分形网络结构性质建模与动力学

1、会计学1分形网络结构性质建模与动力学分形网络结构性质建模与动力学Introduction to Fractal Networks1Structural Properites of Fractal Networks 2 Modelling of Fractal Networks3Impact of Fractality on Dynamics4Conclusion and outlook5u Small-world Effectu Scale-free PropertyORu Fractal Behavior Small world seemingly contradicts fractalit

2、y!Many real networks are fractal( ) BdBBBNll4BN 3BN 2BN -dBlog(lB)log(NB)fractalNon-fractalBd u Protein interaction networku Hollywood film actor networku Metabolic networku World Wide WebBox-covering method Song, Havlin, and Makse, Nature (2005) Fractal biological networksMetabolicProtein interacti

3、onSong, Havlin, Makse, Nature (2005)( ) BdBBN ll2.3Bd 3.5Bd Three domains of life: archaea, bacteria, eukariaE. coli, H. sapiens, yeast 43 organisms - all scaleFractal information and social networks World Wide Web( ) BdBBN ll6.25Bd 4.15Bd domainHollywood film actors 212,000 actors300,000 web

4、-pagesOther bio networks: Khang and Bremen groups Internet is not fractal!Most efficient box-covering method4 boxes5 boxes10012p Mapping to graph colouring problemp Greedy algorithm to find minimum boxesJ. Stat. Mech. (2007) P03006k=2renormalizations=1/4k=8factor1Gallos et al. PNAS (2007)( ) kdBBs l

5、landfollow the same distribution.: exponent of the boxesRenormalization of WWW with 2022-5-10New scaling relationNumboer of boxesBox sizeDegree of boxBox exponent( ) kdBBs ll2022-5-102022-5-10EPL (Europhysics Letters), 2010, 90:68002.Entropy of spanning trees:The larger the entropy, the larger the n

6、umber of spanning trees.2022-5-10Spanning trees in a non-fractal scale-free webThe entropy for spanning trees in square lattice is 1.16624A counterintuitive conclusion that a network with more spanning trees may be relatively unreliable.EPL, 2010, 90:680022022-5-10Spanning trees in a fractal scale-f

7、ree networkPhysical Review E, 2011, 83:016116Fractality can significantly increase the number of spanning trees in fractal scale-free networks.2022-5-10European Physical Journal B, 2007, 56:259-27.large-worlddisassortative2022-5-10Model: from fractal to non-fractal SF treesEPJ B, 2008, 64:277-283202

8、2-5-10Physical Review E, 2009, 79: 031110.Advantages：u without crossing edgesu always connectedAs q drops from 1 to 0, it undergoes transitions from large to small world and from fractal to non-fractal.2022-5-10p Robustness (Reliability)p Percolationp Disease spreadingp Random walksp Synchronization

9、pGame 2022-5-10Nature Physics, 2002, 2:275-281.European Physical Journal B, 2007, 56:259-27.Relative size of the largest cluster, S, and the average size of the remaining isolated clusters, as a function of t h e r e m o v a l fraction f of the largest hubs.2022-5-10It is in contrast to the conventi

10、onal wisdom that null percolation threshold is an intrinsic nature of scale-free networks.PRE, 2009, 79: 0311102022-5-10SI model:J. Phys. A, 2010, 43:065001SIR model: nonzero epidemic thresholdsJ. Stat. Mech. (2008) P090082022-5-10Research goal: obtain the dependence of average trapping time on the

11、system size N.2022-5-10PRE, 2002, 65: 021105. New J. Phys. 7, 26 (2005)2022-5-10Walks on nonfractal scale-free nets PRE, 2009, 79: 021127.New finding: ATT scales sublinearly with network size.EPL, 2009, 86: 10006.2022-5-10EPL, 2009, 88: 10001.Conclusion: fractality can induce a general slowing down

12、of diffusion.2022-5-10Physical Review E, 2009, 80: 06111127/432022-5-10Journal of Mathematical Physics, 2009, 50: 033514.Journal of Physics A, 2011, 44: 0751022022-5-10Journal of Physics A, 2011, 44: 075102.The networks seem homogeneous in this sense. Fractality plays a dominant role in determining

13、the ATT in fractal scale-free networks.2022-5-10Physical Review E, 2010, 82: 031140.New Journal of Physics, 2009, 11: 103043.2022-5-10 Fractality suppresses the Fractality suppresses the synchoronizability in scale-free synchoronizability in scale-free works. Fractality is unfavouble for emergence F

14、ractality is unfavouble for emergence of cooperation in scale-free networks. of cooperation in scale-free networks. Fractality may have important effect on Fractality may have important effect on other dynamics on scale-free networks.other dynamics on scale-free networks.European Physical Journal B,

15、 2007, 56:259-27.2022-5-10Fractal networks are ubiquitous 1Fractality is related to other properties2Repulsion between hub leads to fractality3Fractality strongly affects dynamics4Thank You!u Small-world Effectu Scale-free PropertyORu Fractal Behavior Small world seemingly contradicts fractality!2022-5-10Model: from fractal to non-fractal SF treesEPJ B, 2008, 64:277-2832022-5-10It is in contrast to the conventional wisdom that null percolation threshold is an intrinsic nature of scale-free networks.PRE, 200