1. Fractal Networks: Structures, Modeling, and Dynamics 章 忠 志 复旦大学计算机科学技术学院 Email: zhangzz@fudan.edu.cnzhangzz@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/~zhangzz/ http://homepage.fudan.edu.cn/~zhangzz/ Blog: http://group.sciencenet.cn/home.php?mod=space&uid=311410 网络结构分析和拓扑识别学术研讨会

2. 复旦大学 Main Contents Introduction to Fractal Networks 1 Structural Properites of Fractal Networks 2 Modelling of Fractal Networks 3 Impact of Fractality on Dynamics 4 Conclusion and outlook 5

3. 复旦大学  Small-world Effect Remarkable properties of networks  Scale-free Property OR  Fractal Behavior Small world seemingly contradicts fractality!

4. 复旦大学 Many real networks are fractal -dB-dB log (l B ) log (N B ) fractal Non-fractal  Protein interaction network  Hollywood film actor network  Metabolic network  World Wide Web Box-covering method Song, Havlin, and Makse, Nature (2005)

5. 复旦大学 Fractal biological networks MetabolicProtein interaction Song, Havlin, Makse, Nature (2005) Three domains of life: archaea, bacteria, eukaria E. coli, H. sapiens, yeast 43 organisms - all scale

6. 复旦大学 Fractal information and social networks World Wide Web nd.edu domain Hollywood film actors 212,000 actors 300,000 web-pages Other bio networks: Khang and Bremen groups Internet is not fractal!

7. 复旦大学 Most efficient box-covering method 4 boxes 5 boxes 1 0 0 1 2  Mapping to graph colouring problem  Greedy algorithm to find minimum boxes J. Stat. Mech. (2007) P03006

8. 复旦大学 Invariant of scaling under renormalization k’=2 renormalization s=1/4 k=8 factor<1 Gallos et al. PNAS (2007) and follow the same distribution. : exponent of the boxes

9. 复旦大学 Renormalization of WWW with

10. 复旦大学 Degree distribution of WWW 2016-6-11

11. 复旦大学 Scaling relations New scaling relation Numboer of boxes Box size Degree of box Box exponent

12. 复旦大学 Structural properties of fractal networks  Fractal networks are disassortative  Betweennees distribution  The number of spanning trees in fractal networks is larger than that in non-fractal networks 2016-6-11

13. 复旦大学 2016-6-11 Brief introduction to spanning trees Number of spanning trees: EPL (Europhysics Letters), 2010, 90:68002. Entropy of spanning trees: The larger the entropy, the larger the number of spanning trees.

14. 复旦大学 2016-6-11 Spanning trees in a non-fractal scale-free web The entropy for spanning trees in square lattice is 1.16624 A counterintuitive conclusion that a network with more spanning trees may be relatively unreliable. EPL, 2010, 90:68002

15. 复旦大学 2016-6-11 Spanning trees in a fractal scale-free network Physical Review E, 2011, 83:016116 Fractality can significantly increase the number of spanning trees in fractal scale-free networks.

16. 复旦大学 A model of fractal scale-free network 2016-6-11 European Physical Journal B, 2007, 56:259-27. large-world disassortative

17. 复旦大学 2016-6-11 Model: from fractal to non-fractal SF trees EPJ B, 2008, 64:277-283

18. 复旦大学 2016-6-11 A model for fractal and nonfractal scale-free networks with identical degree sequences Physical Review E, 2009, 79: 031110. Advantages ：  without crossing edges  always connected As q drops from 1 to 0, it undergoes transitions from large to small world and from fractal to non-fractal.

19. 复旦大学 Impact of fractality on dynamics 2016-6-11  Robustness (Reliability)  Percolation  Disease spreading  Random walks  Synchronization  Game � ……

20. 复旦大学 Fractal networks are more robust under intentional attack 2016-6-11 Nature 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 the removal fraction f of the largest hubs.

21. 复旦大学 Bond percolation: nonzero thresholds 2016-6-11 It is in contrast to the conventional wisdom that null percolation threshold is an intrinsic nature of scale-free networks. PRE, 2009, 79: 031110

22. 复旦大学 Fractality can resist disease spread 2016-6-11 SI model: J. Phys. A, 2010, 43:065001 SIR model: nonzero epidemic thresholds J. Stat. Mech. (2008) P09008

23. 复旦大学 2016-6-11 Trapping problem: Random walks on graphs with an immobile trap  Trapping time (TT) for node i denoted by  Average trapping time (ATT) Research goal : obtain the dependence of average trapping time on the system size N.

24. 复旦大学 2016-6-11 Previous work: ATT on Sierpinski gasket and complete graph PRE, 2002, 65: 021105. New J. Phys. 7, 26 (2005)

25. 复旦大学 2016-6-11 Walks on nonfractal scale-free nets PRE, 2009, 79: 021127. New finding: ATT scales sublinearly with network size. EPL, 2009, 86: 10006.

26. 复旦大学 2016-6-11 Walks on a fractal scale-free tree EPL, 2009, 88: 10001. Conclusion: fractality can induce a general slowing down of diffusion.

27. 复旦大学 2016-6-11 Trapping time in fractal and non-fractal scale- free networks with identical degree sequences Physical Review E, 2009, 80: 061111 27/43

28. 复旦大学 2016-6-11 Significant impact of trap position on ATT in non-fractal scale-free trees Journal of Mathematical Physics, 2009, 50: 033514. Journal of Physics A, 2011, 44: 075102

29. 复旦大学 2016-6-11 No essential impact of trap position on MTT in fractal scale-free trees Journal of Physics A, 2011, 44: 075102. The networks seem homogeneous in this sense. Fractality plays a dominant role in determining the ATT in fractal scale-free networks.

30. 复旦大学 2016-6-11 No qualitative effect of trap location on MFPT in extended T-graphs Physical Review E, 2010, 82: 031140. New Journal of Physics, 2009, 11: 103043.

31. 复旦大学 Impact of fractality on synchronization and game 2016-6-11 ◊ Fractality suppresses the synchoronizability in scale-free networks. ◊ Fractality is unfavouble for emergence of cooperation in scale-free networks. ◊ Fractality may have important effect on other dynamics on scale-free networks. European Physical Journal B, 2007, 56:259-27.

32. 2016-6-11 Summary and outlook Fractal networks are ubiquitous 1 Fractality is related to other properties 2 Repulsion between hub leads to fractality 3 Fractality strongly affects dynamics 4

33. Thank You!