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Mining the Structure and Evolution of the AirportNetwork of China over the Past Twenty YearsZhengbin Dong 1 , Wenjie Wu 2,3 , Xiujun Ma 1,* , Kunqing Xie 1 , and Fengjun Jin 21Key Laboratory of Machine Perception, Ministry of Education, Peking University,100871 Beijing{dongzhengbin,maxj,kunqing} of Geographic Sciences and Natural Resources Research,Chinese Academy of Sciences,100101 Beijing{wuwj.07s,jinfj} School of the Chinese Academy of Sciences,100101 BeijingAbstract. In this paper we study the Airport Network of China (ANC), whichrepresents China’s domestic civil aviation infrastructure, as a complex network.We mine the structure and evolution of ANC over the past twenty years by usingthe real aviation data in the year of 1984, 1993 and 2006. The main contributionscan be summarized as three-fold: firstly, we analyze ANC by using thecomplex network analysis method and find that ANC is a typical small worldnetwork with high clustering coefficient and small diameter; secondly, we findthat the evolution of ANC over the past twenty years meets the densificationlaw and shrinking/stabilizing diameter law; lastly, some interesting patterns ofairports in ANC are found by the visual data mining, such as Circle Pattern,Province Capital Pattern and Star Pattern.Keywords: Airport Network of China, ANC, network structure, network evolution,network diameter, degree distribution, clustering coefficient, betweenesscentrality.1 IntroductionTransportation infrastructures are of crucial importance to the development of a countryand are important indicators of its economic growth. They form the backboneof tourism industry, support movement of goods and people across the country,thereby driving the national economy [1]. Roadways, railways and airways are themajor means of transport in China, although contribution of airways is small comparedto that of the other two. The civil aviation in China has been developed veryfast since the Reform and Opening of China in 1980s. There are great changes in thestructure of civil aviation in China. Understanding of the civil aviation system and itschanges over the past twenty years is important for reasons of policy, administrationand efficiency.* Corresponding author.R. Huang et al. (Eds.): ADMA 2009, LNAI 5678, pp. 104–115, 2009.© Springer-Verlag Berlin Heidelberg 2009

Mining the Structure and Evolution of the Airport Network of China 105Complex Network Analysis is a novel method for mining the network data. Duringthe past few years, complex network analysis has been used to study many real-lifecomplex systems. Examples include the Internet, the World Wide Web, email networks,peer-to-peer networks, scientific co-authorship networks [2], human sexualnetwork [3], mobile social network [4] and etc. These researches have shown someubiquitous properties about the real-life social networks: the small-world effect, thepower-law and heavy-tails distributions, the scale-free network, the small diameterand etc.There have been some researches that focus on the transportation systems, includingthe railways [5] and the airways [1, 6-10]. The World-wide Airport Network (WAN),as the global airways system, has been studied from many different aspects, such as theproperties of topological structure, community structure, traffic dynamics and modelingmethods. Paper [6] studies the structure properties of WAN and concludes thatWAN is a small-world network. The same result is found in paper [7] that WAN is ascale-free small-world network. It is also found that in WAN the most connected citiesare not necessarily the most central because of the multi-community structure of WAN.Moreover, the community of WAN has been detected and the results show that communitystructure cannot be explained solely based on geographical constraints and thatgeopolitical considerations have to be taken into account. Beyond the topologicalproperties, WAN has been studied [9] as a complex weighted network, where theweight is the traffic flow amount – strength of interactions between the cities. Thecorrelations among weighted quantities and the topological structure of WAN are investigatedfor the first time. A model with geo-political constraints is proposed [10] toexplain the evolution and growth of WAN. Beyond the study of the global airportnetwork, there are also some researches on the regional airport network. For example,the paper [1] studies the Airport Network of India (ANI), which represents India’sdomestic civil aviation infrastructure. It is found that ANI is a small-world networkcharacterized by a truncated power-law degree distribution and has a signature of hierarchy.The Airport Network of China (ANC), a network much smaller than WAN, isalso analyzed [8] for its topology and traffic dynamics. Its topology was found to behaving small-world network features and a two-regime power-law degree distribution.The Airport Network of China (ANC) is a crucial part of WAN and its detail structureand properties may have its own features. The evolution of ANC is also veryimportant and useful for understanding the growth of economic in China. Moreover,the knowledge about the structure and evolution of ANC can be used to modify thecurrent policy and improve the efficiency. So in this paper we mine the structureproperties and the evolution of ANC by using the real-life aviation data in 1984, 1993and 2006. The data is from the Civil Aviation Administration of China and the durationis about twenty years. The main contributions of this paper can be summarized asthree-fold: firstly, we find that ANC is a typical small world network with small diameterand high clustering coefficient; secondly, we find that the evolution process ofANC over the past twenty years meets the densification law and shrinking/stabilizingdiameter, i.e., the airlines between the cities growth much fast than the growth of theairport cities and the diameter of ANC shrinks year by year; lastly, we investigatesome important city in detail and find some very interesting patterns of them in ANC,such as Star Pattern of Urumqi or Kunming.

106 Z. Dong et al.The paper is organized as follows: in the next section, we describe the dataset andANC. In section 3, we analyze the global structure and the evolution of ANC overpast twenty years. In section 4, we analyze ANC from the city level by the visualmining method and investigate an anomaly in ANC. We finally summarize our workand discuss the future research directions in section 5.2 Data Modeling for Airport Network of China (ANC)The Airport Network of China (ANC) comprises domestic airports of China and airlinesconnecting them. There is traffic flow on each airline. In this paper we use anundirected binary graph to represent ANC without considering the traffic flow. Let anundirected binary graph be G = {(V, E) | V is a set of nodes, E is a set of edges.E ⊆ V×V, an edge e = (i, j) connects two nodes i and j and i, j∈V, e∈E}. In ANC,the nodes of the network represent the airports and the edges between the pairs ofnodes represent the airlines between the cities.Table 1. The aviation data of ANC in 1984, 1993 and 2006YearAirport Airline Passenger Traffic (10 4 )Number Number1984 60 156 3911993 82 422 33852006 91 471 15968In this paper we choose the aviation data of 1984, 1993 and 2006 to analyzethe structure and evolution of ANC. The data (Table 1) is from the Civil AviationAdministration of China.3 Mining the Global Structure and Evolution of ANCIn this section we will use some important network metrics to analyze the globalstructure and evolution of ANC over the past twenty years.3.1 Degree DistributionThe degree of a node v in a network, represented as d(v), is the number of connectionsor edges the node has to other nodes. Let N(v) = {u | (v, u)∈E and v, u∈V}, which isa set of the neighbor nodes of v in the graph G. so d(v) is the size of set N(v). Thedegree distribution p(k) of a network is then defined to be the fraction of nodes in thenetwork with degree k. Thus if there are n nodes in total in a network and n k of themhave degree k, we have p(k) = n k /n.The degree distribution is very important in studying both real networks, such asthe Internet and social networks, and the theoretical networks. The simplest model ofnetwork is the ER random model [11] introduced by Erdos and Renyi. The degree

Mining the Structure and Evolution of the Airport Network of China 107distribution of the network generated by the ER random model follows Poisson distributionbut it is found that many real networks follow the heavy-tail distribution suchas power-law [2-3]: p(k)~k -r , where r is a constant whose value is typically in therange 2

108 Z. Dong et al.The diameter of the network is the length of the longest shortest path, which is importantbecause it quantifies how far apart the farthest two nodes in the graph are.In Table 2 we give the average shortest path length and the diameter of ANC in1984, 1993 and 2006. The length of average shortest path and diameter becomeshorter and shorter over the past twenty years, which indicates that ANC is becomingvery dense. As a result, the efficiency of ANC is improved because there is no need totake several flights from one place to another by air in ANC.Table 2. The average shortest path length and diameter of ANC in 1984, 1993 and 2006Year Airport Average Shortest DiameterNumber Path Length1984 60 2.5 51993 82 2.15 42006 91 2.22 43.3 Clustering CoefficientThe clustering coefficient of a vertex in a network quantifies how close the vertex andits neighbors are to being a clique (complete graph). This measure is first introducedby Duncan J. Watts and Steven Strogatz in 1998 [14] to determine whether a networkis a small-world network.The clustering coefficient of node v, noted as C v , measures the extent of the interconnectivitybetween the neighbors of node v and is the ratio of the number of edgesbetween the nodes in the direct neighborhood to the number of edges that could possiblyexist among them, C v can be defined as:∪ ei ji, j∈N( v)(1)2 ( , )Cv= : e( i, j)∈Edv ()( dv () −1)where d(v) is the degree of node v and N(v) is the set of the neighbor nodes of v,which have been defined in section 3.1.After the clustering coefficient of a node is defined, then we give the definition ofclustering coefficient of a network, which is the average of the clustering coefficientsof all nodes in the graph:1 n Cin i = 1C = ∑ (2)The clustering coefficient distribution p(C) of a network, like the degree disturbing,is defined to be the fraction of nodes in the network with clustering coefficient C.Thus if there are n nodes in total in a network and n k of them have clustering coefficientC, we have p(C) = n k /n. The average clustering coefficient of degree k, noted asC(k), is defined as the average value of clustering coefficient of nodes with degree k.

Mining the Structure and Evolution of the Airport Network of China 109Fig. 2. There are two figures: (a) the clustering coefficient distribution of ANC in 1984, 1993and 2006; (b) the average clustering coefficient versus degree of ANC in 1984, 1993 and 2006In Fig. 2(a) the distribution of clustering coefficient of ANC in 1984, 1993 and2006 is plotted. We can get two conclusions that: 1) most value of clustering coefficientof node is zero (the percent of each years is more than 0.3) because of the greatnumber of degree one in ANC. The clustering coefficient of node with degree one iszero; 2) the distribution does not fit the right-skewed law like other real-life network,such as scientific co-authorship networks [2] and mobile social network [4], whichfits the left-skewed distribution. In other networks with right-skewed distribution,most nodes have small clustering coefficient and few nodes have large value, but inANC the number of nodes which value is grater than 0.9 is relatively higher. Thereason for this interesting result is that ANC is very dense as a whole and the airportsconnect each other very close while other networks are sparse as a whole but verydense in some local region.The average clustering coefficient versus degree of ANC in 1984, 1993 and 2006 isplotted in Fig. 2(b). The figure indicates that the node with lower degree have highervalues of clustering coefficient while the node with higher degree have lower value.The value of C(k) decays from 1.0 to lower than 0.2. The reason for this result is thatin ANC the high degree nodes, i.e., the hub airports, connect other low degree nodes.For example, Beijing, the capital of China, connects almost all other cities in China,so its degree is high and clustering coefficient is low.The network clustering coefficient of ANC in 1984, 1993 and 2006 are 0.38, 0.48and 0.54 separately, which also indicates like other metrics that ANC is becomingvery dense over the past twenty years.The small-world network is the network with two properties: 1) a small averageshortest path length and 2) a large clustering coefficient, which is proposed by DuncanJ. Watts and Steven Strogatz in 1998 [14]. In ANC the diameter is very smallcompared to its size (see section 3.2) and the clustering coefficient is much larger thanthe random network, so we can conclude that ANC is a typical small-world networklike other airport network, such as WAN [6] and ANI [1].

110 Z. Dong et al.3.4 Betweenness CentralityCentrality is a core concept for the analysis of social networks, and betweenness isone of the most prominent measures of centrality. It was introduced independently byAnthonisse (1971) [15] and Freeman (1977) [16], and measures the degree to which avertex is in a position of brokerage by summing up the fractions of shortest pathsbetween other pairs of vertices that pass through it. Betweenness is therefore classifiedas a measure of mediation in Borgatti and Everett (2006) [17].The formal definition of betweenness centrality in a network is as below [18]:denote by σ (,) st the number of shortest paths (sometimes referred as geodesics)from s to t and σ (, st| v)be the number of shortest number from s to t passingthrough some vertex v other than s, t. If s = t, let σ (,) st = 1, and if v∈{,} s tσ (, st| v) = 0. Then the betweenness c () v of a vertex v can be defined to be:st , ∈VB, letσ (, st| v)cB() v = ∑ (3)σ (,) stwhere 0 00 = by convention. The measure is therefore usually interpreted as the degreeto which a vertex has control over pair-wise connections between other vertices,based on the assumption that the importance of connections is equally divided amongall shortest paths for each pair.In airport network the betweeness is a indicator of “central” of a vertex and the vertexwith high betweenness is very important from the angle of flight change because itis on the position of brokerage between other pairs of vertices.We use cB( k)to represent the average value of betweenness of the vertices withthe same degree k. In order to compare the distribution of cB( k)in ANC in threeyears, we scale the value by dividing the maximum value in each year and then thescaled values of cB( k)are among 0 and 1. In Fig.3 we plot the distribution of cB( k )in 1984, 1993 and 2006.We find that the cB( k)distributions of ANC after scaling follow the exponentialdistributions and the fit curves in Fig.3 are plotted by using the exponential functionas below:y y AeRx 0=0+ (4)The exponential distribution of cB( k)means that there is a strong relation betweendegree and betweenness: the higher the degree the higher value of betweenness. Inaddition, in ANC the most-connected node (with highest degree) is the most-centralnode (with highest betweeness), which is different with the result of WAN [7]. InWAN, the most connected cities are not necessarily the most central. The reason forthis interesting pattern is that WAN has multi-community structure while ANC doesnot have this structure. The node connecting different communities will have higher

Mining the Structure and Evolution of the Airport Network of China 111betweeness. However in ANC, all the nodes are connected closely and form only onelarge group, which can also be proved by the abnormal left-skewed distribution ofclustering coefficient in section 3.3. The central of the community of ANC is thecapital of China, i.e., Beijing. As a result, it has the highest degree and betweenness.It is clear that there is an obvious anomalous point (see the green arrow) of cB( k )in 2006, which represents a very important pattern in ANC and we will discuss it indetail in section 4.Fig. 3. The cB( k)distribution of ANC in 1984, 1993 and 2006 and the curves are the fit resultsby using exponential function3.5 Evolution of ANCWe have analyzed the global structure of ANC in section from 3.1 to 3.4. We also getsome knowledge about the evolution of ANC over the past twenty years: the numberof airports and the airlines are increasing year by year and the whole network of ANCis becoming denser. But what is the relation between the number of nodes and thenumber of edges overtime and what is the law of diameter change?The conventional wisdom or intuition of these two questions is that: 1) constantaverage degree, i.e., the number of edges grows linearly with the number of nodes; 2)slowly growing diameter, i.e., as the network grows the distances between nodesgrow. However the real-life network does not follow these two laws, J. Leskovec(2005) [19] find: that 1) networks are denser over time and the number of edgesgrows faster than the number of nodes – average degree is increasing, which followthe power-law. This result is called densification power law; 2) the diameter actuallyexhibits a gradual decrease as the network grows, which is called shrinking/stabilizesdiameter law.In ANC, the evolution law over past twenty years obviously follows the densificationpower law and shrinking/stabilizes diameter law, which can be proved by theincrease of average degree in section 3.1 and the decrease of diameter in section 3.2.

112 Z. Dong et al.4 City Level Pattern Detecting of ANCIn this section, we will analyze some important cities of ANC and find some interestingpatterns in node level by using the visualization of ANC. First, we will visualizeANC of three years and discuss some obvious patterns that can be observed. Then wediscuss the anomalous point at section Visualization of ANCVisualization is an important tool to analyze the properties of network when the sizeof network is relative small. In this paper we use Pajek [20] to visualize ANC, whichis a very famous and powerful network analysis software.The visualization of ANC in 1984, 1993 and 2006 is plotted in Fig. 4, where thered nodes represent the provincial capital cities and the blue nodes represent the nonprovincialcapital cities. The size of the node represents the degree of the node, whichis scaled by the maximum value of each year.There are some very interesting patterns can be observed from the Fig. 4, wediscuss three examples.Circle Pattern. We can see that there is no obvious community structure in ANC,while ANC is connected as a whole and from a circle. The centers of the circle aresome major cities, for example, Beijing, Shanghai, Guangzhou and etc. These centersconnect very close at the central of circle and other cities connect center cities at thesurrounding. The more close to the center the more connections of the city.Province Capital Pattern. This pattern reflects the differences of some provincecapital and non-province capital cities. In the year of 1984, Chongqing is not yet aprovince capital city, but we can see from Fig. 4(a) that it is very close to the center ofANC and its degree is relative high. However, Tianjin and Haikou are on the fringe ofANC in 1984 although they are province capital cities all the time. The similar examplescan be found in 1993 (Shenzhen VS Lhasa) and 2006 (Qingdao VS Lhasa), sowhat the reason for this pattern? This pattern indicates that the development of airportand airline has not much relation with the geopolitical. Some cities that are very importantfrom the view of geopolitical are not very important from the view of networkof airport and vice versa. We think that the major factor for the importance of airportis the economy of the city. If the economy of a city is very good, it will have the greatdemand of traffic flow by air because of the great growth of people and amount ofexchange goods, then the city becomes an important airport in ANC. Shenzhen is agood example to illustrate our point. In 1984 Shenzhen is still a small village andthere is no airport at all. In 2006, Shenzhen become a very major airport because ofthe development of economy since the Reform and Opening in 1980s. AlthoughShenzhen is very close to another major city in geography, Guangzhou, but it is nothingto prevent the development of Shenzhen airport, which indicates that the economyfactor is beyond the geography factor.Star Pattern. This pattern can be observed obviously in 1984 and 2006. We willdiscuss this pattern in detail in next section.

Mining the Structure and Evolution of the Airport Network of China 113ChifengFuyangNingboXilinhotShijiazhuangHuangshanYantaiHohhotChangchun HefeiShashiChangzhiQingdao DalianTaiyuanGuilinEnshiXichangWuhanUrumqiShanghai JinanHailarBeijingYichangZhengzhouYinchuan ChengduHarbinSimaoShenyangXiamen NanyangHangzhouLhasaChangshaKunmingGanzhouNanjingChongqingNanchangAnkangLanzhouXi’anFuzhouNanningGuangzhouGuiyangJingdezhenJiuquanShantouHanzhongTianjinYan’an MeizhouHaikouGelmuZhuhaiZhanjiangSanyaZhanjiang JinghongSimaoMudanjiangHengyangYanjiShashiNanchongQinhuangdaoSanyaDalianLhasaMeizhouNanningTaiyuanJiayuguan ChangchunLuxiQingdao KunmingHangzhouJinanQiqiharShenyangGuiyangLanzhouGuilinYichangLianyungangShanghai Chengdu UrumqiGuangzhouXiamen Ningbo ChangshaChangzhiYiningLuoyangShenzhenHaikou ChongqingFuzhouWuhu HuhhotHefeiBeijingHarbin NanjingWenzhouHuizhouShiyan YinchuanYantaiXi’an ShantouWuhan ZhengzhouWuxi NanchangTianjin BeihaiJiningJilin ChangzhouKurle LiangpingChangdeNanyang XiangfanHanzhongTonghuaHuangshanFoshanJiujiangEnshiWeihaiXiningXuzhou NantongXingtai BangbuPajekPajek(a) Year of 1984 (b) Year of 1993YulinJinhuaAletaiJiuquanJiuzhaigouPanzhihuaLinyiHetianBeihaiZhanjiangLhasaAksuLuzhou ShijiazhuangXichangEnshiYiningYinchuanTianjinChengduUrumqi LanzhouKashiXi’anNanning Lijiang ZhaotongXiningZhengzhouGuangzhouSanya WuxiKurleZhangjiajieLincangChongqing GuiyangHuangshanChangshaHuhhot QingdaoHaikouSimaoShenzhenLiuzhouDalian TaiyuanShanghaiWuhan KunmingNanjing FuzhouTaizhouHefeiLuxiBeijingShantouHangzhou NanchangYibinHarbinChangzhouJinan Guilin Dali BaoshanBaotouShenyang WenzhouChangchunJinghongMudanjiangNingboXiamenYichangLuoyangQuanzhouYantaiZhuhai JingdezhenLianyungangNanping XuzhouWeihaiChangzhiYanjiZhoushanHailarNantongYunchengMianyangPajek(c) Year of 2006Fig. 4. The visualization of ANC in 1984, 1993 and 20064.2 Anomaly of Betweeness CentralityIn section 3.4 we find an anomalous point of betweenness in 2006, this anomalouspoint indicates a very important pattern in ANC that will be discussed in this section.We analyze the value of betweenness in 2006 and find that the anomalous point isKunming. The degree of Kunming in 2006 is 33 and it is not very high compared tothe maximum value of 54 of Beijing, but the betwwenness value is very high in 2006.As a result, it produces an obvious anomalous point in Fig. 3 in section 3.4.

114 Z. Dong et al.What is the pattern of Kunming in 2006 with mediate degree but high betweenness?We can see it from the Fig. 4(c) that Kunming connected many nodes with onedegree, such as Zhaotong, Lincang, Simao, Baoshan and etc. These nodes with onedegree connect to other nodes in ANC through Kunming and so the betweeness ofKunming is much higher according to the definition of betweenness. We call thispattern Star Pattern. There is another obvious Star Pattern in 2006, i.e., Urumqi. Thereason for Urumqi is not an anomalous point is that the average value of betweennesswith the same degree hides the high value of Urumqi. The Star Pattern can alsobe found in other years, for example, Xi’an and Guangzhou in 1984 (see Fig. 4(a)),Beijing in 1993 (see Fig. 4(b)).The next question is that what is the mechanism of forming the Star Pattern? Wethink there are two factors to form this pattern. One is the development of economyand another is the geography of some cites. Let us take Urumqi as an example. In1993 there is no Star Pattern for Urumqi but there is in 2006. We can observe that in2006, Urumqi connects some small cities, such as Kurle, Kashi and Aksu, which arenot in 1993. That is the result of the development of economy of these small cities,i.e., they need to connect to other airports in ANC for the economy reason. But whythey connect Urumqi but not Beijing or other cities? That is the result of geography.Urumqi is the capital of Xinjiang and the nearest major city for these small cities. Sothe Star Pattern is the result of development of economy and the limit of geography.The Star Pattern make the major city like Urumqi be the center of the regional area.5 ConclusionIn this paper we mine the structure and evolution of Airport Network of China (ANC)over the past twenty years by using the complex network analysis method. We findthat ANC is a typical small-world network with high clustering coefficient and smalldiameter although the degree distribution does not follow power-law like other airportnetworks. And the evolution of ANC follows the densification law and shrinking/stabilizingdiameter law. In addition, we find some very interesting patterns inANC with the help of visual mining, such as the Star Pattern, Circle Pattern and etc.In the future we will continue to study the properties of ANC from the view ofcomplex network and pay more attentions to the weight analysis of ANC.AcknowledgmentsThe author is grateful to Prof. Fengjun Jing and Master Wenjie Wu of Institute ofGeographic Sciences and Natural Resources Research, CAS, for help in obtaining theaviation data. The author is thankful to Dr. Xiujun Ma and Master Wenjie Wu forsome useful discussions.This work is supported by the National Natural Science Foundation of China(NSFC) under Grant No.40801046 & 60874082 and the Key Project of NationalNatural Science Foundation of China (NSFC) under Grant No.40635026.

Mining the Structure and Evolution of the Airport Network of China 115References1. Bagler, G.: Analysis of the Airport Network of India as a complex weighted network.arXiv:cond-mat/0409773 (2004)2. Newman, M.E.J.: Who is the best connected scientist? A study of scientific co-authorshipnetworks. Phys., Rev. E64, 06131–06132 (2001)3. Fredrik, L., Christofer, R.E., Luis, A.N.A., et al.: The web of human sexual contacts.Nature 411, 907–908 (2001)4. Dong, Z.B., Song, G.J., Xie, K.Q., Wang, J.Y.: An experimental study of large-scalemobile social network. In: Proc. of WWW 2009, Madrid, Spain, April 20-24 (2009)5. Sen, P., Dasupta, S., Chatterjee, A., et al.: Small-world properties of the Indian railwaynetwork. Phys., Rev. E67, 036106 (2003)6. Amaral, L.A.N., Scala, A., Barthelemy, M., Stanley, H.E.: Classes of small-world networks.Proc. Natl. Acad. Sci (USA) 97, 11149 (2000)7. Guimera, R., Mossa, S., Turtschi, A., Amaral, L.A.N.: The worldwide air transportationnetwork: anomalous centrality, community structure, and cities’ global roles. Proc. Natl.Acad. Sci. (USA) 102, 7794 (2005)8. Li, W., Cai, X.: Statistical analysis of airport network of China. Phys. Rev. E69, 046106(2004)9. Barrat, A., Barthelemy, M., Pastor-Satorras, R., et al.: The architecture of complexweighted networks. Proc. Natl. Acad. Sci. (USA) 101, 3747 (2004)10. Guimera, R., Amaral, L.A.N.: Modeling the world-wide airport network. Eur. Phys.J.B 38, 381 (2004)11. Erdos, P., Renyi, A.: On random graphs. Publ. Math. (Debrecen) 6, 290 (1959)12. Albert, R., Barabasi, A.L.: Statistical mechanics of complex networks. Rev. Mod.Phys. 74, 47–97 (2002)13. Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data.arXiv:076.1062v1 (2007)14. Watts, D.J., Steven, S.: Collective dynamics of ‘small-world’ networks. Nature 393,440–442 (1998)15. Anthonisse, J.M.: The rush in a directed graph. Tech. Rep. BN 9/71 (1971)16. Freeman, L.C.: A set of measures of centrality based upon betweenness. Sociometry 40,35–41 (1977)17. Borgatti, S.P., Everett, M.G.: A graph-theoretic perspective on centrality. Social Networks28, 466–484 (2006)18. Brandes, U.: On variants of shortest-path betweenness centrality and their generic computation.Social Networks 30(2), 36–45 (2008)19. Leskovec, J., Kleinber, J., Faloutsos, C.: Graphs over time: densification laws, shrinkingdiameters and possible explanaionts. In: ACM KDD (2005)20. Batagelj, V., Mavar, A.: Pajek: Program for large network analysis. Connections (1998)

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