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Scaling and Fractals Presented by: Shlomo Havlin, Bar-Ilan University Introduction Since about 1980, extensive research was held in complex systems in many areas including: Earthquakes, climate, rivers and coastlines Galaxies-density and structure Neurons structure and networks Heartbeat dynamics, DNA structure Euclidian geometry cannot describe such complex structures, it deals with straight lines: triangles: and circles: However, Nature does not have such structures: trees are not triangles mountains are not cones clouds do not look like balls

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Crystal Translational Symmetry

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Self-similarity = Revolution in science In every scientific discipline (biology, chemistry, physics) the assumption of “characteristic length scale” is a basic concept. For example, in crystalline solid materials it is the distance between atoms. Or the mean free path in gas state. In fractals there is no characteristic length as happens in many natural systems. Trees for example do not have branches of a specific length. Instead one has many length scales starting with the smallest leaves until branches of almost the size of the tree. Why Nature chose fractals? – trees of many length scales are more stable against storms. Fractal geometry was developed mainly due to development of computers – graphics and powerful computers were necessary. Fractal geometry is much more suitable for computers than Euclidian geometry due to the recursive language of both fractals and computers. To plot a fractal in a computer is usually much easier than to plot a circle.

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Astrophysics – distribution of stars and galaxies Biology – DNA molecules, proteins structure and networks, neural cells Medicine – heartbeat dynamics, cancer diagnostics, Alzheimer disease Geology – earthquakes, oil recovery, rivers, climate Economics – stock market changes, currency dynamics, companies Nano materials – electron wave function Technology – compressing pictures, background for movies, Internet. Applications in many fields

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Identifying cancer growth Plaques in Alzheimer disease Artificial landscape for movies

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Fractal geometry describes Nature better than classical geometry. Two types of fractals: deterministic and random. Deterministic fractals Ideal fractals having self-similarity. Every small part of the picture when magnified properly, is the same as the whole picture. Self-similarity is an important property of fractals To better understand fractals, we discuss several examples: Fractals

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3D Sierpinski gasket 2D Sierpinski gasket

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1 This is a fractal for Sierpisnki gasket with lower cut off

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Definition of fractal dimension generalization to non-integer dimension Solution: Example: Koch curve - non integer – between 1 and 2 dimensions. Koch curve is not a line (d=1) but does not fill a plane (d=2). Non integer dimension between 1 and 2 dimensions. Example: Sierpinski gasket

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Model for lightening

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Box covering method for random fractals Draw a lattice of squares of different sizes For each count the number of boxes needed to cover the fractal increases with decreasing     The fractal dimension is obtained from     Plotting vs on log-log graph – the slope is

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Complex Networks and Fractals A Network is a structure of N nodes and M edges (or 2M links ) Called also graph – in Mathematics Complex systems can be described and understood using networks Internet: nodes represent computers links the connecting cables Social systems: nodes represent people links their relations Cellular systems: nodes represent molecules links their interactions Weighted networks each link has a weight determining the strength or cost of the link

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Internet Network Faloutsos et. al., SIGCOMM ’99

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WWW-Network Barabasi et al (1999) Nodes-pages Links – html Scale-free No characteristic degree

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Box covering method Generate boxes where all nodes are within a distance We obtain for WWW, social networks, cellular networks, etc. Calculate number of boxes, , of size needed to cover the network or Self similarity Direct Measurements of Fractal Dimention C. M. Song et al, Nature, 433, 392 (2005)

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Self similarity of WWW, Actors, PIN and Cellular Networks

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Cellular and Metabolic Networks

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Scaling and Box Covering Approach Hierarchy of WWW NOW, REGARD EACH BOX AS A SINGLE NODE (SUPERNODE) AND ASK WHAT IS THE DEGREE DISRIBUTION OF THE NETWORK OF BOXES (SUPERNODES) AT DIFFERENT SCALES ? NETWORKS OF SUPERNODES RENORMALIZATION:

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Self similarity of WWW The degree distribution after renormalization does not change 0

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After Renormalization: With the same ! Where From which follows: HOW FAMILIES OF VARIOUS SIZES ARE LINKED? THE SCALING TRANSFORMATION OF THE DEGREE DISTRIBUTION Hierarchy of Scale Free Fractal Dimension:

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Renormalization and self-similarity of WWW network with

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NOT ALL REAL NETWORKS OR MODELS ARE FRACTALS! BA-model There exist fractals and non-fractals networks

Books : 

Books 1. B.B. Mandelbrot: The Fractal Geometry of Nature (Freeman, San Francisco 1982). 2. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, 1971) 3. A. Bunde and S. Havlin (eds): Fractals and Disordered Systems (2nd Ed, Springer, Berlin 1996); Fractals in Science (Springer, Berlin 1994). 4. T. Vicsek: Fractal Growth Phenomena (World Scientific, Singapore 1992). 5. J. Feder: Fractals (Plenum, NY 1988). 6. H.O. Peitgen, H. Jurgens and D. Saupe: Chaos and Fractals (Springer, NY 1992). 7. P. Bak, How Nature Works (Copernicus, NY 1996). 8. James Gleick, Chaos (Penguin books, NY 1997). 9. P. Meakin, Fractals, Scaling and Growth far from Equilibrium (Cambridge University press, 1998). 10. D. ben Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, 2000). 11. A. L. Barabasi, Linked (Plume books, 2003); R. Albert, A.-L. Barabasi, Rev. Mod. Phys. 74 (2002) 47. 12. R. Pastor-Satorras, A. Vespignani, Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge University Press, 2004). 13. S. N. Dorogovtsev, J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and www (Physics) (Oxford University Press, 2003).