Complex Networks - Theory and Applicationsale/dsta/dsta-5/caldarelli-lecture-1.pdf · IntroductionStatistical Physics SpiritData Why Why are we here? you have to :D IMT wants to become

Introduction Statistical Physics

Complex NetworksTheory and Applications

Guido CALDARELLI

1IMT Alti Studi Lucca, Italy

http://www.guidocaldarelli.com/index.php/lectures

Guido Caldarelli Course on Complex Networks Lecture # 1

Introduction Statistical Physics Spirit Data

Why

Why are we here?you have to :D

IMT wants to become a key institution at European Scale forthe analysis of networks.We all believe this is one of the necessary courses for you

It will provide you new tools for your researchIt will give you an advantage in the job marketspeaking of which, you’d better publish a lot



Why

Why are we here?you have to :D

IMT wants to become a key institution at European Scale forthe analysis of networks.We all believe this is one of the necessary courses for you

It will provide you new tools for your researchIt will give you an advantage in the job marketspeaking of which, you’d better publish a lot



Who

Who can follow this course?Essentially everyone.Some mathematical skills are necessary though

A Very Short Introduction to NetworksG. Caldarelli & M. Catanzaro, OUP (2012)

Networks, Crowds and MarketsD. Easley & J Kleinberg, CUP (2010)

Scale-Free NetworksG. Caldarelli, OUP (2007)



How

How can I follow?DO NOT TAKE NOTES

Pajek software (http://pajek.imfm.si/)R. Albert & A.-L. BarabásiStatistical Mechanics of Complex NetworksReview of Modern Physics 74 47-97 (2002)Mark E. J. Newman.The structure and function of complex networks.SIAM Review, 45 167-256 (2003).



What

What exactly am I going to learn?

Goal of this course is to make you expert of complex networksGoal of this lecture is to make you used with the tools used

How to spot connections in different systemsHow to find most important parts in a systemHow to model social dynamical processes



Fenomenology

Why should I matter?The present computer revolution makes networks crucial

Social Networks have been created (Web blogs, Twitter,Facebook)Computer memories made possible storing everything(Finance, Economics (trade data), News)How to model social dynamical processes



Finance

This is the ownership network of companies traded in Italian Stock Exchange (green persons, red other companies)



Social

Blogosphere in US. Red are Republican bloggers, Blu Democrats bloggers



Trade

The structure of trade between countries, arrows are net flow of money from on country to another



Internet

The structure of internet. Vertices are IP addresses, edges are direct connections.


Introduction Statistical Physics Space Time Social

Foundation

Different disciplinesStatistical Physics has something to contribute to this newscience

Scale InvarianceThe look for universalitySimple Toy models



Symmetries and quantities

Physics has a long tradition of quest for conserved quantities

In an isolated systemE1 + E2 = E ′1 + E ′2~p1 + ~p2 = ~p′1 +

~p′2



Scale Invariance

One of the most recent discoveries is that many systems innature display scale-invariance. This quantity can be used as aconserved quantity.



Power Laws

Power laws are the mathematical signature of scale invariance.Given a relation f (x) = axk , scaling the argument x by aconstant factor c causes only a proportionate scaling of thefunction itself. That is,

f (cx) = a(cx)k = ck f (x) ∝ f (x).

To the right is the long tail, and to the leftare the few that dominate



Scale Invariance in space: Fractals

70’s -80’s has been the time of geometrical scale-invariance80’s-90’s has been the time of temporal scale-invariance00’s-10’s has been the time of topological scale-invariance?? ?? up to you



Plain Dimension

MeasureCovering an object with a unit measure.

Measuring a line with a stick and thenwith another 1/2 long. The measure willdouble.This holds in any dimension. Measure asquare then measure again with a box 1/2the length of the original, you will find 4times as many squares.

N(ε) =1εD→ D =

ln N(ε)

ln 1/ε



Fractal Dimension

MeasureCovering an object with a unit measure.

D =ln N(ε)

ln 1/ε→ D =

ln 3ln 2' 1.58



Fractal Dimension: a little experiment

Real FractalsNatural Fractals are usuallydefined by looking at thevariation of mass with respectto the size of the object.

Let us consider an ordinary A4 sheet. (A4 = 0.210 m X

0.297 m) Good quality printing paper weighs 80g/m2 This

means that one A4 weighs 0.297*0.21*80 g = 4.9896 g

Fold A4 sheet (M ' 5 g)Fold 1/2 A4 (M ' 2.5 g)Fold 1/4 of A4 (M '1.25 g)

Measure the radius of theobjects.



Scale Invariance in Time: Avalanches

Self-organized criticality (SOC) is a property of (classes of)dynamical systems which have a critical point as an attractor.Their macroscopic behaviour thus displays the spatial and/ortemporal scale-invariance characteristic of the critical point of aphase transition, but without the need to tune controlparameters to precise values.



Real Avalanches

Some granular systemmay display anavalanche dynamicswhen they form piles.Various statistical models(BTW) have beenstudied for that

Per Bak, Chao Tang, and Kurt Wiesenfeld PRL 59 381-384 (1987)



Punctuated Equilibrium

In biology since Charles Darwin it was noticed that mostspecies appear suddenly without showing gradual change

During the Cambrian Explosion, earlyorganisms evolved into manydifferent, more complex formspersisting today. Arthropoda,Brachiopoda, Echinodermataoriginated and diversified quicklythereafter.

Gould, S. J. & N. Eldredge “Punctuated equilibrium comes of age". Nature 366 223-227 (1993)



Pareto’s law

Power laws in economics were introduced by the seminal andpioneering work of Vilfredo Pareto. He noticed that in a varietyof different societies, regardless of countries or ages, holds

Pareto’s lawthe distribution of incomes and wealth follows a typical law

N(X > x) ∝ x−a

where N(X > x) is the number of income earners whoseincome is larger than x .This law is often indicated as the 80-20 rule (80 % of wealth inthe hands of 20% population)



Pareto’s law II

A plot realized on the income of billionaires (as far as declared)



Zipf’s law I

Zipf’s lawZipf’s law states that given some corpus of text, the frequencyof any word is inversely proportional to its rank in the frequencytable

P(r) ∝ r−b



Zipf’s law II

Zipf’s law holds also for avariety of different datafrom stocks to cities.Here on the left a map ofthe ranking of the mostpopulated cities



Zipf and Pareto

Actually the two laws can be put in relation if we consider thatthe expected value of a variable Xr of rank r is

E [Xr ] ' Ar−b,

we then have r variables with expected value ≥ A ∗ r−b:

P[X ≥ Ar−b] ' Br

Changing variables we get: P[X ≥ y ] ' y−(1/b). Taking thederivative we obtain

Pr [X = y ] ' y−(1+(1/b)) = y−a.

a = 11+b



Benford’s law I

Check yourself with data from stock exchange. Collect the firstdigit and do the histogram of their frequency



Benford’s law II

Almost unexpectedly this is what you obtain


Documents

Complex Networks - Theory and Applicationsale/dsta/dsta-5/caldarelli-lecture-1.pdf · IntroductionStatistical Physics SpiritData Why Why are we here? you have to :D IMT wants to become