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Insight to the express transport network
Computer Physics Communications 180 (2009) 1511–1515
Contents lists available at ScienceDirect
Computer Physics Communications www.elsevier.com/locate/cpc
Insight to the express transport network Hua Yang a,b , Yuchao Nie a,b , Hongbin Zhang c,d , Zengru Di a,b , Ying Fan a,b,∗ a
Department of Systems Science, Beijing Normal University, Beijing 100875, P.R. China Center for Complexity Research, Beijing Normal University, Beijing 100875, P.R. China c School of Science, China Agricultural University, Beijing 100083, P.R. China d CITIC Group, Beijing 100004, P.R. China b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 February 2009 Received in revised form 9 April 2009 Accepted 13 April 2009 Available online 16 April 2009 PACS: 89.40.-a 89.75.Fb Keywords: Express transport network Statistical properties Spatial structure Evolution model
The express delivery industry is developing rapidly in recent years and has attracted attention in many fields. Express shipment service requires that parcels be delivered in a limited time with a low operation cost, which requests a high level and efficient express transport network (ETN). The ETN is constructed based on the public transport networks, especially the airline network. It is similar to the airline network in some aspects, while it has its own feature. With the complex network theory, the topological properties of the ETN are analyzed deeply. We find that the ETN has the small-world property, with disassortative mixing behavior and rich club phenomenon. It also shows difference from the airline network in some features, such as edge density and average shortest path. Analysis on the corresponding distance-weighted network shows that the distance distribution displays a truncated power-law behavior. At last, an evolving model, which takes both geographical constraint and preference attachment into account, is proposed. The model shows similar properties with the empirical results. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Express delivery service process involves the pickup process and delivery of parcels, it emphasizes on service quality, especially the timeliness of service (e.g., 24 hours, 48 hours). Customers adopting this expedited service pay a premium and, in return, receive reliable service meeting their standards. Express delivery service providers are interested in the express transport network which cannot only satisfy the customers’ needs, but also maintain a low operation cost. Express transport network (ETN) is constructed based on the public transport networks, including the airline network, the ground (car or train) transport network, and the sea transportation network. Generally, the ETN has low operating cost and high efficiency, it can be regarded as a high efficient subgraph of the public transport network. Many researchers have already studied the ETN from the perspective of operational research and also acquired many achievements. These research cared more about the design of the network and solution algorithms of the design problem. In recent years, understanding the structure and function of complex networks has become the foundation for explaining many different real world complex systems, including biological, tech-
*
Corresponding author at: Department of Systems Science, Beijing Normal University, Beijing 100875, P.R. China. E-mail address: [email protected] (Y. Fan). 0010-4655/$ – see front matter doi:10.1016/j.cpc.2009.04.004
©
2009 Elsevier B.V. All rights reserved.
nological, and social systems [1–3]. From the view of complex network, all kinds of transportation systems can be described as complex networks. During the past several years, complex network has been used to study transportation systems like roads or railway transportation network [4] and airline network. Recently, the worldwide airline network (WAN) has been studied from topological as well as traffic dynamics perspective [5–7], many national domestic airline networks also have been analyzed, including the airline network of China (ANC) [8], Italian airline network [9], U.S. intercity passenger air transportation network [10], India airline network [11]. As we know, the ETN is based on the airline networks mainly, but it is different from airline networks in network structure and function. It is necessary to analyze such a kind of empirical network to find the similarity and difference between the ETN and airline network, which will benefits the design and optimization of the express delivery system in the future. In this paper, we study the ETN using complex network theory. First, we regard the network as an unweighted network to investigate its topological properties. This network is formed by considering whether or not a pair of cities is connected by direct delivery route line. Then take the distance between cities into account and investigate the corresponding weighted network. Our analysis shows that the ETN has small-world property which also exists in airline networks. We also find the ETN has a disassortative mixing pattern with obvious rich club phenomenon. Different from other domestic airline networks, the ETN has a
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H. Yang et al. / Computer Physics Communications 180 (2009) 1511–1515
Fig. 1. The route line map of the ETN.
lower clustering coefficient and a higher average shortest path. The analysis in the distance-weighted network shows that distance between cities in the ETN shows a truncated power-law distribution. At last, an evolution model considering both geographical constraint and preference attachment, is proposed to express the evolution mechanism of the real network. 2. The express transport network Data. The network data is collected from a Chinese logistics company. It contains 301 cities and 421 routes managed by the company in 2007. In this network, a city with more than one express service sector is considered as a node, and edge represents the delivery route between cities. Unweighted express transport network. We build an unweighted binary adjacency matrix A ( N , N ), whose elements ai j take value 1 if there exist a route line from city i to city j and 0 otherwise. Fig. 1 is the route line map of the ETN. First, the statistic properties of the ETN are analyzed. Its average shortest path is 3.6, and the average clustering coefficient is 0.134. Compared with the corresponding random network whose average shortest path is 5.546 and clustering coefficient is 0.0093, the ETN has a lower average shortest path and higher average clustering coefficient, it indicates that the ETN is a small-world network which is similar with many airline networks [7–11]. We use the method proposed by Clauset and Newman in [12] to verify the degree distribution and other distributions in this paper. The degree distribution of the ETN displays a broad distribution, as shown in Fig. 2. We compare the ETN with many airline networks, as show in Table 1. It is found that the ETN is different from airline networks in some topological properties. The ETN has a lower average clustering coefficient and a higher average shortest path than all the domestic airline networks, this difference can be explained by the fact that the ETN is sparser than airline networks, and these differences in network structure may be related to different network functions. In the ETN, cargo can be transferred many times as long as it reach the destination on time. The ETN is constructed from
Fig. 2. The accumulative distribution of degree distribution (the line slope is −2.26).
the overall optimization, it emphasizes the trade off between efficiency and cost; while in the airline network, people prefer flying to the target directly or least transfers when traveling, the airline network has to provide travelers many airlines to make more convenience. Further more, the average shortest path in every domestic airline network is close to 2, meaning that passengers need to be transferred once on average when flying between domestic cities. In the world airline network, because of the geographical constraints and the geopolitical constraints [7], passengers need to be transferred more than three times on average. In fact, when passenger flies between two cities in different countries, firstly he needs to fly from the origin city to the national hub airport, then from the hub airport to hub airport located in the other country, and at last, from the hub airport to the destination city. This global fly process is similar with the express delivery process, maybe that is why the edge density and average shortest path in these two networks are similar. Communities of the ETN. We use the GN algorithm proposed by Girvan and Newman [13] to detect the community structure of this network. With the maximum modularity Q value is 0.688, this network is divided into 16 communities, which is shown in Fig. 3(a). Compared with the real enterpriser partition by area, we find that this network contains a community composed by a lot of hub cities and other 15 communities which are consistent with these geographic location. The biggest community (see Fig. 3(b)) contains many cities which are spatial faraway but interact tightly, it indicates that the network may has rich club phenomenon. Mixing pattern and rich club phenomenon. The ETN’s assortative coefficient α is −0.32, it describes the ETN is a disassortative network [14], which means that in the ETN, high-degree nodes tend to connect with low-degree nodes more likely. Previous studies have found that many social networks have assortative mixing pattern, which makes the network more reliable for target attack and some other benefits, while the technical and biological networks are dis-
Table 1 Basic topological properties of the ETN and airline networks, including the number of nodes, the number of edges, the edge density, the average shortest path, the average clustering coefficient in network. Network
Nodes number
Edges number
Edge density
Average shortest path
Average clustering coefficient
World airport network China airport network Italian airport network Indian airport network U.S air transport network Express transport network
3883 128 42 79 272 301
27051 1165 310 442 6566 421
0.004 0.143 0.360 0.143 0.178 0.009
4.4 2.01 1.98–2.14 1.26 1.9 3.6
0.62 0.733 0.07–0.10 0.657 0.75 0.134
H. Yang et al. / Computer Physics Communications 180 (2009) 1511–1515
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Fig. 3. (a) The community structure of the ETN, different signs represent different communities. (b) The biggest community in the ETN.
Fig. 4. (a) The index
ρ (k) with degree k, it indicates the ETN has the rich club phenomenon at high degree. (b) The interconnections among the richest 6 nodes in the ETN.
assortative network, which is sensitive to the target attack [15]. As we know, the ETN, airline network, Internet and other disassortative networks are prevalent in the real world, the advantages of this kind of networks need to be more discussed in the future. The rich club phenomenon shows that in a network, highdegree nodes are tightly interconnected with other rich nodes and form a core group [16]. Usually, the index ρ (k) = ϕ (k)/ϕran (k) is used to evaluate the rich club phenomenon [17]. Here, ϕ (k) = 2E >k /( N >k ( N >k − 1)) is the rich club coefficient of the origin network, N >k is the number of nodes whose degree are larger than k, E >k is number of edges among the N >k nodes in the real network, N >k ( N >k − 1)/2 represents the maximum possible number of edges among the N >k nodes. ϕran (k) is the rich club coefficient of the maximally random network which has the same degree distribution P (k) with the origin network. When the value of ρ (k) is larger than 1, it concludes the rich club phenomenon. Here we detect the rich club phenomenon in the ETN. Fig. 4(a) shows that the ETN has a obvious rich club phenomenon at the high degree. In the ETN, hub cities usually connect with many small cities, and interconnect with other hub cities tightly. For example, we show the interconnection among nodes whose degree is larger than 25, these 6 cities are nearly full connected, as shown in Fig. 4(b). Analysis of the distance-weighted network. Different from the preceding research on airline networks, we do not take the traffic flow as the weight of the network, but the distance instead. The spatial structure of network is very important and has been concerned by some works, such as the effect of spatial structure on the tradeoff between efficiency and cost [18]. Here we first investigate the spatial structure of the ETN, and then propose a model to
reproduce both the topological properties and the spatial structure of the real ETN. We found the distance in ETN displays a truncated power-law distribution (see Fig. 5(a)). The power-law distance distribution also exists in some airline networks, in the Japanese airline network [19], even there is an exponential decay in domestic flights, the distance distribution follows power-law when international flights are added; In the U.S. air network [10], the distribution of link distance has a power-law tail, the power law distance distribution is interesting and should be concerned. We also calculate the nodes’ strength in ETN, it also displays the power-law feature, as shown in Fig. 5(b). Evolution Model of the ETN. In fact, the ETN is shaped by the geographical constraint, both the topological properties and spatial structure will be considered when modeling it. Based on the classical BA model [20], we propose a modified evolution model named evolving model with geographical constraint. This model generates not only the basic topological properties of the unweighted network, but also the spatial structure of the ETN. The evolution model is as below: 1. Start with a small number m0 of fully connected nodes. The initial m0 modes are considered to be the hub cities in the entire network. This initialization can make the model results more reliable. 2. At every step, a new node is added with m edges to the existing nodes whose distance from the new node are less than L. The probability of the new node i to connect with an existing node j is: P (i , j ) = Π j / i = j Π j , where Π j = k j /di j . (k j is the degree of node j, di j is the distance from node i to j.)
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Fig. 5. (a) The accumulative distribution of distance in the ETN (the line slope is −1.87). (b) The accumulative distribution of the nodes’ strength in the ETN (the line slope is −1.76).
Fig. 6. (a) The accumulative distribution of degree in the model (the line slope is −2.14). (b) The accumulative distribution of distance in the model (the line slope is −1.76). (c) The accumulative distribution of nodes’ strength in the model (the line slope is −1.73). (d) The index ρ (k) with degree k in the model, it indicates the model has the rich club phenomenon at high degree.
3. At every step, with P probability, a new edge is added between two existing nodes. The probability of edge created be tween two existing nodes i and j is P (i , j ) = Πi j / i = j Πi j , where Πi j = ki k j /di j . (ki , k j are separately the degree of node i and j, the di j is the distance from node i to j.) We choose m0 = 6, the distance between any pair of nodes di j is calculated as the Euclidean distance based on the Longitude and Latitude of city node. In the evolution process, only the existing nodes, whose distance from the new node are less than L, have chance to connect with the new node. This evolution process finishes until the number of nodes reaches 301. Considering the scale of the model be close to the real ETN, in every step, m = 1 edge is
added for every new node, and the probability P = 0.4. By parameter testing, the value of L is decided to be 10, which makes the model matches the real ETN best. The model has same properties with real ETN. The degree distribution, the distance distribution and nodes’ strength distribution are consistent with the origin ETN qualitatively. We also test the mixing pattern and rich club phenomenon of the model, the assortative coefficient is −0.42, and it also has rich club phenomenon, as shown in Fig. 6. It is worth mentioning that this model results are identical to the real ETN not only on the topological properties, but also the spatial structure, as reflected with the distance distribution and nodes’ strength distribution.
H. Yang et al. / Computer Physics Communications 180 (2009) 1511–1515
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3. Conclusions
[2] S.N. Dorogovtsev, J.F.F. Mendes, Evolution of networks, Adv. Phys. 51 (2002) 1079.
In this paper, we analyze the characteristics of the ETN from the perspective of complex network. First, we investigate the ETN as an unweighed network, in which the nodes represent the cities and the links between the nodes stand for the express routes. The analysis on the unweighed express network shows that the ETN is a small-world network. We compare the ETN with several airport networks and find some similarities and difference with the airline networks. We divided the ETN into 16 communities, the community structure verify the geographical features of the express network. Next, we calculate the mixing pattern value and analyze the rich club phenomenon, the disassortative mixing behavior make a description of the hub-existing in the network, the rich club phenomenon state that the hub cities connect more tightly and form a core club. In the distance-weighted network, the distance displays a truncated power-law behavior and strength has a powerlaw distribution. At last, we propose an evolution model which considers both geographical constraint and preference attachment. The model results are consistent well with the empirical results not only in the unweighted network but also the corresponding distance-weighted network. Our results indicate that considering the spatial structure in the model will be more reasonable and reliable, the spatial structure is important in the spatial networks and should be taken into account when modeling real complex systems.
[3] M.E.J. Newman, The structure and function of complex networks, SIAM Rev. 45 (2003) 167. [4] P. Sen, S. Dasgupta, A. Chatterjee, P.A. Sreeram, G. Mukherjee, S.S. Manna, Phys. Rev. E 67 (2003) 036106. [5] L.A.N. Amaral, A. Scala, M. Barthelemy, H.E. Stanley, Classes of small-world networks, Proc. Natl. Acad. Sci. (USA) 97 (2000) 11149–11152. [6] R. Guimer, L.A.N. Amaral, Modeling the world-wide airport network, European Phys. J. B 38 (2004) 381–385. [7] R. Guimer, S. Mossa, A. Turtschi, L.A.N. Amaral, The worldwide air transportation network: Anomalous centrality, community structure, and cities global roles, Proc. Natl. Acad. Sci. (USA) 102 (2005) 7794–7799. [8] W. Li, X. Cai, Statistical analysis of airport network of China, Phys. Rev. E 69 (2004) 46106. [9] M. Guida, F. Maria, A topology of the Italian airport network: A scale-free small-world network with a fractal structure, Chaos, Solitons and Fractals 31 (2007) 527–536. [10] Z. Xu, R. Harriss, Exploring the structure of the U.S. intercity passenger air transportation network: A weighted complex network approach Robert Harriss, GeoJournal 73 (2008) 87–102. [11] G. Bagler, Analysis of the airport network of India as a complex weighted network, Physica A 387 (2008) 2972–2980. [12] A. Clauset, C.R. Shalizi, M.E.J. Newman, Power-law distributions in empirical data, SIAM Rev. (2009), in press, arXiv:0706.1062. [13] M. Girvan, M.E.J. Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99 (2002) 7821–7826. [14] M.E.J. Newman, Mixing patterns in networks, Phys. Rev. E 67 (2003) 026126. [15] R. Albert, H. Jeong, A. Barabasi, Error and attack tolerance of complex networks, Nature 406 (2000) 378–381.
Acknowledgements
[16] S. Zhou, R.J. Mondragon, The rich-club phenomenon in the Internet topology, IEEE Commun. Lett. 8 (2004) 180–182.
This work is partially supported by 985 Projet and NSFC under the grant Nos. 70771011, 60534080 and 70471080. The authors would like to thank the reviewers for their useful comments.
[17] V. Colizza, A. Flammini, M.A. Serrano, A. Vespignani, Detecting rich-club ordering in complex networks, Nat. Phys. 2 (2006) 110–115.
References [1] R. Albert, A.-L. Barabasi, Statistical mechanics of complex networks, Rev. Modern Phys. 74 (2002) 47.
[18] M.T. Gastner, M.E.J. Newman, The spatial structure of networks, European Phys. J. B 49 (2006) 247–252. [19] Y. Hayashi, A review of recent studies of geographical scale-free networks, IPSJ Digital Courier 2 (2006) 155–164. [20] A.-L. Barabasi, R. Albert, H. Jeong, Mean-field theory for scale-free random networks, Physica A 272 (1999).
Contents lists available at ScienceDirect
Computer Physics Communications www.elsevier.com/locate/cpc
Insight to the express transport network Hua Yang a,b , Yuchao Nie a,b , Hongbin Zhang c,d , Zengru Di a,b , Ying Fan a,b,∗ a
Department of Systems Science, Beijing Normal University, Beijing 100875, P.R. China Center for Complexity Research, Beijing Normal University, Beijing 100875, P.R. China c School of Science, China Agricultural University, Beijing 100083, P.R. China d CITIC Group, Beijing 100004, P.R. China b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 February 2009 Received in revised form 9 April 2009 Accepted 13 April 2009 Available online 16 April 2009 PACS: 89.40.-a 89.75.Fb Keywords: Express transport network Statistical properties Spatial structure Evolution model
The express delivery industry is developing rapidly in recent years and has attracted attention in many fields. Express shipment service requires that parcels be delivered in a limited time with a low operation cost, which requests a high level and efficient express transport network (ETN). The ETN is constructed based on the public transport networks, especially the airline network. It is similar to the airline network in some aspects, while it has its own feature. With the complex network theory, the topological properties of the ETN are analyzed deeply. We find that the ETN has the small-world property, with disassortative mixing behavior and rich club phenomenon. It also shows difference from the airline network in some features, such as edge density and average shortest path. Analysis on the corresponding distance-weighted network shows that the distance distribution displays a truncated power-law behavior. At last, an evolving model, which takes both geographical constraint and preference attachment into account, is proposed. The model shows similar properties with the empirical results. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Express delivery service process involves the pickup process and delivery of parcels, it emphasizes on service quality, especially the timeliness of service (e.g., 24 hours, 48 hours). Customers adopting this expedited service pay a premium and, in return, receive reliable service meeting their standards. Express delivery service providers are interested in the express transport network which cannot only satisfy the customers’ needs, but also maintain a low operation cost. Express transport network (ETN) is constructed based on the public transport networks, including the airline network, the ground (car or train) transport network, and the sea transportation network. Generally, the ETN has low operating cost and high efficiency, it can be regarded as a high efficient subgraph of the public transport network. Many researchers have already studied the ETN from the perspective of operational research and also acquired many achievements. These research cared more about the design of the network and solution algorithms of the design problem. In recent years, understanding the structure and function of complex networks has become the foundation for explaining many different real world complex systems, including biological, tech-
*
Corresponding author at: Department of Systems Science, Beijing Normal University, Beijing 100875, P.R. China. E-mail address: [email protected] (Y. Fan). 0010-4655/$ – see front matter doi:10.1016/j.cpc.2009.04.004
©
2009 Elsevier B.V. All rights reserved.
nological, and social systems [1–3]. From the view of complex network, all kinds of transportation systems can be described as complex networks. During the past several years, complex network has been used to study transportation systems like roads or railway transportation network [4] and airline network. Recently, the worldwide airline network (WAN) has been studied from topological as well as traffic dynamics perspective [5–7], many national domestic airline networks also have been analyzed, including the airline network of China (ANC) [8], Italian airline network [9], U.S. intercity passenger air transportation network [10], India airline network [11]. As we know, the ETN is based on the airline networks mainly, but it is different from airline networks in network structure and function. It is necessary to analyze such a kind of empirical network to find the similarity and difference between the ETN and airline network, which will benefits the design and optimization of the express delivery system in the future. In this paper, we study the ETN using complex network theory. First, we regard the network as an unweighted network to investigate its topological properties. This network is formed by considering whether or not a pair of cities is connected by direct delivery route line. Then take the distance between cities into account and investigate the corresponding weighted network. Our analysis shows that the ETN has small-world property which also exists in airline networks. We also find the ETN has a disassortative mixing pattern with obvious rich club phenomenon. Different from other domestic airline networks, the ETN has a
1512
H. Yang et al. / Computer Physics Communications 180 (2009) 1511–1515
Fig. 1. The route line map of the ETN.
lower clustering coefficient and a higher average shortest path. The analysis in the distance-weighted network shows that distance between cities in the ETN shows a truncated power-law distribution. At last, an evolution model considering both geographical constraint and preference attachment, is proposed to express the evolution mechanism of the real network. 2. The express transport network Data. The network data is collected from a Chinese logistics company. It contains 301 cities and 421 routes managed by the company in 2007. In this network, a city with more than one express service sector is considered as a node, and edge represents the delivery route between cities. Unweighted express transport network. We build an unweighted binary adjacency matrix A ( N , N ), whose elements ai j take value 1 if there exist a route line from city i to city j and 0 otherwise. Fig. 1 is the route line map of the ETN. First, the statistic properties of the ETN are analyzed. Its average shortest path is 3.6, and the average clustering coefficient is 0.134. Compared with the corresponding random network whose average shortest path is 5.546 and clustering coefficient is 0.0093, the ETN has a lower average shortest path and higher average clustering coefficient, it indicates that the ETN is a small-world network which is similar with many airline networks [7–11]. We use the method proposed by Clauset and Newman in [12] to verify the degree distribution and other distributions in this paper. The degree distribution of the ETN displays a broad distribution, as shown in Fig. 2. We compare the ETN with many airline networks, as show in Table 1. It is found that the ETN is different from airline networks in some topological properties. The ETN has a lower average clustering coefficient and a higher average shortest path than all the domestic airline networks, this difference can be explained by the fact that the ETN is sparser than airline networks, and these differences in network structure may be related to different network functions. In the ETN, cargo can be transferred many times as long as it reach the destination on time. The ETN is constructed from
Fig. 2. The accumulative distribution of degree distribution (the line slope is −2.26).
the overall optimization, it emphasizes the trade off between efficiency and cost; while in the airline network, people prefer flying to the target directly or least transfers when traveling, the airline network has to provide travelers many airlines to make more convenience. Further more, the average shortest path in every domestic airline network is close to 2, meaning that passengers need to be transferred once on average when flying between domestic cities. In the world airline network, because of the geographical constraints and the geopolitical constraints [7], passengers need to be transferred more than three times on average. In fact, when passenger flies between two cities in different countries, firstly he needs to fly from the origin city to the national hub airport, then from the hub airport to hub airport located in the other country, and at last, from the hub airport to the destination city. This global fly process is similar with the express delivery process, maybe that is why the edge density and average shortest path in these two networks are similar. Communities of the ETN. We use the GN algorithm proposed by Girvan and Newman [13] to detect the community structure of this network. With the maximum modularity Q value is 0.688, this network is divided into 16 communities, which is shown in Fig. 3(a). Compared with the real enterpriser partition by area, we find that this network contains a community composed by a lot of hub cities and other 15 communities which are consistent with these geographic location. The biggest community (see Fig. 3(b)) contains many cities which are spatial faraway but interact tightly, it indicates that the network may has rich club phenomenon. Mixing pattern and rich club phenomenon. The ETN’s assortative coefficient α is −0.32, it describes the ETN is a disassortative network [14], which means that in the ETN, high-degree nodes tend to connect with low-degree nodes more likely. Previous studies have found that many social networks have assortative mixing pattern, which makes the network more reliable for target attack and some other benefits, while the technical and biological networks are dis-
Table 1 Basic topological properties of the ETN and airline networks, including the number of nodes, the number of edges, the edge density, the average shortest path, the average clustering coefficient in network. Network
Nodes number
Edges number
Edge density
Average shortest path
Average clustering coefficient
World airport network China airport network Italian airport network Indian airport network U.S air transport network Express transport network
3883 128 42 79 272 301
27051 1165 310 442 6566 421
0.004 0.143 0.360 0.143 0.178 0.009
4.4 2.01 1.98–2.14 1.26 1.9 3.6
0.62 0.733 0.07–0.10 0.657 0.75 0.134
H. Yang et al. / Computer Physics Communications 180 (2009) 1511–1515
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Fig. 3. (a) The community structure of the ETN, different signs represent different communities. (b) The biggest community in the ETN.
Fig. 4. (a) The index
ρ (k) with degree k, it indicates the ETN has the rich club phenomenon at high degree. (b) The interconnections among the richest 6 nodes in the ETN.
assortative network, which is sensitive to the target attack [15]. As we know, the ETN, airline network, Internet and other disassortative networks are prevalent in the real world, the advantages of this kind of networks need to be more discussed in the future. The rich club phenomenon shows that in a network, highdegree nodes are tightly interconnected with other rich nodes and form a core group [16]. Usually, the index ρ (k) = ϕ (k)/ϕran (k) is used to evaluate the rich club phenomenon [17]. Here, ϕ (k) = 2E >k /( N >k ( N >k − 1)) is the rich club coefficient of the origin network, N >k is the number of nodes whose degree are larger than k, E >k is number of edges among the N >k nodes in the real network, N >k ( N >k − 1)/2 represents the maximum possible number of edges among the N >k nodes. ϕran (k) is the rich club coefficient of the maximally random network which has the same degree distribution P (k) with the origin network. When the value of ρ (k) is larger than 1, it concludes the rich club phenomenon. Here we detect the rich club phenomenon in the ETN. Fig. 4(a) shows that the ETN has a obvious rich club phenomenon at the high degree. In the ETN, hub cities usually connect with many small cities, and interconnect with other hub cities tightly. For example, we show the interconnection among nodes whose degree is larger than 25, these 6 cities are nearly full connected, as shown in Fig. 4(b). Analysis of the distance-weighted network. Different from the preceding research on airline networks, we do not take the traffic flow as the weight of the network, but the distance instead. The spatial structure of network is very important and has been concerned by some works, such as the effect of spatial structure on the tradeoff between efficiency and cost [18]. Here we first investigate the spatial structure of the ETN, and then propose a model to
reproduce both the topological properties and the spatial structure of the real ETN. We found the distance in ETN displays a truncated power-law distribution (see Fig. 5(a)). The power-law distance distribution also exists in some airline networks, in the Japanese airline network [19], even there is an exponential decay in domestic flights, the distance distribution follows power-law when international flights are added; In the U.S. air network [10], the distribution of link distance has a power-law tail, the power law distance distribution is interesting and should be concerned. We also calculate the nodes’ strength in ETN, it also displays the power-law feature, as shown in Fig. 5(b). Evolution Model of the ETN. In fact, the ETN is shaped by the geographical constraint, both the topological properties and spatial structure will be considered when modeling it. Based on the classical BA model [20], we propose a modified evolution model named evolving model with geographical constraint. This model generates not only the basic topological properties of the unweighted network, but also the spatial structure of the ETN. The evolution model is as below: 1. Start with a small number m0 of fully connected nodes. The initial m0 modes are considered to be the hub cities in the entire network. This initialization can make the model results more reliable. 2. At every step, a new node is added with m edges to the existing nodes whose distance from the new node are less than L. The probability of the new node i to connect with an existing node j is: P (i , j ) = Π j / i = j Π j , where Π j = k j /di j . (k j is the degree of node j, di j is the distance from node i to j.)
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Fig. 5. (a) The accumulative distribution of distance in the ETN (the line slope is −1.87). (b) The accumulative distribution of the nodes’ strength in the ETN (the line slope is −1.76).
Fig. 6. (a) The accumulative distribution of degree in the model (the line slope is −2.14). (b) The accumulative distribution of distance in the model (the line slope is −1.76). (c) The accumulative distribution of nodes’ strength in the model (the line slope is −1.73). (d) The index ρ (k) with degree k in the model, it indicates the model has the rich club phenomenon at high degree.
3. At every step, with P probability, a new edge is added between two existing nodes. The probability of edge created be tween two existing nodes i and j is P (i , j ) = Πi j / i = j Πi j , where Πi j = ki k j /di j . (ki , k j are separately the degree of node i and j, the di j is the distance from node i to j.) We choose m0 = 6, the distance between any pair of nodes di j is calculated as the Euclidean distance based on the Longitude and Latitude of city node. In the evolution process, only the existing nodes, whose distance from the new node are less than L, have chance to connect with the new node. This evolution process finishes until the number of nodes reaches 301. Considering the scale of the model be close to the real ETN, in every step, m = 1 edge is
added for every new node, and the probability P = 0.4. By parameter testing, the value of L is decided to be 10, which makes the model matches the real ETN best. The model has same properties with real ETN. The degree distribution, the distance distribution and nodes’ strength distribution are consistent with the origin ETN qualitatively. We also test the mixing pattern and rich club phenomenon of the model, the assortative coefficient is −0.42, and it also has rich club phenomenon, as shown in Fig. 6. It is worth mentioning that this model results are identical to the real ETN not only on the topological properties, but also the spatial structure, as reflected with the distance distribution and nodes’ strength distribution.
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3. Conclusions
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In this paper, we analyze the characteristics of the ETN from the perspective of complex network. First, we investigate the ETN as an unweighed network, in which the nodes represent the cities and the links between the nodes stand for the express routes. The analysis on the unweighed express network shows that the ETN is a small-world network. We compare the ETN with several airport networks and find some similarities and difference with the airline networks. We divided the ETN into 16 communities, the community structure verify the geographical features of the express network. Next, we calculate the mixing pattern value and analyze the rich club phenomenon, the disassortative mixing behavior make a description of the hub-existing in the network, the rich club phenomenon state that the hub cities connect more tightly and form a core club. In the distance-weighted network, the distance displays a truncated power-law behavior and strength has a powerlaw distribution. At last, we propose an evolution model which considers both geographical constraint and preference attachment. The model results are consistent well with the empirical results not only in the unweighted network but also the corresponding distance-weighted network. Our results indicate that considering the spatial structure in the model will be more reasonable and reliable, the spatial structure is important in the spatial networks and should be taken into account when modeling real complex systems.
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Acknowledgements
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This work is partially supported by 985 Projet and NSFC under the grant Nos. 70771011, 60534080 and 70471080. The authors would like to thank the reviewers for their useful comments.
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