Abstract:In order to find the possible generation mechanism of double power law distribution, a spatial network model with dynamic constraints was proposed in this paper based on double-value confined areas. The results of theoretical analysis and simulation showed that the degree distribution of the model presented a single power law in the early stage of evolution, and then evolved into a stable double power law. Furthermore, we made an empirical analysis on Chinese aviation network to verify the validity of the model in reality. The empirical results not only gave the evolution patterns of degree distribution as predicted by theoretical model, but also enabled us to directly observe the evolution trend of the confined area required by model to match its degree distribution. Our theoretical model could provide a possible explanation for the emergence of double power law degree distribution.
徐静, 钱江海. 动态空间约束网络模型及其双段幂律特征[J]. 复杂系统与复杂性科学, 2020, 17(3): 86-93.
XU Jing, QIAN Jianghai. A Spatial Network Model with Dynamic Constraints and Its Double Power Law Feature. Complex Systems and Complexity Science, 2020, 17(3): 86-93.
[1] Lin J Y, Ban Y F. The evolving network structure of US airline system during 1990-2010[J]. Physica A, 2014, 410: 302-312. [2] Li W, Cai X. Statistical analysis of airport network of China[J]. Physical Review E, 2003, 69(4): 046106. [3] Liu H K, Zhou T. Empirical study of Chinese city airline network[J]. Acta Physica Sinica, 2007, 56(1): 106-112. [4] 钱江海. 空间加权网络模型及权度相关性研究[D]. 上海:华东师范大学,2009. Qian Jianghai. The study of spatial weighted network model and strength-degree correlation[D]. Shanghai: East China Normal University, 2009. [5] Barthélemy M, Flammini A. Modeling urban street patterns[J]. Physical Review Letters, 2008, 100(13): 138702. [6] Masucci A P, Smith D, Crooks A, et al. Random planar graphs and the London street network[J]. Eur Phys J B, 2009, 71(2): 259-271. [7] Crucitti P, Latora V, Porta S. Centrality measures in spatial networks of urban streets[J]. Physical Review E, 2006, 73(3): 036125. [8] Yook S H, Jeong H, Barabási A L. Modeling the Internet's large-scale topology[J]. Proc Natl Acad Sci USA, 2002, 99(21): 13382-13386. [9] Ravasz E, Barabási A L. Hierarchical organization in complex networks[J]. Physical Review E, 2003, 67(2): 026112. [10] Albert R, Albert I, Nakarado G L. Structural vulnerability of the North American power grid[J]. Physical Review E, 2004, 69(2): 025103. [11] Solé R V, Rosas-Casals M, Corominas-Murtra B, et al. Robustness of the European power grids under intentional attack[J]. Physical Review E, 2008, 77(2): 026102. [12] Kinney R, Crucitti P, Albert R, et al. Modeling cascading failures in the North American power grid[J]. Eur Phys J B, 2005, 46(1): 101-107. [13] Lambiotte R, Blondel V D, Kerchove C, et al. Geographical dispersal of mobile communication networks[J]. Physica A, 2008, 387(21): 5317-5325. [14] Liben-Nowell D, Nowak J, Kumar R, et al. Geographic routing in social networks[J]. Proc Natl Acad Sci USA, 2005, 102(33): 11623-11628. [15] Okumura Y. Spatial competition and collaboration networks[J]. International Journal of Game Theory, 2012, 41(3): 455-472. [16] Chandra A K, Hajra K B, Das P K, et al. Modeling temporal and spatial features of collaboration networks[J]. Int J of Mod Phys C, 2007, 18(7): 1157-1172. [17] Onel S, Zeid A, Kamarthi S. The structure and analysis of nanotechnology co-author and citation networks[J]. Scientometrics, 2011, 89(1): 119-138. [18] Adamic L A, Huberman B A, Barabási A L, et al. Power-Law distribution of the World Wide Web[J]. Science, 2000, 287(5461): 2115. [19] Progulova T, Gadjiev B. Comparative analysis of collaboration networks[C] // Mohammad-Djafari A. AIP Conference Proceedings of the 30nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering. New York: American Institute of Physics, 2011: 415-422. [20] Montoya J M, Solé R V. Small world patterns in food webs[J]. Journal of Theoretical Biology, 2002, 214(3): 405-412. [21] Motter A E, De Moura A P S, Lai Y C, et al. Topology of the conceptual network of language[J]. Physical Review E, 2002, 65(6): 065102. [22] Scellato S, Cardillo A, Latora V, et al. The backbone of a city[J]. Eur Phys J B, 2006, 50(1/2): 221-225. [23] Dall J, Christensen M. Random geometric graphs[J]. Physical Review E, 2002, 66(1): 016121. [24] Liu F, Zhao Q. An efficient organization mechanism for spatial networks[J]. Physica A, 2006, 366: 608-618. [25] Guan Z H, Ding L, Kong Z M. Multi-radius geographical spatial networks: Statistical characteristics and application to wireless sensor networks[J]. Physica A, 2010, 389(1): 198-204. [26] Waxman B M. Routing of multipoint connections[J]. IEEE J Select Areas Commun, 1988, 1(3): 286-292. [27] Sen P, Banerjee K, Biswas T. Phase transitions in a network with a range-dependent connection probability[J]. Physical Review E, 2002, 66(3): 037102. [28] Kosmidis K, Havlin S, Bunde A. Structural properties of spatially embedded networks[J]. Europhysics Letters, 2008, 82(4): 48005. [29] Barthélemy M. Crossover from scale-free to spatial networks[J]. Europhysics Letters, 2003, 63(6): 915-921. [30] Manna S S, Sen P. Modulated scale-free network in Euclidean space[J]. Physical Review E, 2002, 66(6): 066114. [31] Xie Y B, Zhou T, Bai W J, et al. Geographical networks evolving with optimal policy[J]. Physical Review E, 2007, 75(3): 036106. [32] Bounova G A. Topological evolution of networks: case studies in the US airlines and language Wikipedias[D]. Cambridge: Massachusetts Institute Of Technology, 2009. [33] Chi L P, Cai X. Structural changes caused by error and attack tolerance in US airport network[J]. Int J of Mod Phys B, 2008, 18(17-19): 2394-2400. [34] Zhang J, Cao X B, Du W B, et al. Evolution of Chinese airport network[J]. Physica A, 2010, 389(18): 3922-3931. [35] Dorogovtsev S N, Mendes J F F. Language as an evolving word web[J]. Proc R Soc London B, 2001, 268(1485): 2603-2606. [36] Pinto C M A, Lopes A M, Machado J A T. Double power laws, fractals and self-similarity[J]. Applied Mathematical Modelling, 2014, 38(15/16): 4019-4026. [37] Toda, Akira A. Income dynamics with a stationary double Pareto distribution[J]. Physical Review E, 2011, 83(4): 046122. [38] Yuan W G, Liu Y. A mixing evolution model for bidirectional microblog user networks[J]. Physica A, 2015, 432: 167-179. [39] Reed W J. The Pareto law of incomes—an explanation and an extension[J]. Physica A, 2003, 319: 469-486. [40] Mitzenmacher M. A brief history of generative models for power law and lognormal distributions[J]. Internet Mathematics, 2004, 1(2): 226-251. [41] Han D D, Qian J H, Ma Y G. Emergence of double scaling law in complex systems[J]. Europhysics Letters, 2011, 94(2): 28006. [42] Mitzenmacher M. Dynamic models for file sizes and double pareto distributions[J]. Internet Mathematics, 2004, 1(3): 305-333. [43] Barthélemy M. Spatial networks[J]. Physics Reports, 2013, 499(1-3): 1-101. [44] Karpiarz M, Fronczak P, Fronczak A. International trade network: fractal properties and globalization puzzle[J]. Physical Review Letters, 2014, 113(24): 248701. [45] Disdier A C, Head K. The puzzling persistence of the distance effect on bilateral trade[J]. Review of Economics and Statistics, 2008, 90(1): 37-48. [46] Barabási A L, Albert R. Emergence of scaling in random networks[J]. Science, 1998, 286(5439): 509-511. [47] Barabási G, Bianconi A L. Competition and multiscaling in evolving networks[J]. Europhysics Letters, 2000, 54(1): 37-43. [48] Paleari S, Redondi R, Malighetti P. A comparative study of airport connectivity in China, Europe and US: which network provides the best service to passengers?[J]. Transportation Research, 2010, 46(2): 198-210. [49] Kleinberg J. Navigation in a small world[J]. Nature, 2000, 406: 845. [50] Li G, Reis S D S, Moreira A A, et al. Towards design principles for optimal transport networks[J]. Physical Review Letters, 2010, 104(1): 018701. [51] Chen Q, Qian J H, Zhu L, et al. Optimal transport in time-varying small-world networks[J]. Physical Review E, 2016, 93(3): 032321. [52] Gastner M T, Newman M E J. The spatial structure of networks[J]. European Physical Journal B, 2006, 49(2): 247-252.