China Institute of Atomic Energy, Beijing 102413

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WWWInternet

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The Small World Effect1998 What a small world! Indeed, we all are connected through a short chain of acquaintances. The most popular manifestation of such small world effect is the so-called Six degree of separation concept The Scale-free feature1999

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Erds-Rnyi model (1960)- Democratic- RandomPl Erds (1913-1996)

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Small-world networksWatts & Strogatz, Nature 393, 440 (1998)N = 1000Large clustering coeff. Short average path length

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Scale-free networksBarabasi & Albert, Science 286, 509 (1999)ActorsMoviesWeb-pagesHyper-linksTrans. stationsPower linesNodes:Links:

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Single-scale small-world networksProc Nat Acad Sci USA 97, 11149 (2000)

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Barabsi & Albert, Science 286, 509 (1999): the probability that a node connects to a node with k links is proportional to k. GROWTH: add a new node with m links

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Lifes Complexity Pyramid(Zoltn N. Oltvai and Albert-Lszl Barabsi,Science, 298(2002)763)

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Steven H. Strogatz, Exploring Complex Networks, Nature, 2001,Vol.410, 268-276. R. Albet and A. L. Barabasi, Statistical Mechanics of Complex Networks, Rev. Mod. Phys., 2002, Vol. 74, pp48-97M. E. J. Newman, The Structure Function of Complex Networks, SIAM Review, 2003, Vol.45, No.2, pp167-256.

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(LUHNM)(OL)RA=PA+RP gr=GR/RADA=HP+PA fd=HP/DA(HP) (PA) (PA)(GR)(OL) (RA) (DA)dr = DA / RA(HUHPM)(UHVGM)vg = DVG / RVGRVGDVGII.

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Exponent of node degree power-lawHUHPM-BA HUHPM- BBV HUHPM- TDE2.1 HUHPMFang J Q, Bi Q, Li Y, Advances in Complex Systems, Vol. 10, No. 2 (2007) 117-141Fang J Q, Bi Q, Li Y, et al, Science in China Series G, 2007,50(3):379-396.Fang J Q, Bi Q, Li Y, et al, Chin. Phys. Lett., 2007, 24(1): 279283.Fang J Q, Progress in Nature Science, 2007,17(7):761-774.

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Left: Comparison of Average path length Right: Comparison of Clustering coefficient C( HUHPM model with d/r=1/1 and =4)

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2.2 UHNM r C versus d/r (a) fd =1/1, m = 30; (b) fd =1/1, m = 10; (c) fd =1/10, m = 2.

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UHNM- BBV rc vs drgr.

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d/r r C vs (fd ,gr) (weighted network ) d/r =1/99 d/r=1/1 d/r 19/1

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(a) UHNM (b)MAM (A: )

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Some PapersYong Li , Jin-Qing Fang , Qiao Bi , and Qiang Liu. Entropy Characteristic on Harmonious Unifying Hybrid Preferential NetworksEntropy, 2007, 9: 73-82Fang Jin-Qing, Bi Qiao, Ly, Yong, Advances in theoretical models of network science, Front. Phys. China,2007, 1:109-124.Liu Qiang, Fang Jinqing, Li yong, Commun. Theor. Phys. 2007,47:752-758.Bi Qiao, and Fang Jinqing, Entropy and HUHPM approach for complex networks, Physica A, in press (2007).Wu-jie Yuan Xiao-Su Luo, Pin- Qun Jiang ,Bi-Hong Wang, Jin-Qing Fang, Transition to Chaos in Small World Dynamical Network ,Physica A, 2007in press Sun Weiguan, XU CongXiang, Li Chang-Ping, Fang Jin-qing, Synchroniuzation and Bifurcation of General Complex Dynamical Networks,Commun. Theor.Phys. 2007,47:1073-1075.

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2007177841-857. . , G2007, 3(2)230249.. , 2007, 25(11):23-29()20073239-343()20074361-556

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III. Unified Hybrid Variable Growing Network Model

,In many real-world networks such as the Intemel, World Wide Web, collaboration, citation, telephone exchanges, engineering, society, metabolism, biology, gene regulatory network ( e.g., the network of regulatory proteins that control gene expression in bacteria), etc.. The number of links grow in time in a nonlinear fashion.

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vgUHVSGdrfd, grvg

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:P=Const: 0 p(t) 1

[1] Sen P. Phys. Rev. E, 2004, 69:46107. [2]Mattick J S. Gagen G M. Science, 2005 307:856[3]Gagen G M. Mattick J S. Phys. Rev. E, 2005, 72:16123.[4]David M D S, Jukka P O, Neil F J. arXiv, 2007,physics/0701339.

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III. UHVGM 3.1 P(k) SFSED ( )

00.30.6

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3. 2 drP(k) vg d/r =1/99 d/r=1/1 d/r 19/1

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3.3 C

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3.4rc

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rcdr,fd,gr,vg

rcdrdr=1/49vgfd=0/1gr=1/1drdr=49/1grvgvgvg1/1rcC

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rc,fd0.9/1rcdrrc-1fd1/1rcdrrc

,

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IV.:

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4.

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2005

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1 E-31 E-4

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2005

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5.

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Master Equation of the QID

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QID Fokker-Plank EquationFokker-Plank QID

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QID4

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123Thom

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The 2nd orderThe 5th orderHexagonal nanowire network

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SF-

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Fish Swarming Birds Flocking (

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2Vicsekhubhub

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An Adaptive Velocity ModelHesitate, and move slow!Local order parameter:

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Convergence Probabilitythe probability that a group of N initially randomly distributed agents will finally converge to a global convergence state.

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BA

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EMGKauffman2MGEMG

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.SJTU-WXFIM

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Coefficient of speed reflects the willingness of each agent to move faster or lower along the average direction of its neighbors based on the local degree of direction consensus.*