get conclusion that the degree distributions of all PTNs follow an exponential or power-law distribution. |
Berche et al.[48]
2009 | Istanbul, Taipei, Los Angeles, Paris | Bus, Subway, Tram / Yes | Modified Space L | 4043, 5311, 46244, 4003 | Cumulative “Harness” distribution | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
The “Harness” distribution of the selected PTNs can be described by power-laws. Furthermore, to mimic a harness effect, one-dimensional and two-dimensional models of randomly placed routes modeled by different types of walks are studied, thus getting that in one dimension an analytic treatment is successful, but not in the two dimension. |
Soh et al.[49]
2010 | Singapore | Bus, Urban rail transit / Yes | Modified Space L | 4139 | Degree and cumulative degree distribution, Strength and strength distribution, Weighted clustering coefficient, Degree-degree correlations, Eigenvector centrality, Average shortest path length, Network assortativity coefficient | With Modified Space L representation method, the edge weight is the actual passenger flow based on the card data.
The topological centralities of the URTN are equal during weekdays and weekends, but weighted centralities can differ significantly, particularly for nodes within the central business district. The BTN appears to possess topological hierarchy, with a clustering spectrum that decays with degree following a power-law; and its weighted degree similarity spectrum illustrates a slightly disassortative behavior. |
Wang et al.[50]
2011 | Hangzhou | Bus / No | Space L, Space P, Space R | 1404 (L,P);
328 (R) | Degree and cumulative degree distribution, Average shortest path length, Clustering coefficient, Community characteristic, Spreading characteristic | The BTN is analyzed as a small-world network which has an exponential degree distribution, obvious community structure, and strong epidemic spreading capability. |
Yang et al.[51]
2011 | Beijing, Shanghai, Hangzhou | Bus / No | Space P | 1404-6235 | Average number of stations (node) of each bus route (clique), Average number of new / old stations (node) in the new bus route (clique), Network diameter, Average shortest path time coefficient, Average shortest path length, Clustering coefficient, Cumulative distribution of some network properties | Based on the proposed ideal n-depth clique network model and m-depth community, a new BTN model is established. This new model is proved to be closer to the real BTN, and can be applied directly to real BTN optimization. |
Sui et al.[52]
2012 | Beijing, Hong Kong and other seven cities | Bus / No | Modified Space L | 873-4693 | Degree and degree distribution, Strength and strength distribution, Average length of routes, Average Euclidean distance between starting and ending stations of one route, Average coefficient of space tortuosity, Average stations number of route, Average routes number at station, Average shortest path length, Clustering coefficient | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
Network space evolution is explored by using empirical evidence and a simulation model validated on that data. Based on the statistical analyses on topological and spatial attributes, it can get the conclusion that an evolution network with traffic demands characterized by power-law numerical values which distribute in a mode of concentric circles, tallies well with these nine cities. |
Zhen et al.[53]
2012 | Beijing | Bus / No | Modified Space L | 9618 | Degree, Strength and cumulative strength distribution, Carrying pressure, Regional central node, Average shortest path length, Clustering coefficient | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
Two kinds of “key nodes” by using the high carrying pressure analysis and the extract regional central node analysis are found. |
Zhen et al.[54]
2012 | Chengdu | Bus / No | Modified Space P | 214 | Weighted shortest path length | With Modified Space P representation method, the edge weight is the actual distance between two adjacent stations that is calculated by spatial coordinates embedding.
A shortest path calculation algorithm considering both transfer time and path length is designed, guaranteeing that the path is the shortest based on the premise of having the minimum transfer times. |
Guo et al.[55]
2013 | Beijing, Wuhan, Tianjin, Taiyuan | Bus / No | Space L, Modified Space L | 1136-9182 | Degree and cumulative degree distribution, Strength and cumulative strength distribution, Average shortest path length, Clustering coefficient, Fractal dimensions of network | With Modified Space L representation method, the edge weight is the number of routes passing through this edge.
Considering the geographical embedding on modeling BTN, the network scaling properties is focused on by fractal analysis, as well as scaling property of the correlation between stations and the distribution of station weight. As a result, it can get the conclusions that this modified modeling method can reflect the human movement between stations indirectly, and show the heterogeneous property of human activity between different stations. |
Huang et al.[56]
2013 | Beijing | Bus / No | Space L, Modified Space L | 346 | Degree and degree distribution, Edge weight distribution, Strength and strength distribution, Weighted clustering coefficient, Weighted average shortest path length, Weighted average nearest neighbors’ degree distribution | With Modified Space L representation method, the edge weight is the route passenger flow.
The degree and strength distribution both follow power-law functions, and the weighted clustering coefficient and weighted average shortest path length are relatively small, showing that the weighted PTN is analyzed as a scale-free network. |
Leng et al.[57]
2014 | Beijing | Subway / No | Space L | 64-226 | Degree, Average shortest path length, Clustering coefficient | Based on the analysis of the evolution rule of subway topological characteristics during the high-speed development period (from 2007 to 2013), a new growth model composed by an expanding mode and an intensifying mode is proposed. |
Ding et al.[58]
2015 | Kuala Lumpur | Urban rail
transit / No | Space L | 41-193 | Degree and cumulative degree distribution, Average shortest path length, Network efficiency, Clustering coefficient, Degree centrality, Closeness centrality, Betweenness centrality | Through analyzing the evolution of topological structure and future growth of the URTN from 1995 to 2017, it can get conclusions that the number of stations and edges are linearly related with each other and follows a heavy-tailed distribution; the cumulative distributions of the degree and the average shortest path length are respectively exponential distribution and normal distribution; the cumulative distributions of the closeness centrality and the betweenness centrality are respectively normal distribution and exponential distribution. These conclusions reveal that the growth of URTN may be based on the stations with the shortest path length; the network protection should focus on the stations with the largest degree or betweenness. |
Luo et al.[59]
2015 | Beijing | Bus, Subway / Yes | Space L, Space P | - | Degree and cumulative degree distribution, Average shortest path length, Network efficiency, Clustering coefficient | A composite network of subway and bus is established, thus showing that the composite network has the characteristics of small-world and scale-free network. Additionally, it had proven that the high-efficiency connection and coordination between subway and bus is the basis for fully using the public transportation system. |
Alessandretti et al.[60]
2016 | Paris, Toulouse, Nantes, Strasbourg | Bus, Urban rail transit and Car / Yes | A method combined by user-based multi-edge and Space P | - | Time distributions for each distance, Travel time factor, Pattern extraction, Pattern extraction | With the combined representation method, the edge weight is represented by a time function and corresponds to the average travel time on the actual route.
A novel user-based representation of PTN is proposed for providing better design policies for future developments. This novel methodology combines representations, accounting for the presence of multiple routes 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. |
Dimitrov et al.[61]
2016 | Auckland | Bus, Train, Ferry / Yes | Space L | 6039 | Degree and degree distribution, Average shortest path length, Clustering coefficient | A new method for examining and analyzing topological characteristics of PTN through a combination of computer programming, statistical data and large-network analyses is proposed, thus getting that the PTN is best fitted by an exponential rather than a power-law function, verifying that the PTN is neither random nor scale-free network, but a mixture of the two types networks. |
Xu et al.[62]
2016 | Huangshi | Bus / No | Space L, Space P | 398 | Degree and degree distribution, Average shortest path length, Clustering coefficient | Through analyzing the topological structure characteristics of BTN in small and medium-sized cities, it can get conclusion that the network has both a preferential growth mechanism and a random connection factor in the evolution process. |
Xu et al.[63]
2016 | Beijing | Subway / No | Modified Space L | 203 | Flow weight distribution, Node throughflow distribution, Flow hierarchical organization | With Modified Space L representation method, the edge weight is the actual passenger flow based on the card data.
Through establishing the subway passenger flow weighted network using a trip dataset collected from smart subway card transactions, the network with a power-law characteristic is obtained; the temporal patterns of in-transit and delayed passengers are identified by characterizing the flow weight and node throughflow distributions; the hierarchical flow organizations are also examined, based on the number of passengers flowing into the stations, as well as the spatial flow distributions. |
Regt et al.[64]
2017 | London, Manchester and other two cities | Bus, Subway / Yes | Space L | 2580-16397 | Degree and degree distribution, Relative size of maximum connectivity cluster, Average shortest path length, Measure of efficiency in terms of path length, Mean shortest travel time, Assortativity, Clustering coefficient, Fractal dimension | A comprehensive analysis for PTN that is either properties in geospatial or in topological space, is examined. |
Yang et al.[65]
2018 | Hangzhou, Ningbo | Bus, Public bicycle / Yes | Modified Space L, Modified Space P | Bus: 2943, 3261.
Public bicycle: 3866, 2493 | Average shortest path length, Degree and degree distribution, Edge length distribution, Transfer characteristic | With Modified Space L and Modified Space P representation method, the edge weights are both the actual distance between two adjacent stations that is calculated by spatial coordinates embedding.
A multi-layer coupled spatial network model considering the geographical information on bus stations, bus routes and public bicycle stations is proposed. Additionally, it can get the conclusions that the short-distance cycling and short-distance walking can reduce the average path length of PTN, decrease the average trip time of passengers, and improve the PTN operation efficiency. |
Zhang et al.[66]
2018 | Jinan | Bus / No | Space P | 883 | Degree and degree distribution, Betweenness, Clustering coefficient | From the perspective of graph theory and discrete mathematics, an effective method to calculate the transfer times between stations based on reachable matrix is proposed, and the minimum transfer time between stations is finally obtained. |
Zhang et al.[67]
2018 | Jinan | Bus / No | Space R | 195 3 | Degree and cumulative degree distribution, Betweenness, Clustering coefficient, Average shortest path length | The established bus transit route network is analyzed as a typical small-world network with a relatively large average clustering coefficient. Subsequently, based on a comprehensive understanding of network complexity, link prediction in this network is conducted, thus an auxiliary optimization method for complex bus transit route network is established. |
Shanmukhappa et al.[68]
2018 | Hong Kong, London and Bengaluru | Bus / No | Modified Space L | 4065, 20192, 5662 | Node weight, edge weight, Degree and degree distribution, Clustering coefficient, Average shortest path length, Eigen vector centrality, Hub and authority centrality, Betweenness, Figure of merit, Network efficiency | A supernode graph considering spatial embedding is proposed for modeling the BTN, which makes the scale-free and small-world behavior become evident as compared to conventional graph representation for Hong Kong’s BTN, and makes significant improve in clustering, reduce in path length, and increase in centrality values for all cities’ BTNs. Additionally, a static demand estimation procedure is proposed for assigning station weights by considering the points of interests (POIs) and the population distribution, which aids in better identifying the geographically significant stations. |
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