ws1_caldarelli

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Lectures on Complex Networks


G. CaldarelliCNR-INFM Centre SMC Dep. Physics University Sapienza Rome, Italyhttp://www.guidocaldarelli.com


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http://www.scale-freenetworks.com

INTRODUCTION


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INTRODUCTION

Free Resources on the web

R. Albert, A.-L. Barabasi STATISTICAL MECHANICS OF COMPLEX NETWORKSReview of Modern Physics 74, 47 (2002)http://arxiv.org/abs/cond-mat/0106096

M.E.J. Newman THE STRUCTURE AND FUNCTION OF COMPLEX NETWORKS,SIAM Review45, 167-256 (2003)http://arxiv.org/abs/cond-mat/03030516

R. Diestel GRAPH THEORY, Springer-Verlag (2005)http://www.math.ubc.ca/~solymosi/443/GraphTheoryIII.pdf

Networks Visualization and Analysis PAJEK, Springer-Verlag (2005)http://pajek.imfm.si/doku.php

Bibliography I collected http://www.citeulike.org/user/gcalda/tag/book


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INTRODUCTION

Dissemination on Networks

A.-L. Barabsi Linked, Perseus (2001)

M. Buchanan Nexus, W.W. Norton (2003)

D. Watts Six Degrees: the science of a Connected Age W.W.Norton (2004)

M. Buchanan The Social Atom, W. W. Norton (2007)


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INTRODUCTION

Resources in your library

G. Caldarelli, A. Vespignani (eds) Handbook of Graphs andNetworks, Wiley (2003)

U Brandes T Erlebach, Network Analysis - MethodologicalFoundations, LNCS Tutorial 3418, Springer Verlag, (2005)

M.E.J Newman, A.-L. Barabsi, D. Watts The Structure andDynamics of Networks, Princeton University Press (2006)

G. Caldarelli, A.Vespignani (eds) Large Scale Structure andDynamics of Complex Networks, World Scientific Press (2007)


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INTRODUCTION

Summary

This first lecture wants to provide a basic knowledge of network theory11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4


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GRAPH THEORY

1.1.2 Basic Quantities (degree)11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

Degree k(In-degree kin and out-degree kout) = number of edges (oriented) per vertex

Distance d= number of edges amongst two vertices ( in the connected region !)

Diameter D = Maximum of the distances ( in the connected region !)

A Graph is an objectcomposed by verticesand edges


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GRAPH THEORY

1.1 Definition and Adjacency Matrix

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4


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GRAPH THEORY

Through the adjacency matrix we can write11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

=

=nj

iji ak

,1

=

=nj

ij

in

i ak

,1

=

=nj

ji

out

i ak

,1

DEGREE

WEIGHTED DEGREE (Strength) =

=nj

w

ij

w

i ak,1

1.1.2 Basic Quantities (degree)

SIMPLE ORIENTED WEIGHTED


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GRAPH THEORY

One way to visualize the behaviour of the degree in anetwork (especially for large ones) is to check the behaviourof the degree frequency distribution P(k)

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

1.1.2 Basic Quantities (degree)


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GRAPH THEORY

1.1.3 Basic Quantities (distance)

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

Through the adjacency matrix we can write

= ijPk,l

klij ad minDISTANCE

WEIGHTED DISTANCE

= ijPk,l

w

klij ad min

= ijPk,l

w

kl

ija

d1

min

DISTANCE INVERSE OF

WEIGHTSDISTANCE SUM OF WEIGHTS


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GRAPH THEORY

1.1.3 Basic Quantities (distance)

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

Similarly to the degree, one usually plots the histogram ofthe frequency density P(d) of distances d


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GRAPH THEORY

1.1.4 Basic Quantities (Clustering)

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4


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GRAPH THEORY

1.1.4 Basic Quantities (Clustering)

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

Through the adjacency matrix we can write

( )

=

jk

jkikij

ii

i aaa

kk

C

2/1

1CLUSTERING

DIRECTED (or even worse WEIGHTED) CLUSTERING

J Saramki et al. Phys Rev E75 027105 (2007).


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GRAPH THEORY

1.2 Trees

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4


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GRAPH THEORY

1.2 Trees

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

Trees are particularly important, since they appear in

Girvan-Newman Algorithm for communitiesTaxonomy of speciesTaxonomy of information


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GRAPH THEORY

1.2 Trees

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

Trees can also be viewed as a way to filter information. In astock exchange the prices of all stocks are correlated, butrestricting only to a Minimum Spanning Tree is a way tovisualize the stronger correlations

In order to build a MinimumSpanning Tree

1) Compute the correlationbetween the N(n-1)/2 vertices

2) Define a distance out of

correlation.3) Rank the distances4) Draw the vertices of the

shortest distance5) Run on the ranking, whenever

you find a new (or two)vertex(-ices), draw it (them)if you do not close a cycle.

6) Stop whenever all the verticeshave been drawn


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GRAPH THEORY

1.2 Trees

11.1

2

1.2

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

1.4

One way to visualize the behaviour of a tree (on top of thedegree) is to check the behaviour of the basin size frequencydistribution P(n).


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11.1

2

GRAPH THEORY

1.3 Vertex Correlation: Assortativity

1.2

1.41.3

1.5

2.12.2

2.3

2.5

2.4

2.6

A way to check the Conditioned Probability P(k|k) that avertex whose degree is k is connected with another vertexwith degree k is given by the measure of the average

degree of the neighbours.


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11.1

2

GRAPH THEORY


1.2

1.4

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6

Similarly, one can define a measure of the assortativity bystarting from the correlation function for degrees of vertices i,j

( ) =ji kk

jijijijiji kPkPkkPkkkkkk,

)()(),(

If we introduce the variance2

22 )()(

=

kk

kkPkPk

We have the ASSORTATIVITY COEFFICIENT r given by

( ) = ji kkjijiji kPkPkkPkkr ,2 )()(),(

1


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11.1

2

GRAPH THEORY


1.2

1.4

1.3

1.5

2.12.2

2.3

2.5

2.4

2.6 M.E.J. Newman, Phys.Rev.Lett89 208701 (2002)


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11.1

2

GRAPH THEORY

1.4 Hierarchical Correlation of Graphs

1.2

1.3

1.4

1.5

2.12.2

2.3

2.5

2.4

2.6 Many Complex Networks are originated or display a hierarchicalform like the one described above (self-similar structure)


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11.1

2

GRAPH THEORY

1.4 Hierarchical Correlation of Graphs

1.2

1.3

1.4

1.5

2.12.2

2.3

2.5

2.4

2.6

kkC )(

A typical case has been found in Metabolic Networks, where thevarious reaction can be assembled in different modules

The key quantity is the assortativity

E.Ravasz, et al. Science 297 1551-1554 (2002)

A. Clauset, C. Moore, M.E.J Newman Nature 453 98-101 (2008)


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GRAPH THEORY

1.5 The Properties of Complex Networks

11.1

2

1.2

1.3

1.5

1.4

2.12.2

2.3

2.5

2.4

2.6

Scale Invariant Degree Distribution P(k)Distribution of distances P(l) peaked around small valuesClustering (with respect to random connections)

Assortativity


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GRAPH THEORY


11.1

2

1.2

1.3

1.5

1.4

2.12.2

2.3

2.5

2.4

2.6

Scale Invariant DegreeDistribution P(k)

b) Actors

d) Neuroscientists

a) WWW

c) Physicists


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GRAPH THEORY


11.1

2

1.2

1.3

1.5

1.4

2.12.2

2.3

2.5

2.4

2.6

Distribution of distances p(l) peaked around small values

A.Vazquez, R Pastor-Satorras, A.Vespignani Phys.Rev.E65 066130 (2002)


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GRAPH THEORY


11.1

2

1.2

1.3

1.5

1.4

2.12.2

2.3

2.5

2.4

2.6

Clustering(with respect torandomconnections)

M.E.J. Newman, M. Girvan, Phys.Rev.E69 026113 (2004)


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GRAPH THEORY


11.1

2

1.2

1.3

1.5

1.4

2.12.2

2.3

2.5

2.4

2.6

Assortativity C(k)~k-or average degree of neighbours

a, In assortative networks, well-connected nodes tend to join to other well-connected nodes, as in many social networks

here illustrated by friendship links in a school in the United States6. b, In disassortative networks, by contrast, well-connected nodes join to a much larger number of less-well-connected nodes. This is typical of biological networks; depictedhere is the web of interactions between proteins in brewer's yeast, Saccharomyces cerevisiae7. Clauset and colleagues'hierarchical random graphs2 provide an easy way to categorize such networks.

S. Redner, Nature 453, 47-48 (2008)


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GRAPH STRUCTURE: COMMUNITIES

2.1 Introduction

2

1.1

1

1.2

1.3

2.1

1.5

1.4

2.2

2.3

2.5

2.4

2.6

The presence of communities in a graph is one of the mostimportant features.Communities are important for Amazon, to run their businesses

For biologists to detect proteins with the same functionFor physicians to detect related diseases


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.5

2.4

2.6


2.2 Motifs

Motifs are the smallest version of communities, it is currentlyunder debate if their presence is important or not in the area of

Complex networks.Mangan, S. and Alon, U. PNAS 100, 1198011985 (2003).Mazurie et al. Genome Biology6 R35 (2005)


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4


2.3 Classes of Vertices

2.5

2.6

Some communites must be defined as made by vertices withsimilar properties (they could share no edge at all).More often the communities are made by vertices sharing manyedges


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4


2.4 Centrality Measures Betwenness and Robustness

2.5

2.6

The concept of centrality is at the basis of the Newman-Girvanmethod for the analysis of communities.

=

=

ilj

Nlj

iib

,1,

)()(

jl

jl

P

P

Girvan, M. and Newman, M.E.J. PNAS, 99, 78217826 (2002).


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4



2.5

2.6

Starting from a graphWe iterativelyCompute betweenness

Cut the edges with thelargest value of it

For the graph above the result is given by the seriesof deletionsE-F, E-I, B-D, A-D, C-F,C-L,

D-H, G-L, D-I, B-E, A-B, H-I,F-L, C-G.


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4



2.5

2.6Guimer, R., Danon, L., Daz-Guilera, A., Giralt, F., and Arenas, A..Physical Review E, 68, 065103 (2002).


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4


2.5 Clustering detection modularity

2.5

2.6

Several way have been proposed in order to check if a divisionis good or not.One of the quantities proposed is the modularityThe starting point is to find a suitable division of the graphinto g subgraphs. To detect whether the division is good or

not, we define a g g matrix E whose entries eij give thefraction of edges that in the original graph connect subgraph ito subgraph j.We want the largest possible number of edges within acommunity and the lowest possible number of edges betweendifferent communities.


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4


2.5 Clustering detection modularity

2.5

2.6

The actual fraction of edges in the subgraph i is given by eii;

=

=gj

iji ef,1

This gives the probability that in a random partitionone random edge has one endvertices in i

In order to define a null case (random partition) wecan compare the diagonal elements of matrix E withthe quantity f

= = giiii feQ

,1

2

The modularity Q is a measure of the validity of a partition of thegraph

M.E.J. Newman, PNAS 103 8577-8582 (2006)


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4


2.6 Communities in Graphs (spectral properties)

2.5

2.6

0110010000000000000

1011100000000000000

1101110000000000000

0110100000000000000

0111010000000000000

1010100001000000000

0000000111000000000

0000001010110000000

0000001101110000000

0000011010110000000

0000000111010000000

0000000111100100000

0000000000000110000

0000000000011011100

0000000000001100010

0000000000000110110

0000000000000101011

0000000000000001101

0000000000000000110

=

nnnn

n

n

aaa

aaaaaa

A

...

............

...

...

21

22221

11211

G S C CO S


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4



2.5

2.6

Spectral analysis is based on the analysis of the following matrices

The Adiacency Matrix A

The Laplacian Matrix L= A - K

The Normal(ized) Matrix N=K-1A

Note that by definition for every node i

=

nk

k

k

K

000

............000

000

2

1

=

=Nj

iji ak,1

GRAPH STRUCTURE COMMUNITIES


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4



2.5

2.6

Laplacian Matrix

=

=

=

=

Nj

njnn

n

Nj

j

n

Nj

j

aaa

aaa

aaa

L

,1

21

2

,1

221

112

,1

1

...

............

...

...

If = L

=

==nj

iiijiji ka,1

2'

n

...

2

1

=

0...//............

/...0/

/.../0

21

22221

11112

nnnn

n

n

kaka

kaka

kaka

N

The elements of matrix N

give the probability with

which one field passesfrom a vertex i to theneighbours.

Normal Matrix

GRAPH STRUCTURE COMMUNITIES


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2

1.1

1

1.2

1.3

2.2

1.5

1.4

2.1

2.3

2.4



2.5

2.6

Given this probabilistic explanation for the matrix N

We have a series of results, for example

One eigenvalue is equal to one and

The eigenvector related is constant.Consider the case of disconnected subclusters:

The matrixNis made of blocks and a general eigenvector will be given by

the space product of blocks eigenvectors (the constant can be different!)

Biological Networks


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Biological Networks

Summary

This third lecture wants to provide an overview of the applications in thefollowing fields

66.1

7

6.2

9.1

9.2

6. Biological Networks

6.1 Protein Interaction Networks

6.2 Metabolic Networks

6.3 Gene Networks

7. Geophysical Networks

8. Ecological Networks

9. Technological Networks

9.1 Internet

9.2 World Wide Web

10. Social Networks

11. Financial Networks

8

6.3

9

10

11

Biological Networks


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Biological Networks

6.1 Protein Interaction Network

Protein interact in various ways during cell life.66.1

7

6.2

9.1

9.2

8

6.3

9

10

11

Uetz et al. Nature 403 623 (2000)H. Jeong et al. Nature, 411, 41 (2002)Giot et al. Science 302, 1727 (2003)Rual et al. Nature 437, 1173 (2005)Ramani et al. Molecular Biology4, 180 (2008)

Biological Networks


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Biological Networks

6.2 Metabolic Networks

Multinetwork analysis of acarbon/light/nitrogen-responsive metabolicnetwork.

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11

Most of the reactions arenot reversible and thenetwork is oriented.

M.L. Gifford et al. Plant Physioloogy on-line Essay 12.2H. Jeong et al. Nature 407 651 (2000).D. Segr et al. PNAS 99 15112 (2002).R. Gumer et al. Nature 433, 895 (2005)

INTRODUCTION


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INTRODUCTION

6.3 Gene Regulatory Networks

A particular class ofreactions in which thereis the expression of agene is at the basis of

GRN.

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11 J. Hasty et al. Nat. Rev. Gen., 2, 268 (2001).T.I. Lee et al. Science 298, 799 (2002)

Geophysical Networks


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Geophysical Networks

Trees and Basin distributions

The case of rivernetworks presents ascale invariance inthe basin size

distributions

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11 Rodrguez-Iturbe, I. and Rinaldo, A.Fractal River Basins: Chance and Self-Organization.Cambridge University Press, Cambridge (1996).

Dodds, P.S. and Rothman, D.H. Physical Review E,

63, 016115, 016116, 016117 (2000).

Ecological Networks


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Ecological Networks

Food Webs data

Most of theactivity onecologicalnetworks is

related tothestatisticalpropertiesof foodwebs

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11 PEaCE LAB http://www.foodwebs.orgR.J. Williams N.D. Martinez Nature 404 180 (2000)

Ecological Networks


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Ecological Networks

Statistical properties of food webs

J.M. Montoya, R.V. Sol,Journal ofTheoretical Biology214 405 (2002)

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11D.B. Stouffer et al. Ecology, 86, 13011311 (2005).

Small-world but notscale-free

Technological Networks


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g

9.1 Internet

Traditionally the analysisof the Internet structure ismade by means oftraceroutes. That is to say,

by exploring all the pathsfrom a given point to allthe possible destinations.

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11 Internet Mapping Project http://www.cheswick.com/ches/map/http://www.caida.orghttp://www.cybergeography.org



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g

9.1 Internet

66.1

7

6.2

9.1

9.2

8

6.3

9

10

11

A. Vazquez, R. Pastor-Satorras and A.Vespignani PRE 65 066130 (2002)

R. Pastor-Satorras and A. VespignaniEvolution and Structure ofInternet: A Statistical Physics Approach.Cambridge University Press (2004)



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9.2 World Wide Web

WWW is probably thelargest networkavailable of the orderof billions of elements.

Different centralityproperties arise in suchstructure.

66.1

7

6.2

9.19.2

8

6.3

9

10

11 A. Broder et al. Computer Network, 33, 309 (2000).D. Donato et al.Journal of Phys A, 41, 224017 (2008).

Social Networks


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Wikipedia

66.1

7

6.2

9.19.2

8

6.3

9

10

11

Social Networks


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Wikipedia

66.1

7

6.2

9.19.2

8

6.3

9

10

11

indegree(empty) and outdegree(filled)

Occurrency distributions for the Wikgraphin English (o) and Portuguese ().

The degree shows fat tailsthat can be approximatedby a power-law function of

the kind P(k) ~ k-

Where the exponent is

the same both forin-degree and out-degree.

V. Zlatic et al. Physical Review E, 74, 016115 (2006).

A. Capocci et al. Physical Review E, 74, 036116 (2006).

Financial Networks


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Stock Correlation Network

66.1

7

6.2

9.19.2

8

6.3

9

10

11

Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-yearperiod 1987-1998 (3030 trading days). The node colour is based on Standard Industrial Classification system.The correspondence is:red for mining cyan for construction yellow for manufacturinggreen for transportation,, light blue for publicelectric,gas and sanitary services administration magenta wholesale tradeblack for retail trade purple for finance and insurance orange for service industrie

Bonanno et al. Physical Review E68 046130 (2003).

Financial Networks


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Stock Ownership Network

66.1

7

6.2

9.19.2

8

6.3

9

10

11Garlaschelli et al. Physica A, 350 491 (2005).

Financial Networks


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Trading Webs

66.1

7

6.2

9.19.2

8

6.3

9

10

11M. Angeles Serrano, M. Boguna Phys Rev. E68 015101 (2003)D. Garlaschelli et al. Physica A, 350 491 (2005).C.A. Hidalgo, et al. Science 317 482 (2007)

Financial Networks


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Banks Credit Network

66.1

7

6.2

9.19.2

8

6.3

9

10

11G. De Masi et al. PRE, 98 208701 (2007)

Banks exchange moneyovernightly, in order tomeet the customer needsof liquidity as well asECB requirements

The network shows a rather peculiar architecture

The banks form a disassortative network where largebanks interact mostly with small ones.

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