64 8 2009 8 ACTA GEOGRAPHICA SINICA Vol.64 No.8 Aug. 2009 1 2 1 3 1 (1. 100101; 2. 70803; 3. 100053) : (2.23) (0.69) ( ) : ; ; ; ; ; 1 ( ) : [1] (Hub-and-Spoke) [2] ; 10 [3] 1980 1998 [4] GIS [5] ; [6] 20 [7 8] [9-13] 2009-02-10; 2009-06-18 (40635026); (20080440065) [Foundation: Key Project of National Natural Science Foundation of China No.40635026; Project of China Postdoctoral Science Foundation No.20080440065] (1981-) E-mail: jiaoewang@163.com 899-910
900 64 (SW Small World) [14] 2 2.1 2007-2008 152 1570 785 1-10 ; 11-20 21-40 ( ) 41-80 81-100 A 0 500km B B A 1 : Fig. 1 Distribution of nodes and airlines (1) in Chinese aviation network (2) 144 (1) (3) 1018 2.2 V E G = (V E) V = {v i : i = 1 2 n} n = V ; E = {e i : i = 1 2 m}e = {e ij : v i v j V} m = E e ij = 1 0 e ii = 0 2.2.1 (Degree) v i k i ( ) <k> p(k) [15] P(k) p(k) (Cumulative Degree Distribution) : P(k) = Σp(k) (1) k ' = k P(k) p(k) (SF Scale-Free) p(k)
8 : 901 p(k) k -λ (2 λ 3) [16 17] v i ( ) N i K(i) = 1 Σk k j k i v j N i K (k) = 1 Σ K (k) N (k) k (Degree-degree N(k) vj Vki=k kk i [16 17] Correlation) : ; 2.2.2 (Average Path Length) d ij ( [18] ) L : L = 1 Σd 1 ij (2) i > j n(n - 1) 2 D; v i L i = (n - 1) Σd ij v j V [18] 2.2.3 (Clustering Coefficient ) v i v i N i E i = Σe jk e kh v i : vjvkvh Ni C i = E i k i (k i - 1)/2 C i ( ) ( ) 1 0 1 0 1 (3) C = 1 ΣC n i (4) vi V C (Clustering-degree Correlation) k (C(k) = ΣC i ) [16 17] N(k) vj Vki=k vi Vki=k 3 [15 18-20] ( )
902 64 3.1 1998 Watts Strogatz Nature (WS ) [18] : (ρ) 0 1 1 : ( ) [19 20] 3.2 1999 Barab 仳 si Albert 1 Tab. 1 The comparison of three indices Science in various network models ( [19] )! (BA ) [15] "# - $%& ( )*+ - +-./012345678 1 3.3 ; [2-5] [21] : [22] [5] 4 4.1 (2) 50.9% 7 1 2 1 ( )
8 : 903 16% 32% 1 2 15 6 12 6 5.56% (8 ) 7 9 2.08% (3 ) 6 14.14 14 3 ; 0 14 14 2 14 Fig. 2 Distribution of nodal degree in Chinese aviation network 100 100 ( 3) : 92 : 1 0 500km (92) (82) (79) 3 (65) (58) (58) Fig. 3 Spatial distribution of nodal degree (57) (56) in Chinese aviation network (36) (35) 20 22 100 80 (30) (26) 60 40 20 (P (k) = 0.705e -0.047k R 2 = 0.977) (4) 0 2/3 4 10 /% /%
904 64 1/3 (48 ) 10 1701 83.5% ( 20 81.4%) [5] (0.84) 1 2 4.2 2 2 Tab. 2 Percentage of flight routes among the number of shortest paths in China (%) (%) 1 2036 0 9.89 9.89 2 12200 1 59.25 69.14 3 5976 2 29.02 98.16 4 362 3 1.76 99.92 5 18 4 0.09 100.00 98% 70% 10% (2) 2.23 2.23 (L ER lnn/ln<k> 1.88) 5 ( ) ; (5) : 1 1 2 : 0.73 : 0.26 0 500km 5 Fig. 5 Spatial pattern of accessibility in Chinese aviation network
8 : 905 ( 22 ) 4.3 (3) (3) 0.69 1 54 37.5% 2~10 23 0 16% 1 : (0.18) (0.23) (0.25) Moran I(d) 1% Moran 0.2 (Z = 2.9) 1% ( ) 6 : 1 : 0 0500km 6 Fig. 6 Spatial pattern of the clustering coefficient in Chinese aviation network ( ) 3 Tab. 3 Correlative coefficient of indices in Chinese aviation network 4.4-0.43-0.15 0.92 0.09 (3) 0.84-0.24 0.71 (0.84) (0.71) (-0.24) 4.4.1-0.43 (7)
906 64 (92) 19.4 82 21.7 79 23.6 4 (62.2) 5 6 2 2 ( 87) 7 Fig. 7 Distribution of degree-degree correlation in Chinese aviation network 4 ; (k < 10) ; (k 10) [23] (14) ; (14) [24] Barrat k < 30 k 30 : 1 2 3 4.4.2 ( 8) -0.15 (0.41) (-0. 8 92) 2 Fig. 8 Distribution of clustering-degree correlation in Chinese 0 aviation network
8 : 907 1 54 1 2-10 : 1 2 36 (66.7%) 25 (46.3%) 23 (42.6%) 15 (27.8%); 15 (27.8%) 3 : 4.4.3 0.92 (9) 58 6 (57) (56) 0.60 8 (0.624 5) (0.622 6) (0.619 7) : 1 2 9 4.4.4 Fig. 9 Distribution of degree-accessibility correlation in Chinese 0.09 aviation network (0.46) (0.61) (10) (-0.92) 10 Fig. 10 Distribution of clustering-accessibility correlation in Chinese aviation network
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910 64 Spatial Structural Characteristics of Chinese Aviation Network Based on Complex Network Theory WANG Jiao'e 1 2 MO Huihui 1 3 JIN Fengjun 1 (1. Institute of Geographic Sciences and Natural Resources Research CAS Beijing 100101 China; 2. Louisiana State University Baton Rouge LA 70803 USA; 3. China Communications and Transportation Association Beijing 100053 China) Abstract: Today air transport plays an important role in transportation systems especially for medium and long distance. Based on complex network theory Chinese aviation network is abstracted as a set of edges (linkages) connecting a set of nodes (cities) and 144 cities and 1018 air routes were chosen for this research operated from winter 2007 to spring 2008. Then this paper analyzes the spatial structure of Chinese aviation network using indices of degree distribution the average path length the clustering coefficient degree-degree correlation and clustering-degree correlation. Degree is the number of edges that a node shares with others and thus symbolizes the importance of the node in the network. Degree distribution is used to reflect the distribution function of degree and donates the statistical characteristics of a network. Average path length is defined as the average number of edges along the shortest paths for all possible node-pairs in the network and it is a measurememt of the efficiency of transportation network. The clustering coefficient of a node is the ratio of actual edges to maximal edges between nodes which are directly connected with the node. The clustering coefficient of the whole network is the average of all individuals. As for an airport network the average path length and the clustering coefficient are the two most important parameters reflecting network properties and configurations. The results show that Chinese aviation network has a relatively small average path length of 2.23 and a relatively large cluster coefficient of 0.69. More importantly the accumulative degree distribution follows an exponential expression with significance of 0.977. Therefore Chinese aviation network shows the characteristic of a "Small World" network. Due to most new airports' preferences for direct connections with the three top-level national hubs: Beijing Shanghai and Guangzhou the network hierarchy has no distinct difference with the exception of the top one. Also the spatial distribution of Chinese aviation network is imbalanced according to the indices of degree clustering coefficient and accessibility index. Besides the correlation coefficients of these indices mentioned above are analyzed. The results show that there are negative degree correlation nonlinear clustering-degree correlation positive accessibility-degree correlation and slightly positive clustering-accessibility correlation. In conclusion with the rapid development of air transport demand driven by the market economy Chinese aviation network will further evolve to a combined model of "Scale Free" and "Small World" networks and its spatial structure will be more complex. Key words: aviation network; spatial structure; complexity; small world network; scale free network; China