Fractal Networks: Structures, Modeling, and Dynamics 章 忠 志 复旦大学计算机科学技术学院 Homepage: Blog: 网络结构分析和拓扑识别学术研讨会

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

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

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)

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

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!

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

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

复旦大学 Renormalization of WWW with

复旦大学 Degree distribution of WWW

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

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

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

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

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

复旦大学 A model of fractal scale-free network European Physical Journal B, 2007, 56: large-world disassortative

复旦大学 Model: from fractal to non-fractal SF trees EPJ B, 2008, 64:

复旦大学 A model for fractal and nonfractal scale-free networks with identical degree sequences Physical Review E, 2009, 79: 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.

复旦大学 Impact of fractality on dynamics  Robustness (Reliability)  Percolation  Disease spreading  Random walks  Synchronization  Game � ……

复旦大学 Fractal networks are more robust under intentional attack Nature Physics, 2002, 2: European Physical Journal B, 2007, 56: 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.

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

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

复旦大学 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.

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

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

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

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

复旦大学 Significant impact of trap position on ATT in non-fractal scale-free trees Journal of Mathematical Physics, 2009, 50: Journal of Physics A, 2011, 44:

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

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

复旦大学 Impact of fractality on synchronization and game ◊ 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:

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

Thank You!