Physica A388(2009)2061-2071 Contents lists available at ScienceDirect Physica A ELSEVIER journal homepage:www.elsevier.com/locate/physa Empirical analysis of the worldwide maritime transportation network Yihong Hu3,Daoli Zhu a.b.* School of Management,Fudan University.Shanghai 200433.China b Shanghai Logistics Institute,Shanghai 200433.China ARTICLE INFO ABSTRACT Article history: In this paper we present an empirical study of the worldwide maritime transportation Received 12 June 2008 network(WMN)in which the nodes are ports and links are container liners connecting the Received in revised form 14 September 2008 ports.Using the different representations of network topology-the spaces L and P.we Available online 16 December 2008 study the statistical properties of WMN including degree distribution,degree correlations, weight distribution,strength distribution,average shortest path length,line length distribution and centrality measures.We find that WMN is a small-world network with Keywords: Scaling law power law behavior.Important nodes are identified based on different centrality measures Transportation network Through analyzing weighted clustering coefficient and weighted average nearest neighbors Complex system degree,we reveal the hierarchy structure and rich-club phenomenon in the network. 2008 Elsevier B.V.All rights reserved. 1.Introduction The recent few years have witnessed a great devotion to exploration and understanding of underlying mechanism of complex systems [1-5]as diverse as the Internet [6,7].social networks [8]and biological networks [9].As critical infrastructure,transportation networks are widely studied.Examples include airline [10-15].ship [16].bus [17-20]. subway 21]and railway 22,23]networks. Maritime transportation plays an important role in the world merchandise trade and economics development.Most of the large volume cargo between countries like crude oil,iron ore,grain,and lumber are carried by ocean vessels.According to the statistics from United Nations[24],the international seaborne trade continuously increased to 7.4 billion tons in 2006 with a robust annual growth rate of 4.3%.And over 70%of the value of world international seaborne trade is being moved in containers. Container liners have become the primary transportation mode in maritime transport since 1950's.Liner shipping means the container vessels travel along regular routes with fixed rates according to regular schedules.At present most of the shipping companies adopt hub-and-spoke operating structure which consists of hub ports,lateral ports,main lines and branch lines,forming a complex container transportation network system[25]. Compared with other transportation networks,the maritime container liner networks have some distinct features: (1)A great number of the routes of container liners are circular.Container ships call at a series of ports and return to the origin port without revisiting each intermediate port.It's called pendulum service in container transportation.While bus transport networks and railway networks are at the opposite with most of buses or trains running bidirectionally on routes. (2)The network is directed and asymmetric due to circular routes.(3)Lines are divided into main lines and branch lines. Main lines are long haul lines which involves a set of sequential port calls across the oceans.Sometimes long haul lines call at almost 30 ports.Branch lines are short haul lines connecting several ports in one region to serve for main lines. Corresponding author at:School of Management.Fudan University.Shanghai 200433.China.TeL:+862165643072. E-mail address:dlzhu@fudan.edu.cn(D.Zhu). 0378-4371/S-see front matter 2008 Elsevier B.V.All rights reserved. doi10.1016j-physa.2008.12.016
Physica A 388 (2009) 2061–2071 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Empirical analysis of the worldwide maritime transportation network Yihong Hua , Daoli Zhua,b,∗ a School of Management, Fudan University, Shanghai 200433, China b Shanghai Logistics Institute, Shanghai 200433, China a r t i c l e i n f o Article history: Received 12 June 2008 Received in revised form 14 September 2008 Available online 16 December 2008 Keywords: Scaling law Transportation network Complex system a b s t r a c t In this paper we present an empirical study of the worldwide maritime transportation network (WMN) in which the nodes are ports and links are container liners connecting the ports. Using the different representations of network topology — the spaces L and P, we study the statistical properties of WMN including degree distribution, degree correlations, weight distribution, strength distribution, average shortest path length, line length distribution and centrality measures. We find that WMN is a small-world network with power law behavior. Important nodes are identified based on different centrality measures. Through analyzing weighted clustering coefficient and weighted average nearest neighbors degree, we reveal the hierarchy structure and rich-club phenomenon in the network. © 2008 Elsevier B.V. All rights reserved. 1. Introduction The recent few years have witnessed a great devotion to exploration and understanding of underlying mechanism of complex systems [1–5] as diverse as the Internet [6,7], social networks [8] and biological networks [9]. As critical infrastructure, transportation networks are widely studied. Examples include airline [10–15], ship [16], bus [17–20], subway [21] and railway [22,23] networks. Maritime transportation plays an important role in the world merchandise trade and economics development. Most of the large volume cargo between countries like crude oil, iron ore, grain, and lumber are carried by ocean vessels. According to the statistics from United Nations [24], the international seaborne trade continuously increased to 7.4 billion tons in 2006 with a robust annual growth rate of 4.3%. And over 70% of the value of world international seaborne trade is being moved in containers. Container liners have become the primary transportation mode in maritime transport since 1950’s. Liner shipping means the container vessels travel along regular routes with fixed rates according to regular schedules. At present most of the shipping companies adopt hub-and-spoke operating structure which consists of hub ports, lateral ports, main lines and branch lines, forming a complex container transportation network system [25]. Compared with other transportation networks, the maritime container liner networks have some distinct features: (1) A great number of the routes of container liners are circular. Container ships call at a series of ports and return to the origin port without revisiting each intermediate port. It’s called pendulum service in container transportation. While bus transport networks and railway networks are at the opposite with most of buses or trains running bidirectionally on routes. (2) The network is directed and asymmetric due to circular routes. (3) Lines are divided into main lines and branch lines. Main lines are long haul lines which involves a set of sequential port calls across the oceans. Sometimes long haul lines call at almost 30 ports. Branch lines are short haul lines connecting several ports in one region to serve for main lines. ∗ Corresponding author at: School of Management, Fudan University, Shanghai 200433, China. Tel.: +86 2165643072. E-mail address: dlzhu@fudan.edu.cn (D. Zhu). 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.12.016
2062 Y.u,D.Zhu/Physica A388(2009)2061-2071 Table 1 Number of sea ports by major geographic region. Region No of sea ports Africa 96 Asia and Middle East 251 Europe 311 North America 61 Latin America Oceania 63 Total 878 4 5+ 4 6+ 2 (a) (b) Line A Line B 5 16 6+ 2 1 (C) (d)" 8+ Fig.1.Description of the space L and the space P.(a)and(b)are the undirected representations in the space L and the space P.respectively.(c)and (d) are the directed representations in the space L and the space P,respectively.In the space L,a link is created between consecutive stops in one route.In the space P all ports that belong to the same route are connected.Line A(solid line)and line B(dashed line)are two different pendulum routes sharing one common node:the port No.1. We construct the worldwide maritime transportation network(WMN)using two different network representations. namely the spaces L and P,and analyze basic topological properties.Our result shows that the degree distribution follows a truncated power-law distribution in the space L and an exponential decay distribution in the space P.With small average shortest path length 2.66 and high clustering coefficient 0.7 in the space P,we claim that WMN is a small world network. We also check the weighted network and find the network has hierarchy structure and"rich-club"phenomenon.Centrality measures are found to have strong correlations with each other The rest of the paper is organized as follows:in Section 2,we introduce the database and set up the network using two different network representations.In Section 3 various topological properties are studied including degree distribution, degree correlations,shortest path length,weight distribution and strength distribution,etc.Section 4 discloses the hierarchy structure by studying the weighted and unweighted clustering and degree correlations.Centrality measures correlations and central nodes'geographical distribution are studied in Section 5.Section 6 gives the conclusion. 2.Construction of the network We get the original data from an authoritative container industry database named Cl-online[26]which provides up-to- date statistics of all the 878 sea ports in the world and 1802 container lines served by 434 ship companies covering all the big ship companies such as Maersk,MSC,Evergreen,Cosco,etc.The ports are distributed in different regions and we list the number of ports in each region in Table 1. To construct the worldwide maritime transportation network from the above data,we have to introduce the concept of spaces L and P as presented in Fig.1.The idea of spaces L and P is first proposed in a general form in [22]and later widely used in the study of public bus transportation networks and railway networks.The space L consists of nodes being ports and links created between consecutive stops in one route.Degree k in the space L represents the number of directions passengers or cargoes can travel at a given port.The shortest path length in the space L is the number of stops one has to pass to travel between any two ports.In the space P,two arbitrary ports are connected if there is a container line traveling between both ports.Therefore,degree k in the space P is the number of nodes which can be reached without changing the line.The shortest
2062 Y. Hu, D. Zhu / Physica A 388 (2009) 2061–2071 Table 1 Number of sea ports by major geographic region. Region No of sea ports Africa 96 Asia and Middle East 251 Europe 311 North America 61 Latin America 96 Oceania 63 Total 878 Fig. 1. Description of the space L and the space P. (a) and (b) are the undirected representations in the space L and the space P, respectively. (c) and (d) are the directed representations in the space L and the space P, respectively. In the space L, a link is created between consecutive stops in one route. In the space P all ports that belong to the same route are connected. Line A (solid line) and line B (dashed line) are two different pendulum routes sharing one common node: the port No. 1. We construct the worldwide maritime transportation network (WMN) using two different network representations, namely the spaces L and P, and analyze basic topological properties. Our result shows that the degree distribution follows a truncated power-law distribution in the space L and an exponential decay distribution in the space P. With small average shortest path length 2.66 and high clustering coefficient 0.7 in the space P, we claim that WMN is a small world network. We also check the weighted network and find the network has hierarchy structure and ‘‘rich-club’’ phenomenon. Centrality measures are found to have strong correlations with each other. The rest of the paper is organized as follows: in Section 2, we introduce the database and set up the network using two different network representations. In Section 3 various topological properties are studied including degree distribution, degree correlations, shortest path length, weight distribution and strength distribution, etc. Section 4 discloses the hierarchy structure by studying the weighted and unweighted clustering and degree correlations. Centrality measures correlations and central nodes’ geographical distribution are studied in Section 5. Section 6 gives the conclusion. 2. Construction of the network We get the original data from an authoritative container industry database named CI-online [26] which provides up-todate statistics of all the 878 sea ports in the world and 1802 container lines served by 434 ship companies covering all the big ship companies such as Maersk, MSC, Evergreen, Cosco, etc. The ports are distributed in different regions and we list the number of ports in each region in Table 1. To construct the worldwide maritime transportation network from the above data, we have to introduce the concept of spaces L and P as presented in Fig. 1. The idea of spaces L and P is first proposed in a general form in [22] and later widely used in the study of public bus transportation networks and railway networks. The space L consists of nodes being ports and links created between consecutive stops in one route. Degree k in the space L represents the number of directions passengers or cargoes can travel at a given port. The shortest path length in the space L is the number of stops one has to pass to travel between any two ports. In the space P, two arbitrary ports are connected if there is a container line traveling between both ports. Therefore, degree k in the space P is the number of nodes which can be reached without changing the line. The shortest
Y.Hu,D.Zhu Physica A 388 (2009)2061-2071 2063 path length between any two nodes in the space P represents the transfer number plus one from one node to another and thus is shorter than that in the space L. Since WMN is a directed network,we extend the concept of spaces L and P to directed networks according to [16].See Fig.1.Line A and B are two different pendulum routes crossing at the port No.1.(a)and(b)is the undirected network representation.(c)and(d)is the respective directed version. Based on the above concepts we establish the network under two spaces represented by asymmetrical adjacent matrices A,AP and weight matrices WL,WP.The element aj of the adjacent matrix A equals to 1 if there is a link from node i to j or 0 otherwise.The element w of weight matrix W is the number of container lines traveling from port i to port j. We need to define the quantities used in this weighed and directed network.We employ k(i)and kout (i)to denote in- degree,out-degree of node i in the space L,and k(i)to represent undirected degree in the space L.Similarly k(i).kou(i) and k (i)are employed in the space P.Hence we have 国= (1) j 收ar①=∑ (2) m①=∑d店+c) (3) which also holds for the space P. Strength is defined as the total weight of vertex connections [11].It is also divided into in-strength and out-strength.In the space L the in-strength of node i is denoted by s(i)and out-strength is denoted by S(i).Undirected strength(total strength)is represented by S(i).They can be calculated according to the following equations: S=∑喷 (4) sa-∑时 (5) 决 Sn①=∑(w听+w克) (6) which also holds for (i).Sur(i).S (i)in the space P. Other quantities like clustering coefficient and average nearest neighbors degree also have different versions in directed and weighted WMN.We employ c and c to denote the unweighted clustering coefficient of node i in the space L and P, respectively.Analogouslyandare used to denote the average nearest neighbors degree of nodein the space Land P.respectively.For weighted WMN we add superscript W to the above quantities and consequently they become and r 3.Topological properties 3.1.Degree distribution and degree correlations First we examine the degree distributions in two spaces.Fig.2 shows that in-degree,out-degree and undirected degree distributions in the space L all follow truncated power-law distributions with nearly the same exponents.In-degree and out-degree obey the function P(k)~k-17 before k =20.When k>20 their distribution curves bend down to the function P(k)~k-2.95.Unweighted degree in the space L has the same exponents of-1.7 and-2.95 but the critical point becomes k=30.Truncated power-law degree distributions are often observed in other transportation networks like the worldwide air transportation network [12].China airport network[13].US airport network[14]and the Italian airport network[15].It is explained in Ref.[10]that the connection cost prevents adding new links to large degree nodes.Analogous cost constraints also exist in the maritime transport network.Congestion in hub ports often makes ships wait outside for available berth for several days,which can cost ships extremely high expense.Consequently new links are not encouraged to connect to those busy ports. While in the space P three degree distributions all follow exponential distributions P(k)~e-ak.The parameters are estimated to be o =0.0117 for in-degree,and a =0.0085 for out-degree,o =0.0086 for unweighted degree.The property that degrees obey truncated power-law distributions in the space L and exponential distributions in the space p is identical to public transportation networks[17,18]and railway networks [22].Particularly the Indian railway network [22] has exponential degree distributions with the parameter 0.0085 almost the same with in-degree and out-degree distribution in WMN
Y. Hu, D. Zhu / Physica A 388 (2009) 2061–2071 2063 path length between any two nodes in the space P represents the transfer number plus one from one node to another and thus is shorter than that in the space L. Since WMN is a directed network, we extend the concept of spaces L and P to directed networks according to [16]. See Fig. 1. Line A and B are two different pendulum routes crossing at the port No. 1. (a) and (b) is the undirected network representation. (c) and (d) is the respective directed version. Based on the above concepts we establish the network under two spaces represented by asymmetrical adjacent matrices A L , A P and weight matrices WL , WP . The element aij of the adjacent matrix A equals to 1 if there is a link from node i to j or 0 otherwise. The element wij of weight matrix W is the number of container lines traveling from port i to port j. We need to define the quantities used in this weighed and directed network. We employ k L in(i) and k L out(i) to denote indegree, out-degree of node i in the space L, and k L un(i) to represent undirected degree in the space L. Similarly k P in(i), k P out(i) and k P un(i) are employed in the space P. Hence we have k L in(i) = X j6=i a L ji (1) k L out(i) = X j6=i a L ij (2) k L un(i) = X j6=i (a L ij + a L ji) (3) which also holds for the space P. Strength is defined as the total weight of vertex connections [11]. It is also divided into in-strength and out-strength. In the space L the in-strength of node i is denoted by S L in(i) and out-strength is denoted by S L out(i). Undirected strength (total strength) is represented by S L un(i). They can be calculated according to the following equations: S L in(i) = X j6=i w L ji (4) S L out(i) = X j6=i w L ij (5) S L un(i) = X j6=i (wL ij + w L ji) (6) which also holds for S P in(i), S P out(i), S P un(i) in the space P. Other quantities like clustering coefficient and average nearest neighbors degree also have different versions in directed and weighted WMN. We employ c L i and c P i to denote the unweighted clustering coefficient of node i in the space L and P, respectively. Analogously, k L nn,i and k P nn,i are used to denote the average nearest neighbors degree of node i in the space L and P, respectively. For weighted WMN we add superscript W to the above quantities and consequently they become c WL i , c WP i , k WL nn,i and k WP nn,i . 3. Topological properties 3.1. Degree distribution and degree correlations First we examine the degree distributions in two spaces. Fig. 2 shows that in-degree, out-degree and undirected degree distributions in the space L all follow truncated power-law distributions with nearly the same exponents. In-degree and out-degree obey the function P(k) ∼ k −1.7 before k = 20. When k > 20 their distribution curves bend down to the function P(k) ∼ k −2.95. Unweighted degree in the space L has the same exponents of −1.7 and −2.95 but the critical point becomes k = 30. Truncated power-law degree distributions are often observed in other transportation networks like the worldwide air transportation network [12], China airport network [13], US airport network [14] and the Italian airport network [15]. It is explained in Ref. [10] that the connection cost prevents adding new links to large degree nodes. Analogous cost constraints also exist in the maritime transport network. Congestion in hub ports often makes ships wait outside for available berth for several days, which can cost ships extremely high expense. Consequently new links are not encouraged to connect to those busy ports. While in the space P three degree distributions all follow exponential distributions P(k) ∼ e −αk . The parameters are estimated to be α = 0.0117 for in-degree, and α = 0.0085 for out-degree, α = 0.0086 for unweighted degree. The property that degrees obey truncated power-law distributions in the space L and exponential distributions in the space P is identical to public transportation networks [17,18] and railway networks [22]. Particularly the Indian railway network [22] has exponential degree distributions with the parameter 0.0085 almost the same with in-degree and out-degree distribution in WMN
2064 Y.L,D.Zhu/Physica A388(2009)2061-2071 a 10 b 10 10 10 oundirected 10 o in-degree 10 Aout-degree undirected out 10 10 10 10' 10 100 200 300 K T d 口Space P o Space L 10 10 10 103 40D000D四 100 10 10 2 46810.12141618202224262830 In-degree Length Fig.2.(a)Cumulative degree distributions of degree in the space L obey truncated power-law distributions with almost the same exponents.The turning points are at k 20 and k 30,respectively.(b)Cumulative degree distributions of degree in the space P all follow exponential distributions.(c)Positive correlations between in-degree and out-degree.In two spaces they have nonlinear relations:(6 and ke().(d)Cumulative probability distribution of line length.It can be approximated by a straight line in semi-log plot indicating an exponential decay estimated to be P(~e13. Next,the relation between in-degree and out-degree is studied.Fig.2(c)is a plot of out-degree kout vs.in-degree kin.They have positive and nonlinear correlations under two spaces and are estimated to be:(k andk(k Evidently the in-out degree correlation is very strong. 3.2.Line length Let's denote line length,i.e.the number of stops in one line,as I.In Fig.2(d)the probability distribution of line length P(D) can be approximated as a straight line in the semi-log picture representing an exponential decay distribution P(D)~e with the parameter o=0.13.It indicates there are much more short haul lines than long haul lines in maritime transporta- tion.Long haul lines use large vessels and travel long distance from one region to another region while short haul lines as branch lines travel between several neighboring ports and provide cargo to main lines.For example,the line consisting of the following ports:Shanghai-Busan-Osaka-Nagoya-Tokyo-Shimizu-Los Angeles-Charleston-Norfolk-New York- Antwerp-Bremerhaven-Thamesport-Rotterdam-Le Havre-New York-Norfolk-Charleston-Colon-Los Angeles-Oakland- Tokyo-Osaka-Shanghai,is a typical long haul line connecting main ports in Asia and Europe,calling at ports for 24 times. 3.3.Shortest path length The frequency distributions of shortest path lengths d in the spaces L and P are plotted in Fig.3.The distribution in the space L has a wider range than in the space P.The average shortest path length is 3.6 in the space L and 2.66 in the space P (see Table 2).This means generally in the whole world the cargo need to transfer for no more than 2 times to get to the destination.Compared with the network size N 878,the shortest path length is relatively small
2064 Y. Hu, D. Zhu / Physica A 388 (2009) 2061–2071 Fig. 2. (a) Cumulative degree distributions of degree in the space L obey truncated power-law distributions with almost the same exponents. The turning points are at k = 20 and k = 30, respectively. (b) Cumulative degree distributions of degree in the space P all follow exponential distributions. (c) Positive correlations between in-degree and out-degree. In two spaces they have nonlinear relations: k L out ∼ (k L in) 0.96 and k P out ∼ (k P in) 0.90. (d) Cumulative probability distribution of line length. It can be approximated by a straight line in semi-log plot indicating an exponential decay estimated to be P(l) ∼ e −0.13 . Next, the relation between in-degree and out-degree is studied. Fig. 2(c) is a plot of out-degree kout vs. in-degree kin. They have positive and nonlinear correlations under two spaces and are estimated to be: k L out ∼ (k L in) 0.96 and k P out ∼ (k P in) 0.90 . Evidently the in-out degree correlation is very strong. 3.2. Line length Let’s denote line length, i. e. the number of stops in one line, as l. In Fig. 2(d) the probability distribution of line length P(l) can be approximated as a straight line in the semi-log picture representing an exponential decay distribution P(l) ∼ e −αl with the parameter α = 0.13. It indicates there are much more short haul lines than long haul lines in maritime transportation. Long haul lines use large vessels and travel long distance from one region to another region while short haul lines as branch lines travel between several neighboring ports and provide cargo to main lines. For example, the line consisting of the following ports: Shanghai–Busan–Osaka–Nagoya–Tokyo–Shimizu–Los Angeles–Charleston–Norfolk–New York– Antwerp–Bremerhaven–Thamesport–Rotterdam–Le Havre–New York–Norfolk–Charleston–Colon–Los Angeles–Oakland– Tokyo–Osaka–Shanghai, is a typical long haul line connecting main ports in Asia and Europe, calling at ports for 24 times. 3.3. Shortest path length The frequency distributions of shortest path lengths d in the spaces L and P are plotted in Fig. 3. The distribution in the space L has a wider range than in the space P. The average shortest path length is 3.6 in the space L and 2.66 in the space P (see Table 2). This means generally in the whole world the cargo need to transfer for no more than 2 times to get to the destination. Compared with the network size N = 878, the shortest path length is relatively small
Y.Hu,D.Zhu Physica A 388(2009)2061-2071 2065 400000 --Space P -●-Space L 300000 200000 100000 2 3 56 7891011 Fig.3.Frequency distributions of shortest path length under two spaces.The distribution in the space L has a wider range than in the space P. Table 2 Basic parameters for spaces L and P.n is the number of nodes and m is the number of links.(k)is the average undirected degree.(C)is the average unweighted clustering coefficient.()is the average shortest path length. Space n m (kun) (C) ⑨ Space L 878 7955 9.04 0.4002 3.60 Space p 878 24967 28.44 0.7061 2.66 3.4.Weight and strength distribution Usually traffic on the transportation network is not equally distributed.Some links have more traffic flow than others and therefore play a more important role in the functioning of the whole network.Weight should be addressed especially in transportation networks.Here we study four properties of weighted WMN:weight distribution,strength distribution, in-out strength relations and the relations between strength and degree.The results are displayed in Fig.4. First we examine weight distributions.In Fig.4(a)two weight distribution curves are approximately straight declining lines before w=40.The power-law distributions are estimated to be P(w)~w-0.95 in the space P and P(w)~w-0.92 in the space L. Next,Fig.4(b)shows the undirected strength distributions under two spaces both obey power-law behavior with the same parameter.The functions are estimated to be P(s)~s-1.3. And we also analyze the relations between in-strength and out-strength in two spaces.As we can see from Fig.4(c).in- out strength relations under the spaces L and P are positively correlated.The fitted straight line is estimated to be y ox with o 1.03 for space P and a =1.00 for space L.The linear relations between in-out strength indicate the balance of cargo traffic in and out of ports. Finally,an important feature of weighted WMN,the relations between strength and degree,is investigated.Under two spaces the relations between undirected strength and undirected degree are both nonlinear with the slope of the line approximately 1.3,which means the strength increase quicker than the increase of degree.This often occurs in the transportation networks and has its implication in the reality.It's easy for the port with many container lines to attract more lines to connect the port and thus to increase the traffic more quickly. 4.Hierarchy structure In this section,we explore the network structure of WMN through studying both the weighed and unweighted versions of clustering coefficient and average nearest neighbors degree.Hierarchy structure and "rich-club"phenomena are unveiled. We conjecture this kind of structure is related to ship companies'optimal behavior to minimize the transportation cost known as the hub-and-spoke model in transportation industry. 4.1.Clustering Clustering coefficient ci is used to measure local cohesiveness of the network in the neighborhood of the vertex.It indicates to what extent two individuals with a common friend are likely to know each other.And C(k)is defined as clustering coefficient averaged over all vertices with degree k. We plot C(k)in Fig.5(a)in log-log scale.They don't obey power-law distributions as those observed in public transport networks [17,20]or in the ship transport network 16.Either in the space L or in the space P,C(k)exhibits a highly nontrivial
Y. Hu, D. Zhu / Physica A 388 (2009) 2061–2071 2065 Fig. 3. Frequency distributions of shortest path length under two spaces. The distribution in the space L has a wider range than in the space P. Table 2 Basic parameters for spaces L and P. n is the number of nodes and m is the number of links. hki is the average undirected degree. hCi is the average unweighted clustering coefficient. hli is the average shortest path length. Space n m hkuni hCi hli Space L 878 7955 9.04 0.4002 3.60 Space P 878 24967 28.44 0.7061 2.66 3.4. Weight and strength distribution Usually traffic on the transportation network is not equally distributed. Some links have more traffic flow than others and therefore play a more important role in the functioning of the whole network. Weight should be addressed especially in transportation networks. Here we study four properties of weighted WMN: weight distribution, strength distribution, in-out strength relations and the relations between strength and degree. The results are displayed in Fig. 4. First we examine weight distributions. In Fig. 4(a) two weight distribution curves are approximately straight declining lines before w = 40. The power-law distributions are estimated to be P(w) ∼ w−0.95 in the space P and P(w) ∼ w−0.92 in the space L. Next, Fig. 4(b) shows the undirected strength distributions under two spaces both obey power-law behavior with the same parameter. The functions are estimated to be P(s) ∼ s −1.3 . And we also analyze the relations between in-strength and out-strength in two spaces. As we can see from Fig. 4(c), inout strength relations under the spaces L and P are positively correlated. The fitted straight line is estimated to be y = αx with α = 1.03 for space P and α = 1.00 for space L. The linear relations between in-out strength indicate the balance of cargo traffic in and out of ports. Finally, an important feature of weighted WMN, the relations between strength and degree, is investigated. Under two spaces the relations between undirected strength and undirected degree are both nonlinear with the slope of the line approximately 1.3, which means the strength increase quicker than the increase of degree. This often occurs in the transportation networks and has its implication in the reality. It’s easy for the port with many container lines to attract more lines to connect the port and thus to increase the traffic more quickly. 4. Hierarchy structure In this section, we explore the network structure of WMN through studying both the weighed and unweighted versions of clustering coefficient and average nearest neighbors degree. Hierarchy structure and ‘‘rich-club’’ phenomena are unveiled. We conjecture this kind of structure is related to ship companies’ optimal behavior to minimize the transportation cost known as the hub-and-spoke model in transportation industry. 4.1. Clustering Clustering coefficient ci is used to measure local cohesiveness of the network in the neighborhood of the vertex. It indicates to what extent two individuals with a common friend are likely to know each other. And C(k) is defined as clustering coefficient averaged over all vertices with degree k. We plot C(k) in Fig. 5(a) in log–log scale. They don’t obey power-law distributions as those observed in public transport networks [17,20] or in the ship transport network [16]. Either in the space L or in the space P, C(k) exhibits a highly nontrivial