Evolution and Evaluation of the Guangzhou Metro Network Topology Based on an Integration of Complex Network Analysis and GIS

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Article Evolution and Evaluation of the Guangzhou Metro Network Topology Based on an Integration of Complex Network Analysis and GIS

Shaopei Chen and Dachang Zhuang *

School of Public Administration, Guangdong University of Finance and Economics, Guangzhou 510320, China; [email protected] * Correspondence: [email protected]

Received: 11 December 2019; Accepted: 9 January 2020; Published: 10 January 2020

Abstract: This paper takes the metro network of Guangzhou as a case study, and provides a quantitative analysis of the historical development of the network from 1999 to 2018. Particularly, the evolution of the topological structure of the Guangzhou Metro Network (GMN) is evaluated and characterized through the integration of geographic information system (GIS) and complex network analysis. The results show that: (1) The metro network of Guangzhou possesses the basic characteristics of small-world network, (2) with the development of GMN, the network complexity is increased and the spatial dispersion of the nodes tends to ease, but the average travel time and transfer rate continues to rise up, leading to the decreasing of the network transmission efficiency and the scattering of the nodes, (3) a good fault tolerance of the overall metro network of Guangzhou is revealed, but the spatial variance is observed, (4) the peak of degree centrality (DC) of the nodes is gradually moving northward along “Kecun Station–Guangzhou railway station–Jiahe Wanggang station”, while the peak of betweenness centrality (BC) is changing from “Kecun station” to “Jiahe Wanggang station”, and Jiahe Wanggang station has evolved into the most critical node in the current metro network of Guangzhou. In conclusion, this study should provide the scientific basis and significant decision-making support to the planning and operation management of GMN.

Keywords: metro network; complex network; topological structure; statistical analysis; GIS

1. Introduction With the development of modern transportation technology, the emerging countries and their major cities all over the world are seeking a well-developed urban rail transit network system to alleviate the increasingly serious traffic problems, especially to optimize land use spatial structure and transport accessibility in the urban system [1–3]. In China, for example, since the city of Beijing became the first city to have a metro line in 1969, there are now 35 metropolises with a metro network with a total mileage of nearly 4600 km. Among them, the top three metropolises with the largest metro network mileage are Shanghai, Beijing, and Guangzhou. Since 1997, the Guangzhou Metro Network (GMN) has been developing rapidly, while the economy of the city has been booming continuously. Currently, the metro network system in Guangzhou is composed of 16 lines (service routes) and 218 stations. As an important part of the modern public transport service system, the urban metro network has many advantages, such as large traffic volume, high speed, long distance, less pollution, and intensive land use, compared with the conventional public transport modes, such as bus and tram. Certainly, achieving a reliable metro network structure and reasonable spatial network distribution pattern should play a significant role in improving the quality of urban public transport services and promoting the urban sustainable development, especially in a developing country context [4].

Sustainability 2020, 12, 538; doi:10.3390/su12020538 www.mdpi.com/journal/sustainability Sustainability 2020, 12, 538 2 of 18

With the development of network science theory in the field of transport study, the complex network theory has become an important tool for modeling and characterizing the transport network topology. In the context of complex network theory [5,6], more and more scholars build transport network models based on Space-L and Space-P by using the entities (such as stops) of a complex network system and the interaction or association between the entities (such as service routes). Moreover, the graph theory and statistical physics are applied to explore the statistical law of the geometric and topological property in a transport network [7–10], with an important aim of providing the scientific references and decision-making support to the transport network planning. At present, the complex network theory has evolved into a new and extensive scientific paradigm of network system analysis and design. Many studies have focused on the analysis of the complexity, characteristics, and topological properties of urban rail transit network, such as a metro network, mainly considering the following aspects: (1) Measuring and characterizing the statistical characteristics such as node degree distribution of an urban metro network, and verifying its “small world” or/and “scale-free” phenomenon [11–13], (2) according to the geographical spatial characteristics of a given urban rail transit network, the powerful spatial analysis tools, such as geography information system (GIS), are integrated into the complex network analysis to evaluate the spatial network topological structure [14,15], (3) through measuring and quantifying the key topological parameters, such as degree centrality (DC), betweenness centrality (BC), and clustering coefficient (CC), the statistical characteristics of nodes in network are assessed thoroughly [16–21], (4) discussing the dynamic characteristics or the emergence behavior of the whole metro network while it is changing (such as being attacked), including vulnerability, reliability, robustness, etc. [22–27], furthermore, implementing the comparison between the urban metro network complexity characteristics in different cities, and then investigating the topology and classification of the given networks [19,21,28,29]. In summary, the existing research has carried out a more in-depth study of the topology and complexity characteristics of the transportation network. Especially, under the support of GIS, which is one of the innovative approaches that characterize the spatial correlation between spatial entities [30,31], the complex network analysis for the spatial topological characteristics of the transportation network shows a more extensive prospect. However, the empirical analysis of the topology evolution of a given metro network in its complete growing period is less involved, making it difficult to recognize the characteristics and issues in each developmental stage during the growth of a network. In addition, as a typical real-life network, although the basic topological properties and characteristics of an urban metro network can be intuitively revealed through the complex network analysis, in the absence of time, distance, time of transfer or other network service information, it is hard to learn clearly about the performance of network service, especially under a specific topological structure. Indeed, for urban planners, how to improve the service level and quality of a transport network is a crucial issue in urban planning. Therefore, on the basis of the complex network theory, this paper introduces the time cost, time of transfer, and other information of the travel paths among metro stations, and then establishes origin–destination (OD) matrixes and weighted network models during the growth of the Guangzhou Metro Network. Based on the complexity analysis of GMN, furthermore, this paper focuses on evaluating and characterizing the evolution of the spatial network topology and related issues in different network development stages by integrating with GIS. This study would serve as the scientific basis and decision-making support for the planning and operation management of the Guangzhou Metro Network.

2. Methodology

2.1. Modeling of Urban Metro Network In the context of the network science theory, the modeling of an urban public transport network (such as a metro network) is generally implemented by using the methods of Space-L and Space-P. In the network modeling based on Space-L, a station is identified as one node in a network, while there Sustainability 2020, 12, 538 3 of 18 Sustainability 2020, 12, 538 3 of 18 is at least one service route (i.e., a metro line) between two adjacent nodes, these two nodes should be should be connected with an edge. As a network modeling of Space‐P, a station is also defined as one connected with an edge. As a network modeling of Space-P, a station is also defined as one node in a node in a network, if there is at least one direct route (without transfer) between any two nodes, there network, if there is at least one direct route (without transfer) between any two nodes, there is an edge is an edge between them. As shown in Figure 1, the actual shape of a public transport network is between them. As shown in Figure1, the actual shape of a public transport network is illustrated, and illustrated, and the corresponding network modeling based on Space‐L and Space‐P is also presented, the corresponding network modeling based on Space-L and Space-P is also presented, respectively. respectively. Obviously, the Space‐P network model is built on the service routes of a public transport Obviously, the Space-P network model is built on the service routes of a public transport network, network, and currently is widely applied to investigate the transportability between two nodes in a and currently is widely applied to investigate the transportability between two nodes in a network. network. Therefore, compared with the Space‐L model, the network structure modeling based on Therefore, compared with the Space-L model, the network structure modeling based on Space-P can Space‐P can more realistically reflect the service level and quality of an urban public transport more realistically reflect the service level and quality of an urban public transport network. network.

A B C Line 1 A-B-E-F Line 2 C-B-E-F Line 3 D-B-F Line 4 A-B-F D E F

(a) prototype of urban public transport network

A B C A B C

D E F D E F

(b) Space-L network topology (c) Space- P network structure modelling topology structure modelling Figure 1. Public transport network modeling based on Space-L and Space-P. Figure 1. Public transport network modeling based on Space‐L and Space‐P. A large number of studies have illustrated that complex network analysis is an efficient and convenientA large way number to understand of studies the have topology illustrated of a public that transport complex network.network Nevertheless,analysis is an due efficient to the lackand convenientof the locations way (geographical to understand characteristics) the topology of of a stations, public transport and particularly network. the Nevertheless, impedance information, due to the lacksuch asof travelthe locations time or time (geographical of transfer betweencharacteristics) stations, of the stations, outcome and derived particularly from the the analysis impedance cannot information,identify the servicesuch as leveltravel and time quality or time of of a transfer public transport between stations, network the under outcome its specific derived topological from the analysisstructure. cannot According identify to a the questionnaire service level survey and quality on the of residents’ a public travel transport behavior network by using under metro its specific mode topologicalin Guangzhou structure. made byAccording our research to a questionnaire team in 2018, survey the residents on the inresidents’ Guangzhou travel are behavior most concerned by using withmetro the mode convenience in Guangzhou of travel made by by using our metroresearch mode, team which in 2018, specifically the residents implies in Guangzhou their psychological are most concernedchoice of less with travel the timeconvenience and time of of travel route changeby using (i.e., metro transfer). mode, It needswhich tospecifically be emphasized implies that their the psychologicalconcerns of shorter choice time of ofless transfer travel are time also and based time on of the route priority change choice (i.e., of lesstransfer). travel time.It needs From to thebe emphasizedresidents’ perspective, that the concerns 30 min of (without shorter transfer)time of transfer to travel are by also metro based is the on mostthe priority ideal time choice cost, of andless travelthe acceptable time. From time the cost residents’ is 30 to perspective, 50 min (or/and 30 min one (without time of transfer), transfer) to but travel more by than metro 50 minis the or most/and idealover twotime times cost, ofand transfer the acceptable are the most time unacceptable cost is 30 to choice. 50 min Therefore, (or/and one in ordertime of to transfer), attain the but time more cost thanand time 50 min of transferor/and over required two times to travel of transfer between are two the stations most unacceptable in GMN, the choice. spatial dataTherefore, modeling in order and toanalysis attain toolthe time of GIS, cost i.e., and ArcGIS time of 10.2 transfer is applied. required In the to travel environment between of two GIS, stations the shortest in GMN, path the algorithm spatial dataof Dijkstra modeling is used and to analysis analyze thetool shortest of GIS, pathi.e., ArcGIS of travel 10.2 time, is andapplied. the path’s In the time environment of transfer of and GIS, length the shortestbetween path any twoalgorithm nodes. of Therefore, Dijkstra is an used origin–destination to analyze the shortest (OD) matrix path of can travel be accomplished time, and the onpath’s the timebasis of of transfer a given and quantity length of between OD, such any as traveltwo nodes. time orTherefore, time of transfer. an origin–destination If the total number (OD) ofmatrix nodes can is beN inaccomplished a network, it on implies the basis that of there a given are NquantityN pairs of OD, of nodes. such as Therefore, travel time the or OD time matrix of transfer. (named If time the total number of nodes is N in a network, it× implies that there are N × N pairs of nodes. Therefore, the matrix) based on the minimum travel time of each pair of nodes can be built, and represented by: OD matrix (named time matrix) based on the minimum travel time of each pair of nodes can be built, and represented by: Time Matrix = [tij]N N (1) × Time Matrix = [tij]N×N (1) In the OD matrix, tij is the time cost of the shortest travel path between nodes i and j, when i = j, tijIn= the0, thatOD matrix, is, the travel tij is the time time between cost of nodesthe shortest i and travel i is excluded. path between The time nodes cost i and for waitingj, when i and = j, tij = 0, that is, the travel time between nodes i and i is excluded. The time cost for waiting and transferring during the travel between nodes i and j is not included in tij, as it is difficult to be gotten. Sustainability 2020, 12, 538 4 of 18

Sustainability 2020, 12, 538 4 of 18 transferring during the travel between nodes i and j is not included in tij, as it is diﬃcult to be gotten. Moreover, the OD matrix (name as transfer matrix) based on the time of transfer of each pair of nodes Moreover, the OD matrix (name as transfer matrix) based on the time of transfer of each pair of nodes can be attained and identiﬁed as: can be attained and identified as:

TransferTransfer Matrix Matrix= =[r [rijij]]NN×NN (2)(2) × rij is the times of transfer (route change) between nodes i and j, if i = j, rij = 0. In terms of the transfer rij is the times of transfer (route change) between nodes i and j, if i = j, rij = 0. In terms of the transfer matrix,matrix, the the average time of transfertransfer ofof allall paths paths between between any any i i * * 0028 0028 two two nodes nodes in in the the network network can can be becalculated calculated and and represented represented as θasas θ follows.as follows. N N P rij i=1,j=1 rij (3) θ= i=1,j=1 θ = (3) NPNP NPNP is is the the total total number number of of the the paths paths among among the the nodes nodes in network. Furthermore,Furthermore, the the ratio ratio of of the the number number of of paths paths which which need need to to transfer transfer to to the the total total number number of of all all pathspaths in in the the network network can can be be attained, attained, and described as δδ. ρ δ=δ = (4)(4) NPNP ρρ isis the the number number of of the the paths paths with with the the time time of of transfer transfer is is more than than 0. InIn the the Space Space-P‐P network network model, model, the the edge edge between between nodes nodes i i and and j j shows shows that that there there is is at at least least one one directdirect route (metro line) betweenbetween twotwo nodes. nodes. This This implies implies that that the the time time of of transfer transfer is is 0 for0 for each each edge edge in inthe the network. network. In thisIn paper,this paper, however, however, through through assigning assigning the travel the time travel to each time edge to andeach considering edge and consideringthat the round-trip that the time round is the‐trip same time for is eachthe same edge, for an undirectedeach edge, weightedan undirected network weighted model basednetwork on modelSpace-P based is constructed. on Space‐P As is constructed. shown in Figure As shown2, each in metro Figure station 2, each is metro deﬁned station by a node is defined in GMN. by a GMN node incan GMN. be represented GMN can be as v,represented v V and as V isv, the v ∈ vertex V and set. V is While the vertex two metroset. While stations two aremetro connected stations by are a ∈ connectedmetro line by (without a metro transfer), line (without there istransfer), an edge there between is an them, edge and between represented them, and as s, represented s S, S is the as edge s, s ∈ ∈set. S, S As is a the result, edge the set. Guangzhou As a result, Metro the Guangzhou Network can Metro be described Network as can G =be(v,s), described and the as total G = number(v,s), and of thenodes total and number edges of is representednodes and edges as N is and represented M, respectively. as N and M, respectively.

FigureFigure 2. 2. UndirectedUndirected weighted weighted network network model model based based on on Space Space-P.‐P. 2.2. Topology Measuring of the Overall Network 2.2. Topology Measuring of the Overall Network The indexes for evaluating the connectivity of the overall network include network diameter (T), The indexes for evaluating the connectivity of the overall network include network diameter (T), average path length (L), network eﬃciency (E), connection rate (β), loop index (µ), actual loop forming average path length (L), network efficiency (E), connection rate (β), loop index (μ), actual loop rate (α), and actual combination degree (γ). Among these indexes, network diameter (T) is a key index forming rate (α), and actual combination degree (γ). Among these indexes, network diameter (T) is a key index to characterize the size of the network in space, while the average path length (L) reflects the spatial dispersion of nodes in the network. T and L can be calculated as follows:

T= MAX tij i,j   (5) Sustainability 2020, 12, 538 5 of 18 to characterize the size of the network in space, while the average path length (L) reﬂects the spatial dispersion of nodes in the network. T and L can be calculated as follows: n o T =MAX tij (5) i,j

N 1 N 2 X− X L = t (6) N(N 1) ij − i=1 j=i+1 tij is the time cost of the shortest travel path between nodes i and j, where the maximum is the network diameter. N is the total number of nodes. In addition, the average of the reciprocal sum of tij is the network efficiency (E): N 1 N 2 X− X 1 E = (7) N(N 1) tij − i=1 j=i+1 Network efficiency (E) aims to present the average proximity of nodes in an overall network. The larger the value of L of the network, the smaller the value of E, implying that the transmission efficiency of the network is lower, and the passenger flow in the network is more difficult to flow. Therefore, network efficiency (E) can be used to measure the anti-attack and robustness of the network. The indicators of β and µ are used to measure the level of connectivity of the network, while α and γ are to evaluate the development potential of the network. β refers to the average number of edges of nodes in the network, that is, the ratio of the total number of edges (M) to the total number of nodes (N), and can be expressed as β = M/N. β < 1 indicates that the given network is a tree network, while β > 1 shows that the network is a loop network. Furthermore, µ refers to the number of loops in a network, that is µ = M N + q. The larger the value, the more loops, and the more developed − network. q is the number of subgraphs of a network. In an urban metro network, stations are defined as nodes, and a series of nodes form a metro line, so the nodes in the line are connected to each other to constitute a complete subgraph, and different lines form the entire network through the transfer nodes. Therefore, the number of metro lines is the number of subgraphs. α is the ratio of two times of µ to the possible maximum number of loops (i.e., (N 1) (N 2)) in a network. The index of α aims to reveal − × − the actual looping level of the network, expressed as α = 2µ/(N 1) (N 2) = (M N + q)/(2N 5q). − − − − The smaller the value of α, the lower the level of looping, and 1-α indicates the potential for looping. γ is identified by the ratio of the total number of edges (M) to the possible maximum number of edges (3 (N 2)), which reflects the networking level of edges, i.e., γ = M/3 (N 2). × − × − 2.3. Statistical Characteristic Measuring of Node The indicators that reflect the statistical characteristics of nodes in a network in the context of complex network theory include node degree (ND), clustering coefficient (CC), and betweenness centrality (BC). Among these indicators, the number of edges connected to node i is defined as the node degree, i.e., ki = Σsij (j = 1, 2, ... , N). The greater degree denotes that the node with the higher degree centrality (DC), implying that the node is more important in a network. The ratio of the degree of any node to the total number of edges can obtain the distribution characteristics of ND, which is generally described by a degree distribution function, expressed as P (k). P (k) shows the probability of the degree of a node, which is arbitrarily selected in a network, is exactly k. For a given network, a histogram can be used to represent the distribution of ND. In addition, the average ND of a network (denoted as K) can be represented as follows:

N 1 X K = k (8) N i i=1 Sustainability 2020, 12, 538 6 of 18

The clustering coefficient describes the average coupling degree of a node. Assume that node i in a network has Mi edges connected to other nodes, that is, these Mi nodes are identified as the neighbor nodes of node i. There is a possible maximum number (M (M 1)/2) of edges between these i i − M neighbor nodes. Therefore, the ratio of M to M (M 1)/2 is defined as the clustering coefficient of i i i i − the given node, that is: 2M C = i (9) i M (M 1) i i − Moreover, the average clustering coefficient of the network can be calculated as follows:

N 1 X C = C (10) N i i=1

Obviously, the value of C ranges from 0 to 1. While C = 0, all nodes are isolated in the network, that is, there are no edges. When C = 1, it means that any two nodes in the network are connected with edges, that is, the network is a globally coupled network. As mentioned above, the greater the degree of centrality of a node, the more important the node is in a network. However, there are often some nodes with low node centrality in a real-life network, which are very significant in a network. For example, in the urban metro network, although the degrees of some nodes are very small, they may be connecting nodes between lines, if such nodes were removed from the network, the related lines would be interrupted. Therefore, the nodes with lower degree centrality (NC) play a key role in the connectivity of network. For such a node, another global geometric quantity needs to be defined as a measure of its importance, that is, the betweenness centrality. In this paper, the betweenness of a node is denoted as Bi, which refers to the ratio of the number of shortest travel time paths which pass through node i to the total number of paths in network. Indeed, the value of Bi ranges from 0 to 1. The larger the value of Bi, the higher the betweenness centrality of node i, that is, the greater the influence of the node in a network. The formula for calculating the value of Bi is as follows: X p (i) B = mn m, n , i, m , n (11) i p m,n mn

In the formula, pmn (i) is the number of shortest travel time path which passes through node i, and pmn is the total number of shortest time paths in a network.

3. Results and Analysis

3.1. Guangzhou Metro Network and Development Since the ﬁrst metro line (i.e., Line 1) was opened in 1999 in the city of Guangzhou, the Guangzhou Metro Network has developed rapidly in the past 20 years. As shown in Figure3, the spatial evolution of GMN is a process of expanding outwards with the urban central area, including the districts of Yuexiu, Tianhe, Liwan, and Haizhu. With the increasing of network density in the urban central areas, GMN expands service coverage outward. Furthermore, Figure4 shows the growth of the Guangzhou Metro Network (GMN) from 1999 to 2018. It can be seen that the development of the network in the early period (1999–2005) was relatively slow. By 2006, the mileage and the number of stations began to rise rapidly, for example, the network mileage increased from 58.5 to 116 km. By 2018, the network mileage jumped to 478 km, with 16 lines and 218 stations (see Figure5). The rapid development of the urban rail transit network has had a profound impact on the urban mobility pattern of Guangzhou. By the end of 2018, the proportion of residents’ motorized trips reached 61% in public transport in the central urban areas of Guangzhou. Moreover, the daily average passenger volume of public transport modes including metro and bus has reached 16.49 million person-times in 2018, among which the daily average passenger volume of metro mode is more than eight million person-times and the intensity of daily average passenger ﬂow has reached Sustainability 2020, 12, 538 7 of 18

2.19 million person-times of each kilometer, which is higher than that of Beijing (1.79), Shanghai (1.60).

SustainabilityFigure6 shows 2020, 12 the, 538 evolution and comparison of the daily average passenger volumes of metro7 of and 18 Sustainabilitybus in Guangzhou 2020, 12, 538 from 1999 to 2018. Obviously, with the development of GMN, the daily passenger7 of 18 volume of the metro is increasing rapidly, especially in 2016, it exceeded that of the bus, which has shown a decline since 2016.

Figure 3. Spatial structural evolution of the Guangzhou Metro Network. FigureFigure 3. 3.SpatialSpatial structural structural evolution evolution of ofthe the Guangzhou Guangzhou Metro Metro Network. Network.

Figure 4. Development of the Guangzhou Metro Network (data source: The annual report of social Figureand economic 4. Development statistics of Guangzhouthe Guangzhou in 1999–2018). Metro Network (data source: The annual report of social Figure 4. Development of the Guangzhou Metro Network (data source: The annual report of social and economic statistics of Guangzhou in 1999–2018). and economic statistics of Guangzhou in 1999–2018). SustainabilitySustainability 2020, 122020, 538, 12 , 538 8 of 18 8 of 18

Figure 5. The Guangzhou Metro Network in 2018. Figure 5. The Guangzhou Metro Network in 2018. As the construction of a metro line often takes several years to complete, the growth of GMN Theshows rapid a step-by-step development phenomenon of the urban in 1999 rail to transit 2018 (see network Figure4 has). During had a the profound growing impact of the network, on the urban mobilitythere pattern was no of or Guangzhou. only one new metroBy the line end opened of 2018, in some the years, proportion leading toof a residents’ weak inﬂuence motorized on the trips size and structure of the network, on the other hand, it should be emphasized that in some key years, reached 61% in public transport in the central urban areas of Guangzhou. Moreover, the daily average several new metro lines were opened, causing a signiﬁcant change in the network size and structure. passengerTherefore, volume the studyof public of this transport paper focuses modes on the including network data metro analysis and in bus the keyhas time reached points 16.49 (years), million personincluding‐times in the 2018, years among of 1999, which 2003, 2006,the daily 2007, average 2009, 2010, passenger 2013, 2015, volume 2016, 2017, of metro and 2018. mode is more than eight million person‐times and the intensity of daily average passenger flow has reached 2.19 million person‐times of each kilometer, which is higher than that of Beijing (1.79), Shanghai (1.60). Figure 6 shows the evolution and comparison of the daily average passenger volumes of metro and bus in Guangzhou from 1999 to 2018. Obviously, with the development of GMN, the daily passenger volume of the metro is increasing rapidly, especially in 2016, it exceeded that of the bus, which has shown a decline since 2016. As the construction of a metro line often takes several years to complete, the growth of GMN shows a step‐by‐step phenomenon in 1999 to 2018 (see Figure 4). During the growing of the network, there was no or only one new metro line opened in some years, leading to a weak influence on the size and structure of the network, on the other hand, it should be emphasized that in some key years, several new metro lines were opened, causing a significant change in the network size and structure. Therefore, the study of this paper focuses on the network data analysis in the key time points (years), including the years of 1999, 2003, 2006, 2007, 2009, 2010, 2013, 2015, 2016, 2017, and 2018. Sustainability 2020, 12, 538 9 of 18 Sustainability 2020, 12, 538 9 of 18 Sustainability 2020, 12, 538 9 of 18

Figure 6. Evolution of daily average passenger volumes of metro and bus from 1999–2018 (data Figure 6. Evolution of daily average passenger volumes of metro and bus from 1999–2018 (data source: source:Figure The6. Evolution annual report of daily of social average and passengereconomic statisticsvolumes ofof Guangzhou metro and inbus 1999–2018). from 1999–2018 (data The annual report of social and economic statistics of Guangzhou in 1999–2018). source: The annual report of social and economic statistics of Guangzhou in 1999–2018). 3.2. Evaluation and Characterization of the Topology of GMN 3.2. Evaluation and Characterization of the Topology of GMN 3.2. EvaluationIn terms of and the Characterization metro network of models the Topology based of on GMN Space ‐P in the key years, including 1999, 2003, In terms of the metro network models based on Space-P in the key years, including 1999, 2003, 2006, 2007, 2009, 2010, 2013, 2015, 2016, 2017, and 2018, Figure 7 illustrates that the number of edges 2006,In 2007, terms 2009, of the 2010, metro 2013, network 2015, 2016, models 2017, based and on 2018, Space Figure‐P in7 illustratesthe key years, that including the number 1999, of edges 2003, increased rapidly with the augmenting of the number of nodes from 1999 to 2018. Moreover, the 2006,increased 2007, rapidly 2009, 2010, with 2013, the augmenting2015, 2016, 2017, of the and number 2018, Figure of nodes 7 illustrates from 1999 that to the 2018. number Moreover, of edges the number of edges increases much faster than that of nodes. This exhibited a phenomenon of super‐ increasednumber of rapidly edges increases with the much augmenting faster than of thatthe ofnumber nodes. of This nodes exhibited from a1999 phenomenon to 2018. Moreover, of super-linear the linear growth of links in a complex network. Particularly, it is further observed that the number of numbergrowth ofof linksedges in increases a complex much network. faster Particularly,than that of nodes. it is further This observedexhibited thata phenomenon the number of of super edges‐ edges increased linearly with that of nodes (R2 = 0.9952) (see Figure 8). This shows that there is a large linearincreased growth linearly of links with in that a complex of nodes network. (R2 = 0.9952) Particularly, (see Figure it is further8). This observed shows that that there the number is a large of number of nodes with low DC and a small number of nodes with high DC (i.e., HUB nodes), implying edgesnumber increased of nodes linearly with low with DC that and of a nodes small number(R2 = 0.9952) of nodes (see withFigure high 8). This DC (i.e., shows HUB that nodes), there is implying a large that the Guangzhou Metro Network has an effect on the small‐world network. numberthat the of Guangzhou nodes with Metro low DC Network and a small has an number eﬀect on of thenodes small-world with high network.DC (i.e., HUB nodes), implying that the Guangzhou Metro Network has an effect on the small‐world network.

Figure 7. Growth of nodes and edges in the Guangzhou Metro Network. Figure 7. Growth of nodes and edges in the Guangzhou Metro Network. Figure 7. Growth of nodes and edges in the Guangzhou Metro Network. Sustainability 2020, 12, 538 10 of 18 Sustainability 2020, 12, 538 10 of 18

Figure 8. Linear growth relationship between node number and edge number. Figure 8. Linear growth relationship between node number and edge number. As shown in Table1, with the augmenting of network mileage, the network diameter (T) and averageAs path shown length in (L) Table also 1, increase. with the The augmenting values of network of network diameter mileage, did notthe risenetwork sharply diameter but remained (T) and ataverage 0.7–0.8 h path from length 2007 to (L) 2017. also This increase. indicates The that values during of thisnetwork period, diameter the Guangzhou did not Metro rise sharply Network but aimedremained to develop at 0.7–0.8 and h improve from 2007 the to internal 2017. This network indicates instead that of during expanding this period, outward. the With Guangzhou the opening Metro ofNetwork metro lines aimed 14 and to develop 21, which and expanded improve to the the internal north in network 2018, the instead network of diameter expanding increased outward. to 1With h. Inthe addition, opening the of average metro pathlines length 14 and is 21, much which smaller expanded than the to networkthe north diameter, in 2018, and the itnetwork increases diameter more slowlyincreased than to the 1 networkh. In addition, diameter. the average This further path illustrates length is much that GMN smaller meets than the the basic network characteristics diameter, ofand a small-worldit increases more network. slowly As than shown the in network Figure9 diameter., the network This diameter further illustrates and average that path GMN length meets increase the basic 2 logarithmicallycharacteristics with of a thesmall augmented‐world network. network As size shown (i.e., numberin Figure of 9, nodes) the network (the values diameter of R andare 0.9461average andpath 0.8638, length respectively). increase logarithmically It means that whilewith the the augmented complexity ofnetwork GMN increases, size (i.e., thenumber spatial of dispersion nodes) (the ofvalues the nodes of R in2 are the 0.9461 network and will 0.8638, tend respectively). to ease. However, It means the network that while effi theciency complexity (E) exhibits of GMN a downward increases, trend,the spatial reflecting dispersion that as theof the network nodes expands,in the network the average will tend travel to ease. time However, in the network the network is continuously efficiency increasing,(E) exhibits and a downward the connectivity trend, and reflecting accessibility that as of the the network overall network expands, are the gradually average travel deteriorating. time in the network is continuously increasing, and the connectivity and accessibility of the overall network are gradually deteriorating.Table 1. Metrics of topological structure of the Guangzhou Metro Network. In addition,Year Table Mileage 1 shows (KM) that T (Hour)t as a real L‐life (Hour) network, E the urbanβ metro µ network α γ is a typical loop network, and its connection rate (β) is always greater than 1. With the growth of GMN, there are more 1999 18.19 0.4 0.16 0.28 7.5 105 1 2.86 opportunities2003 for convergence 35.57 between 0.4 lines. This 0.15 leads 0.28to a significant 7.74 211 increase 0.49 of 2.76 the loop index (μ), and a more2006 developed 71.68 network. However, 0.5 both 0.16 the actual 0.25 loop 8.36 rate 349(α) and 0.34 the actual 2.91 combination degree (γ) of2007 GMN are 112.56 gradually decreasing. 0.7 Obviously, 0.18 0.23 combining 7.98 416 the characteristic 0.25 2.75 of gradually decreasing 2009the network 146.18 efficiency (E), 0.7 it can explain 0.2 that 0.21 with 9.35 the growth 673 0.22 of GMN, 3.20 the proximity 2010 213.77 0.8 0.21 0.23 9.18 983 0.14 3.11 between nodes2013 in the network 237.87 is declining. 0.8 0.19 0.25 8.86 1055 0.12 3.00 2015 244.22 0.8 0.19 0.25 8.79 1070 0.12 2.98 2016Table 1. 278.25 Metrics of topological 0.8 structure 0.2 of the 0.24 Guangzhou 8.87 1199 Metro 0.11 Network. 3.00 2017 323.16 0.8 0.2 0.24 8.49 1293 0.09 2.86 Year 2018Mileage (KM) 436.8 T (Hour) 1 0.23L (Hour) 0.22 8.38E β μ 1552 0.07 2.82 α γ 1999 18.19 0.4 0.16 0.28 7.5 105 1 2.86 2003 35.57 0.4 0.15 0.28 7.74 211 0.49 2.76 2006In addition, Table71.681 shows that t as0.5 a real-life network,0.16 the urban0.25 metro8.36 network349 is a typical0.34 loop2.91 network,2007 and its connection112.56 rate (β) is always0.7 greater0.18 than 1. With0.23 the growth7.98 of GMN,416 there0.25 are more2.75 opportunities2009 for convergence146.18 between0.7 lines. This leads0.2 to a significant0.21 increase9.35 of673 the loop0.22 index 3.20 (µ), and a2010 more developed213.77 network. However,0.8 both the0.21 actual loop0.23 rate (α9.18) and the983 actual combination0.14 3.11 degree2013 (γ ) of GMN237.87 are gradually decreasing.0.8 Obviously,0.19 combining0.25 the8.86 characteristic 1055 of0.12 gradually 3.00 decreasing2015 the network244.22 efficiency (E),0.8 it can explain0.19 that with0.25 the growth8.79 of GMN,1070 the0.12 proximity 2.98 2016 278.25 0.8 0.2 0.24 8.87 1199 0.11 3.00 between nodes in the network is declining. 2017 323.16 0.8 0.2 0.24 8.49 1293 0.09 2.86 2018 436.8 1 0.23 0.22 8.38 1552 0.07 2.82 Sustainability 2020, 12, 538 11 of 18 Sustainability 2020, 12, 538 11 of 18

(a) Network diameter (D) (b) Average path length (L)

Figure 9. Logarithmic growth of network diameter, average path length and network size.size.

The probability distribution of ND of GMN in each key year illustrates the phenomenon of long tail (see Figure 1010),), and the evolution can be divided into three stages (since there was only one metro line in 1999, the data from this year were removed): (1) The probability distribution of ND from 2003 to 2006 was was a a typical typical power power-law‐law distribution, distribution, indicating indicating the the apparent apparent small small-world‐world characteristics characteristics (see (seeFigure Figure 10i,j), 10 (2)i,j), during (2) during 2007–2013, 2007–2013, with the with growth the growth of the of network, the network, the probability the probability distribution distribution of ND ofdid ND not did have not power have‐law power-law distribution distribution characteristics, characteristics, but tended but to tended be a Poisson to be a distribution, Poisson distribution, showing showingthe characteristics the characteristics of a scale of‐free a scale-free network, network, (3) in 2015–2018, (3) in 2015–2018, the probability the probability distribution distribution of ND of had ND hadmore more obvious obvious characteristics characteristics of power of power law, law, illustrating illustrating the effect the eff ofect a of small a small-world‐world network. network. Table2 shows the average node degree (K), the average clustering coe fficient (C), the average time of transfer (θ), and transfer rate (δ) of GMN in each key year. The results show that in the process of network growth, there is a large average node degree. Furthermore, the values of the average clustering coefficient (C) have been at a high level, illustrating that the metro network in Guangzhou has better fault tolerance, i.e., the alternatives of the path are strong. In particular, the network in 1999 had only one line, so the nodes in the network were connected two by two, and the value of the average clustering coefficient (C) was 1. At this time, the network was a globally coupled network. With the increase in the number of lines and stations, the average clustering coefficient (C) decreased slightly, but the trend of clustering has remained overall.

Table 2. Statistical characteristics of the topology of the Guangzhou Metro Network.

Year K C θ δ (a) 2018 (b) 2017 (c) 2016 1999 15.00 1.00 0 0 2003 15.48 0.98 0.48 0.48 2006 16.72 0.96 0.77 0.64 2007 15.97 0.96 1.06 0.72 2009 18.70 0.94 1.22 0.76 2010 18.37 0.94 1.68 0.84 2013 17.48 0.89 1.94 0.87 2015 17.36 0.89 1.90 0.87 2016 17.53 0.89 2.03 0.88 2017 16.79 0.90 2.25 0.90 2018 16.62 0.91 2.55 0.92

Furthermore, as shown in Table2, the average time of transfer ( θ) and the average transfer rate (δ) of GMN increase with the growth of the network, particularly the increase of the value ofθhas a linear 2 relationship with(d) 2015 the increasing of the number of(e) nodes 2013 (R = 0.982) (see Figure 11().f) Moreover,2010 Table2 shows that the value of δ reached 92% in 2018, which means that 92 of the 100 shortest travel time paths needed to be transferred, and more than two times of transfer were required on average. Obviously, in terms of the characteristics of the gradual decline of α, γ, and E, it further reveals that the larger Sustainability 2020, 12, 538 11 of 18

(a) Network diameter (D) (b) Average path length (L)

Figure 9. Logarithmic growth of network diameter, average path length and network size.

The probability distribution of ND of GMN in each key year illustrates the phenomenon of long tail (see Figure 10), and the evolution can be divided into three stages (since there was only one metro line in 1999, the data from this year were removed): (1) The probability distribution of ND from 2003 Sustainabilityto 2006 2020was, a12 typical, 538 power‐law distribution, indicating the apparent small‐world characteristics (see12 of 18 Figure 10i,j), (2) during 2007–2013, with the growth of the network, the probability distribution of ND did not have power‐law distribution characteristics, but tended to be a Poisson distribution, showing size ofthe the characteristics network, the of morea scale dispersed‐free network, the nodes,(3) in 2015–2018, finally leading the probability to the more distribution difficult of transmission ND had of passengermore obvious flows in characteristics the network. of power law, illustrating the effect of a small‐world network.

Sustainability 2020, 12, 538 12 of 18 (d) 2015 (e) 2013 (f) 2010

(g) 2009 (h) 2007 (i) 2006

(j) 2003

FigureFigure 10. 10.Evolution Evolution of of nodenode degree probability probability distribution. distribution.

Table 2 shows the average node degree (K), the average clustering coefficient (C), the average time of transfer (θ), and transfer rate (δ) of GMN in each key year. The results show that in the process of network growth, there is a large average node degree. Furthermore, the values of the average clustering coefficient (C) have been at a high level, illustrating that the metro network in Guangzhou has better fault tolerance, i.e., the alternatives of the path are strong. In particular, the network in 1999 had only one line, so the nodes in the network were connected two by two, and the value of the average clustering coefficient (C) was 1. At this time, the network was a globally coupled network. With the increase in the number of lines and stations, the average clustering coefficient (C) decreased slightly, but the trend of clustering has remained overall. Furthermore, as shown in Table 2, the average time of transfer (θ) and the average transfer rate (δ) of GMN increase with the growth of the network, particularly the increase of the value ofθhas a linear relationship with the increasing of the number of nodes (R2 = 0.982) (see Figure 11). Moreover, Table 2 shows that the value of δ reached 92% in 2018, which means that 92 of the 100 shortest travel time paths needed to be transferred, and more than two times of transfer were required on average. Obviously, in terms of the characteristics of the gradual decline of α, γ, and E, it further reveals that the larger size of the network, the more dispersed the nodes, finally leading to the more difficult transmission of passenger flows in the network.

Table 2. Statistical characteristics of the topology of the Guangzhou Metro Network.

Year K C θ δ 1999 15.00 1.00 0 0 2003 15.48 0.98 0.48 0.48 2006 16.72 0.96 0.77 0.64 2007 15.97 0.96 1.06 0.72 2009 18.70 0.94 1.22 0.76 Sustainability 2020, 12, 538 13 of 18

2010 18.37 0.94 1.68 0.84 2013 17.48 0.89 1.94 0.87 2015 17.36 0.89 1.90 0.87 2016 17.53 0.89 2.03 0.88 Sustainability20172020 , 12, 538 16.79 0.90 2.25 0.90 13 of 18 2018 16.62 0.91 2.55 0.92

Figure 11. Linear growth relationship between average transfer times and network size. Figure 11. Linear growth relationship between average transfer times and network size. 3.3. Network Topology Spatial Analysis Based on GIS 3.3. Network Topology Spatial Analysis Based on GIS In the environment of ArcGIS 10.2, the degree centrality (DC) and betweenness centrality (BC) of the nodesIn the in environment GMN in the of key ArcGIS years of10.2, 2006, the 2010, degree 2015, centrality and 2018 (DC) were and graded betweenness by using centrality the method (BC) of ofnatural the nodes breakpoint, in GMN and in the then key the years method of 2006, of natural 2010, 2015, neighbor and was2018 applied were graded to interpolate by using the the values method of ofDC natural and BC breakpoint, of each node and to then generate the method a trend of surface natural (see neighbor Figure was12). applied to interpolate the values of DCAs and shown BC of in each Figure node 12, to most generate transfer a trend nodes surface where two (see or Figure more 12). lines meet have a high DC, and if a metroAs line shown has more in Figure nodes 12, (stations) most transfer and transfer nodes nodes, where the two DC or of more its nodes lines becomes meet have greater. a high According DC, and ifto a the metro spatial line and has temporal more nodes evolution (stations) of DC and in transfer GMN, the nodes, center the of DC the of network its nodes is gradually becomes greater. moving Accordingnorth along to the the “Kecun spatial and Station–Guangzhou temporal evolution Railway of DC Station–Jiahein GMN, the center Wanggang of the Station”. network is Particularly, gradually movingthe DC ofnorth transfer along nodes the “Kecun in GMN Station–Guangzhou was significantly higher Railway than Station–Jiahe other nodes inWanggang 2006. Therefore, Station”. a Particularly,degree-centrality the DC peak of wastransfer formed nodes at thein GMN positions was of significantly the transfer higher nodes, than and theother highest nodes peak in 2006. was “KecunTherefore, Station” a degree in‐ 2006.centrality In 2010 peak and was 2015, formed the DC at the of nodespositions in GMN of the formedtransfer two nodes, peak and lines, the highest namely peakLine 2was running “Kecun from Station” north toin south2006. andIn 2010 Line and 5 running 2015, the from DC east of nodes to west, in GMN and they formed meet two at “Guangzhou peak lines, namelyRailway Line Station”. 2 running Therefore, from thenorth position to south of Guangzhouand Line 5 running Railway from Station east was to west, the highest and they peak meet of DC at “Guangzhouin 2010 and 2015. Railway In 2018, Station”. a new Therefore, peak line wasthe position raised along of Guangzhou Line 4 (running Railway from Station north towas south), the highestfurthermore, peak of the DC DC in of 2010 nodes and located 2015. In in the2018, southern a new peak region line has was been raised improved along obviously.Line 4 (running The reason from northis that to the south), opening furthermore, of Line 7 in 2018the DC extends of nodes to the located south to in connect the southern the lines region of 2, 3 has and been 4, which improved finally obviously.are connected The with reason each is other. that the Compared opening withof Line the 7 southern in 2018 extends region, the to the DC south of the to nodes connect located the inlines the ofnorth 2, 3 isand lower 4, which because finally of their are connected relatively highwith spatialeach other. dispersion. Compared This with also the results southern in poor region, network the DCfault of tolerance the nodes in located the northern in the region.north is lower because of their relatively high spatial dispersion. This also resultsIn Figure in poor 12, thenetwork betweenness fault tolerance centrality in the (BC) northern hierarchical region. distributions of the nodes in the GuangzhouIn Figure metro 12, networkthe betweenness in the years centrality of 2006, 2010,(BC) 2015,hierarchical and 2018 distributions is also illustrated. of the Innodes terms in of the the Guangzhouspatial and temporal metro network evolution in the of theyears betweenness of 2006, 2010, centrality 2015, and (BC) 2018 in GMN, is also the illustrated. center of theIn terms network of theis directly spatial changedand temporal from “Kecunevolution Station” of the tobetweenness “Jiahe Wanggang centrality Station”. (BC) in Unlike GMN, the the distribution center of the of networknode degree, is directly nodes withchanged larger from betweenness “Kecun Station” are not just to transfer“Jiahe Wanggang nodes, some Station”. nodes with Unlike smaller the distributionDC have high of BC.node This degree, implies nodes that with such larger nodes betweenness are also significant are not forjust the transfer connectivity nodes, betweensome nodes the withlines. smaller The BC DC of nodeshave high in GMN BC. This all took implies “Kukura that Station”such nodes as theare highest also significant peak in 2006,for the 2010, connectivity and 2015. betweenImportantly, the thelines. network The BC in of 2010 nodes and in 2015 GMN had all two took peak “Kukura lines of BC,Station” one running as the highest along Line peak 8 andin 2006, east 2010,to “Changang and 2015. Station” Importantly, to connect the network to Line in 2, and2010 the and other 2015 running had two west peak to lines “Wanshengwei of BC, one running Station” alongto connect Line to8 and Line east 4, as to the “Changang center of “Kecun Station” Station”. to connect It means to Line that 2, and the nodes the other on the running peak lineswest are to “Wanshengweisignificant for the Station” connectivity to connect of the to network. Line 4, as In the 2018, center the of BC “Kecun of nodes Station”. located It inmeans the northern that the regionnodes onwas the significantly peak lines are higher significant than that for in the other connectivity areas, moreover, of the network. the highest In 2018, peak the changed BC of nodes from “Kecunlocated inStation” the northern to “Jiahe region Wanggang was significantly Station”. Furthermore,higher than that with in “Jiaheother areas, Wanggang moreover, Station” the highest as the center, peak changedone peak from line went“Kecun north Station” to “Xinhe to “Jiahe Station” Wanggang along Line Station”. 14 and Furthermore, the other peak with line “Jiahe south Wanggang to “Kecun Station” along Line 3. It is clear that Jiahe Wanggang Station has evolved into the most critical node in GMN. If the node was blocked, the lines in the northern region would be disconnected from the whole network. Sustainability 2020, 12, 538 14 of 18

Station” as the center, one peak line went north to “Xinhe Station” along Line 14 and the other peak line south to “Kecun Station” along Line 3. It is clear that Jiahe Wanggang Station has evolved into Sustainabilitythe most2020 critical, 12, 538 node in GMN. If the node was blocked, the lines in the northern region would be14 of 18 disconnected from the whole network.

Node degree Betweenness (a) 2018

Node degree Betweenness (b) 2015

Figure 12. Cont. Sustainability 2020, 12, 538 15 of 18 Sustainability 2020, 12, 538 15 of 18

Node degree Betweenness (d) 2006

FigureFigure 12. Spatio-temporal12. Spatio‐temporal evolution evolution of degree of degree centrality centrality and betweennessand betweenness centrality centrality in the in Guangzhou the MetroGuangzhou Network. Metro Network.

4. Conclusions4. Conclusions ThisThis paper paper takes takes the the city city of of Guangzhou Guangzhou as the the study study area area and and uses uses the the complex complex network network theory theory to constructto construct the the OD OD matrixes matrixes of of the the Guangzhou Guangzhou Metro Metro Network Network in the in the key key years years from from 1999 1999to 2018, to 2018, and the corresponding undirected weighted network models based on Space‐P are achieved. And and the corresponding undirected weighted network models based on Space-P are achieved. And thus, thus, the paper focuses on the evolution and evaluation of the topological properties and the paper focuses on the evolution and evaluation of the topological properties and characteristics of characteristics of the metro network of Guangzhou with the integration of geographic information the metro network of Guangzhou with the integration of geographic information system (GIS). The aim of the paper is to provide a scientific basis and decision support for the metro network planning and operation management in Guangzhou. The results derived from this study can be concluded in the following. Firstly, the metro network of Guangzhou illustrates the basic characteristics of a Sustainability 2020, 12, 538 16 of 18 small-world network. In the network, there is a large number of nodes with low degree centrality and a small number of nodes with high degree centrality (i.e., HUB nodes). Moreover, the average path length of the network is much smaller than the network diameter, specifically, it increases more slowly than the network diameter of the network. Secondly, with the growth of the metro network in Guangzhou, while the complexity of the network increases, the spatial dispersion of internal nodes tends to ease. Nevertheless, a large number of nodes spread outward, leading to an increase in the average travel time and the average transfer rate of the network, leading to the decreasing of the network efficiency. Thirdly, the connectivity indexes of α and γ in GMN are gradually decreasing, and the average time of transfer increases linearly with the growth of network. Combining with the gradually decreasing characteristics of the network efficiency, it reveals that the nodes in the network become more dispersed with the growth of the network, and the transmission of passenger flow in the network becomes more difficult. Fourthly, a higher average clustering coefficient and a larger average node degree indicate that the metro network in Guangzhou has better overall fault tolerance, i.e., there are more routes to choose from among the nodes. However, there are spatial differences, particularly the paths to or from the northern region are relatively poor in substitution. Finally, based on the evolution of the degree centrality of nodes, the center of the metro network in Guangzhou is gradually moving north along “Kecun Station–Guangzhou Railway Station–Jiahe Wanggang Station”, but from the evolution of the betweenness centrality of nodes, the metro network center is directly changed from “Kecun Station” to “Jiahe Wanggang Station”. Jiahe Wanggang Station has evolved into the most critical node in the current network. If the node is blocked, all lines in the northern region will be disconnected from the whole network. The above results mentioned in this paper should provide important reference information and decision support for urban planners or metro operation managers of the city of Guangzhou, including what are the topology properties and characteristics of the metro network, how to measure the attack resistance and robustness of the metro network, and how to optimize and strengthen the equilibrium and induction of passenger flow in the metro network based on the importance of a node in the context of complex network analysis. The future work of this paper will further investigate and characterize the accessibility performance and existing issues of the metro network under the specific network topology, through the integration of accessibility measuring models and complex network analysis, and finally, provide theoretical basis and policy suggestions for the scientific planning and construction of the urban metro network in the city of Guangzhou.

Author Contributions: Data curation, S.C.; Formal analysis, S.C.; Investigation, S.C.; Project administration, D.Z.; Writing—original draft, S.C. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by the Guangdong Natural Science Fund (2015A030313626). Conﬂicts of Interest: The authors declare that there are no conﬂicts of interest regarding the publication of this paper.

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