DYNAMICS OF COMPLEX SYSTEMS Self-similar phenomena and Networks Guido Caldarelli CNR-INFM Istituto dei Sistemi Complessi 1/6

1.SELF-SIMILARITY (ORIGIN AND NATURE OF POWER-LAWS) 2.GRAPH THEORY AND DATA 3.SOCIAL AND FINANCIAL NETWORKS 4.MODELS 5.INFORMATION TECHNOLOGY 6.BIOLOGY STRUCTURE OF THE COURSE

STRUCTURE OF THE FIRST LECTURE 1.1) SELF-SIMILARITY AND COMPLEXITY 1.2) DETERMINISTIC FRACTALS 1.3) PHYSICAL FRACTALS 1.4) FRACTALS IN NATURE 1.5) GRAPHS 1.6) POWER-LAW STATISTICS 1.7) SCALE-FREE NETWORKS 1.8) THE ORIGIN OF SCALE FREE NETWORKS 1.9) SMALL WORLD EXPERIMENT

More is different ! quantitatively larger systems are qualitatively different P.W. Anderson Science (1972) Emergence of Complexity is related to 1)Microscopical interactions 2)Co-evolution 3)Self-Organization 1.1 SELF-SIMILARITY and COMPLEXITY

Atom do not show the electrical features of macroscopic materials. Complex rearrangement of electrons in cristals determine these new properties 1.1 SELF-SIMILARITY and COMPLEXITY

Router connections at microscopical level produce the complex Internet structure. 1.1 SELF-SIMILARITY and COMPLEXITY

Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK 1.1 SELF-SIMILARITY and COMPLEXITY

This is a general rule in Complex Structures Through simple microscopical interaction Complex Structures develop long range correlations. Very different systems can be described through Graph Topology River NetworksFood WebsInternet 1.1 SELF-SIMILARITY and COMPLEXITY

NOT all the network shapes seem to work. Almost everywhere we find scale-free networks (highly irregular)

Scale invariant systems or Fractals are self-similar objects. I.e. a fractal is something similar to itself This happens very often in Nature 1.2 DETERMINISTIC FRACTALS

Mathematicians provided the concept of Fractal Dimension The object obtained in the limit has “dimension” less than 2, in particular Where N(  ) is the number of triangles of linear size  needed to cover the structure N(  ) = (1/  ) D 1.2 DETERMINISTIC FRACTALS

N(  ) = 2 k where k is the iteration And  =(1/3) k D=ln(2)/ln(3) = … N(  ) = 8 k where k is the iteration And  =(1/3) k D=ln(8)/ln(3) = … The Cantor Set is the dust of points obtained as the limit of this succession of segments This is already the limit of succession of iterations 1.2 DETERMINISTIC FRACTALS

More generally, Fractals are standard phenomena in Nature, in this case their nature is intrinsically stochastic and not deterministic. 1.3 PHYSICAL FRACTALS

Another way to measure fractal dimension is through mass-length relation M 1  R 1 D M 2  R 2 D 1.3 PHYSICAL FRACTALS

Let us consider an ordinary A4 sheet. A4 format corresponds to m X m Good quality printing paper weighs 80g/m 2 This means that one A4 weighs 0.297*0.21*80 g = g Now fold an A4 sheet of g Then fold one half of A4 (M= g) one fourth of A4 (M= g) ….. Measure the radius of the objects. R (cm)M (g) 3.0 ± ± ± ± ± PHYSICAL FRACTALS

1104 Dome of Anagni (Italy)Viscous fingering (Lenormand) 1.4 FRACTALS IN NATURE

1.4 FRACTALS IN NATURE Dielectric Breakdown

1.4 FRACTALS IN NATURE Electrodeposition

Fella Colorado 1.4 FRACTALS IN NATURE River Networks

The Bridges path Is it possible to find a path visiting ALL the bridges of Königsberg ONLY ONCE? NO! Euler (1736) introduced the first graphs like the one on the right. One can distinguish between passage points and starting/ending points. A passage point must have an EVEN number of edges. Only starting and ending points (max 2) might have ODD number. 1.5 GRAPHS: The origins

· Degree k (in-coming k in e out-going k out ) = number of edges (oriented) per vertex A Graph G(v,e) is made of v vertices and e edges Edges can be Oriented  · Distance d = minimum number of edges between two vertices · Diameter D = Maximum of distances The World Wide Web is like that but made of billions of vertices. We need Statistics ! 1.5 GRAPHS: Definitions

Throw a dice for many times (1200) What happens if dice are unbiased? That histogram shows a Frequency Distribtion. In particular this distribution is (more or less) UNIFORM 1.6 STATISTICS Uniform Distribution

We must expect a similar behaviour, when measuring heights in a population? Uniform distributions describe a case where all the values are equiprobable. WRONG! THE DISTRIBUTION IS CALLED GAUSSIAN 1.6 STATISTICS Gaussian Distribution

In (MANY!) other cases the frequency distribution is different from both The distribution is called SELF-SIMILAR or POWER LAW I.e. what is the frequency of first digit in Stock Prices? 1.6 STATISTICS Power-law Distribution

As for fractals this distribution is self-similar. I.e. it does not change along axis x. This is clear passing to logarithmic scale. A change in the scale of observation means a change from x to x’ where x = ax’ 1.6 STATISTICS Power-law Distribution

IN ALMOST ALL REAL NETWORKS The degree frequency distribution is self-similar Many vertices have small degree, few have large one (hubs) The distance frequency distribution is bell-shaped around a characteristic ``small’’ value (4,5,6) 1.7 SCALE-FREE NETWORKS

Nodes: actors Links: cast jointly N = 212,250 actors  k  = P(k) ~k -  Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999)  = SCALE-FREE NETWORKS

Internet Stock Ownership networks b)Actors d)Neuroscientists a)WWW c) Physicists Social Networks 1.7 SCALE-FREE NETWORKS

Is the phenomenon interesting? Start with 5 vertices Consider all the possible edges We toss a coin to decide if a link must be drawn or not. The degree frequency is not self-similar. Paul Erdös THE ORIGIN OF SCALE-FREE NETWORKS

Is it possible to deliver a message to a stock dealer in Chicago starting from randomly extracted people in Nebraska and Kansas? 1.9 SMALL WORLD EXPERIMENT

On average, less than Six passages !!!! SIX DEGREES OF SEPARATION 1.9 SMALL WORLD EXPERIMENT: Six Degrees of Separation

According to Mark Granovetter, the small world is related to weak links 1.9 SMALL WORLD EXPERIMENT: The structure of communities