Lectures on Complex Networks
G. Caldarelli
CNR-INFM Centre SMC Dep. Physics University “Sapienza” Rome, Italy
http://www.guidocaldarelli.com
Lectures on Complex Networks
INTRODUCTION
http://www.scale-freenetworks.com
Lectures on Complex Networks
INTRODUCTION
Free Resources on the web
•R. Albert, A.-L. Barabasi STATISTICAL MECHANICS OF COMPLEX NETWORKS
Review 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 Review 45, 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
Lectures on Complex Networks
INTRODUCTION
Dissemination on Networks
•A.-L. Barabási 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)
Lectures on Complex Networks
INTRODUCTION
Resources in your library
•S.N. Dorogovtsev, J.F.F. Mendes Evolution of Networks, Oxford
University Press (2003)
•S. Bornholdt, H.G. Schuster(eds) Handbook of Graphs and
Networks, Wiley (2003)
•R.Pastor Satorras, M. Rubi, A. Diaz-Guilera Statistical Mechanics
of Complex Networks Lecture Notes in Physics, Vol. 625 (2003)
•R. Pastor Satorras, A. Vespignani Evolution and Structure of the
Internet Cambridge University Press (2004)
Lectures on Complex Networks
INTRODUCTION
Resources in your library
•G. Caldarelli, A. Vespignani (eds) Handbook of Graphs and
Networks, Wiley (2003)
•U Brandes T Erlebach, Network Analysis - Methodological
Foundations, LNCS Tutorial 3418, Springer Verlag, (2005)
•M.E.J Newman, A.-L. Barabási, D. Watts The Structure and
Dynamics of Networks, Princeton University Press (2006)
•G. Caldarelli, A.Vespignani (eds) Large Scale Structure and
Dynamics of Complex Networks, World Scientific Press (2007)
Lectures on Complex Networks
INTRODUCTION
Summary
1
This first lecture wants to provide a basic knowledge of network theory
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.1.2 Basic Quantities (degree)
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
A Graph is an object
composed by vertices
and edges
2.3
2.4
— Degree k (In-degree kin and out-degree kout ) = number of edges (oriented) per vertex
2.5
— Distance d = number of edges amongst two vertices ( in the connected region !)
— Diameter D = Maximum of the distances ( in the connected region !)
2.6
Lectures on Complex Networks
GRAPH THEORY
1.1 Definition and Adjacency Matrix
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.1.2 Basic Quantities (degree)
1
1.1
1.2
1.3
Through the adjacency matrix we can write
DEGREE
ki =
∑ aij
kiin =
j =1,n
1.4
1.5
WEIGHTED DEGREE (Strength)
out
a
k
∑ ij i =
∑a
j =1,n
j =1,n
kiw =
ji
w
a
∑ ij
j =1,n
2
2.1
2.2
2.3
2.4
2.5
2.6
SIMPLE
ORIENTED
Lectures on Complex Networks
WEIGHTED
GRAPH THEORY
1.1.2 Basic Quantities (degree)
1
1.1
1.2
One way to visualize the behaviour of the degree in a
network (especially for large ones) is to check the behaviour
of the degree frequency distribution P(k)
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.1.3 Basic Quantities (distance)
1
1.1
1.2
Through the adjacency matrix we can write
DISTANCE
1.3
1.4


dij = min  ∑ akl 
k,l∈ Pij 
1.5
2
WEIGHTED DISTANCE
2.1



1 
w
dij = min  ∑ akl  dij = min  ∑ w 
k,l∈ Pij 
k,l∈ Pij akl 
2.2
2.3
DISTANCE ∝
INVERSE OF
WEIGHTS
2.4
2.5
2.6
DISTANCE
∝
SUM OF WEIGHTS
Lectures on Complex Networks
GRAPH THEORY
1.1.3 Basic Quantities (distance)
1
1.1
1.2
Similarly to the degree, one usually plots the histogram of
the frequency density P(d) of distances d
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.1.4 Basic Quantities (Clustering)
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.1.4 Basic Quantities (Clustering)
1
1.1
1.2
1.3
Through the adjacency matrix we can write
CLUSTERING
1
Ci =
aij aik a jk
∑
ki (ki − 1) / 2 jk
1.4
1.5
2
DIRECTED (or even worse WEIGHTED) CLUSTERING
2.1
2.2
2.3
2.4
2.5
2.6
•J Saramäki et al. Phys Rev E 75 027105 (2007).
Lectures on Complex Networks
GRAPH THEORY
1.2 Trees
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.2 Trees
1
1.1
1.2
1.3
Trees are particularly important, since they appear in
•Girvan-Newman Algorithm for communities
•Taxonomy of species
•Taxonomy of information
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.2 Trees
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Trees can also be viewed as a way to filter information. In a
stock exchange the prices of all stocks are correlated, but
restricting only to a Minimum Spanning Tree is a way to
visualize the stronger correlations
In order to build a Minimum
Spanning Tree
1) Compute
the
correlation
between the N(n-1)/2 vertices
2) Define a distance out of
correlation.
3) Rank the distances
4) Draw the vertices of the
shortest distance
5) 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 vertices
have been drawn
Lectures on Complex Networks
GRAPH THEORY
1.2 Trees
1
1.1
1.2
One way to visualize the behaviour of a tree (on top of the
degree) is to check the behaviour of the basin size frequency
distribution P(n).
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.3 Vertex Correlation: Assortativity
1
1.1
1.2
1.3
1.4
A way to check the Conditioned Probability P(k|k’) that a
vertex whose degree is k is connected with another vertex
with degree k’ is given by the measure of the average
degree of the neighbours.
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.3 Vertex Correlation: Assortativity
1
1.1
1.2
1.3
1.4
1.5
2
Similarly, one can define a measure of the assortativity by
starting from the correlation function for degrees of vertices i,j
ki k j − ki k j = ∑ ki k j (P(ki , k j ) − P (ki ) P(k j ) )
ki , k j
If we introduce the variance

σ 2 = ∑ k 2 P(k ) − ∑ kP(k ) 
2.1
2.2
2.3
2.6

k

k
We have the ASSORTATIVITY COEFFICIENT r given by
2.4
2.5

2
r=
1
σ
2
∑ k k (P(k , k ) − P(k ) P(k ))
i
j
i
j
ki , k j
Lectures on Complex Networks
i
j
GRAPH THEORY
1.3 Vertex Correlation: Assortativity
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
M.E.J. Newman, Phys.Rev.Lett 89 208701 (2002)
Lectures on Complex Networks
GRAPH THEORY
1.4 Hierarchical Correlation of Graphs
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Many Complex Networks are originated or display a hierarchical
form like the one described above (self-similar structure)
Lectures on Complex Networks
GRAPH THEORY
1.4 Hierarchical Correlation of Graphs
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
A typical case has been found in Metabolic Networks, where the
various reaction can be assembled in different modules
The key quantity is the assortativity
2.6
C (k ) ∝ k
E.Ravasz, et al. Science 297 1551-1554 (2002)
A. Clauset, C. Moore, M.E.J Newman Nature 453 98-101 (2008)
Lectures on Complex Networks
−φ
GRAPH THEORY
1.5 The Properties of Complex Networks
1
1.1
1.2
1.3
•Scale Invariant Degree Distribution P(k)
•Distribution of distances P(l) peaked around small values
•Clustering (with respect to random connections)
•Assortativity
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH THEORY
1.5 The Properties of Complex Networks
1
1.1
1.2
•Scale
Invariant
Distribution P(k)
Degree
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
a)
c)
Lectures on Complex Networks
WWW
Physicists
b)
d)
Actors
Neuroscientists
GRAPH THEORY
1.5 The Properties of Complex Networks
1
1.1
•Distribution of distances p(l) peaked around small values
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
A.Vazquez, R Pastor-Satorras, A.Vespignani Phys.Rev.E 65 066130 (2002)
Lectures on Complex Networks
GRAPH THEORY
1.5 The Properties of Complex Networks
1
•Clustering
(with respect to
random
connections)
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
M.E.J. Newman, M. Girvan, Phys.Rev.E 69 026113 (2004)
Lectures on Complex Networks
GRAPH THEORY
1.5 The Properties of Complex Networks
1
•Assortativity C(k)~k-φ or average degree of neighbours <Knn>
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
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, wellconnected nodes join to a much larger number of less-well-connected nodes. This is typical of biological networks; depicted
here 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)
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.1 Introduction
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
The presence of communities in a graph is one of the most
important features.
•Communities are important for Amazon, to run their businesses
•For biologists to detect proteins with the same function
•For physicians to detect related diseases
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.2 Motifs
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Motifs are the smallest version of communities, it is currently
under debate if their presence is important or not in the area of
Complex networks.
Mangan, S. and Alon, U. PNAS 100, 11980–11985 (2003).
Mazurie et al. Genome Biology 6 R35 (2005)
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.3 Classes of Vertices
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Some communites must be defined as made by vertices with
similar properties (they could share no edge at all).
More often the communities are made by vertices sharing many
edges
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.4 Centrality Measures Betwenness and Robustness
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
The concept of centrality is at the basis of the Newman-Girvan
method for the analysis of communities.
2.3
2.4
2.5
2.6
b(i ) =
∑
j ,l =1, N
j ≠ l ≠i
P
jl
P
(i )
jl
Girvan, M. and Newman, M.E.J. PNAS, 99, 7821–7826 (2002).
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.4 Centrality Measures Betwenness and Robustness
1
Starting from a graph
We iteratively
•Compute betweenness
•Cut the edges with the
largest value of it
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
For the graph above the result is given by the series
of deletions
E-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.
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.4 Centrality Measures Betwenness and Robustness
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
Guimerà, R., Danon, L., Díaz-Guilera, A., Giralt, F., and Arenas, A..
Physical Review E, 68, 065103 (2002).
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.5 Clustering detection modularity
Several way have been proposed in order to check if a division
is good or not.
One of the quantities proposed is the modularity
The starting point is to find a suitable division of the graph
into g subgraphs. To detect whether the division is ‘good’ or
not, we define a g × g matrix E whose entries eij give the
fraction of edges that in the original graph connect subgraph i
to subgraph j.
We want the largest possible number of edges within a
community and the lowest possible number of edges between
different communities.
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.5 Clustering detection modularity
1
1.1
1.2
The actual fraction of edges in the subgraph i is given by eii;
fi =
∑
j =1 , g
e ij
This gives the probability that in a random partition
one random edge has one endvertices in i
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
In order to define a null case (random partition) we
can compare the diagonal elements of matrix E with
the quantity f
Q =
∑
e ii − f i 2
i =1, g
The modularity Q is a measure of the validity of a partition of the
graph
2.6
M.E.J. Newman, PNAS 103 8577-8582 (2006)
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.6 Communities in Graphs (spectral properties)
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.1
1
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1.2
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1.3
0
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1.4
0
0
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
0
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
 a11 a12

 a21 a22
A=
... ...

a
 n1 an 2
... a1n 

... a2n 
... ... 

... ann 
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.6 Communities in Graphs (spectral properties)
1
1.1
1.2
1.3
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-1 A
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
 k1 0 0 0 


 0 k2 0 0 
K =
... ... ... ... 


0 0 0 k 
n

2.6
Note that by definition for every node i
ki =
∑a
ij
j =1, N
Lectures on Complex Networks
GRAPH STRUCTURE: COMMUNITIES
2.6 Communities in Graphs (spectral properties)
1
1.1
1.2
1.3
1.4
1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
• Laplacian Matrix
 − ∑a1 j
a12
 j=1,N

− ∑a2 j
 a21
j =1, N
L =
...
 ...
 an1
an2





...
a2n 

...
... 
... − ∑anj 
j =1, N

...
a1n
 φ1 
 
φ2 
 ... 
 
φ 
 n
If φ’ = Lφ
φ 'i =
2
a
φ
−
k
φ
=
∇
∑ ij j i i φi
j =1, n
• Normal Matrix
a12 / k1
 0

0
 a21 / k2
N =
...
...

a / k a / k
 n1 n n2 n
... a1n / k1 

... a2n / k2 
... ... 

...
0 
Lectures on Complex Networks
The elements of matrix N
give the probability with
which one field φ passes
from a vertex i to the
neighbours.
GRAPH STRUCTURE: COMMUNITIES
2.6 Communities in Graphs (spectral properties)
1
1.1
1.2
1.3
1.4
1.5
2
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 matrix N is made of blocks and a general eigenvector will be given by
the space product of blocks eigenvectors (the constant can be different!)
2.1
2.2
2.3
2.4
2.5
2.6
Lectures on Complex Networks
Biological Networks
6
6.1
6.2
6.3
7
8
9
9.1
9.2
10
11
Summary
This third lecture wants to provide an overview of the applications in the
following fields
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
Lectures on Complex Networks
Biological Networks
6
6.1
6.1 Protein Interaction Network
Protein interact in various ways during cell life.
6.2
6.3
7
8
9
9.1
9.2
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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 Biology 4, 180 (2008)
Lectures on Complex Networks
Biological Networks
6
6.2 Metabolic Networks
6.1
Multinetwork analysis of a
carbon/light/nitrogenresponsive
metabolic
network.
6.2
6.3
7
8
9
Most of the reactions are
not reversible and the
network is oriented.
9.1
9.2
10
11
M.L. Gifford et al. Plant Physioloogy on-line Essay 12.2
H. Jeong et al. Nature 407 651 (2000).
D. Segré et al. PNAS 99 15112 (2002).
R. Gumerà et al. Nature 433, 895 (2005)
Lectures on Complex Networks
INTRODUCTION
6
6.3 Gene Regulatory Networks
6.1
A particular class of
reactions in which there
is the expression of a
gene is at the basis of
GRN.
6.2
6.3
7
8
9
9.1
9.2
10
11
J. Hasty et al. Nat. Rev. Gen., 2, 268 (2001).
T.I. Lee et al. Science 298, 799 (2002)
Lectures on Complex Networks
Geophysical Networks
6
Trees and Basin distributions
6.1
The case of river
networks presents a
scale invariance in
the basin size
distributions
6.2
6.3
7
8
9
9.1
9.2
10
11
Dodds, P.S. and Rothman, D.H. Physical Review E,
63, 016115, 016116, 016117 (2000).
Rodrıguez-Iturbe, I. and Rinaldo, A.
Fractal River Basins: Chance and Self-Organization.
Cambridge University Press, Cambridge (1996).
Lectures on Complex Networks
Ecological Networks
6
Food Webs data
6.1
Most of the
activity on
ecological
networks is
related to
the
statistical
properties
of food
webs
6.2
6.3
7
8
9
9.1
9.2
10
11
PEaCE LAB http://www.foodwebs.org
R.J. Williams N.D. Martinez Nature 404 180 (2000)
Lectures on Complex Networks
Ecological Networks
6
Statistical properties of food webs
6.1
Small-world but not
scale-free
6.2
6.3
7
8
9
9.1
9.2
10
11
J.M. Montoya, R.V. Solé, Journal of
Theoretical Biology 214 405 (2002)
D.B. Stouffer et al. Ecology, 86, 1301–1311 (2005).
Lectures on Complex Networks
Technological Networks
6
9.1 Internet
6.1
Traditionally the analysis
of the Internet structure is
made by means of
traceroutes. That is to say,
by exploring all the paths
from a given point to all
the possible destinations.
6.2
6.3
7
8
9
9.1
9.2
10
11
Internet Mapping Project http://www.cheswick.com/ches/map/
http://www.caida.org
http://www.cybergeography.org
Lectures on Complex Networks
Technological Networks
6
9.1 Internet
6.1
6.2
6.3
7
8
9
9.1
9.2
10
11
A. Vazquez, R. Pastor-Satorras and A.
Vespignani PRE 65 066130 (2002)
R. Pastor-Satorras and A. Vespignani
Evolution and Structure of
Internet: A Statistical Physics Approach.
Cambridge University Press (2004)
Lectures on Complex Networks
Technological Networks
6
9.2 World Wide Web
6.1
WWW is probably the
largest network
available of the order
of billions of elements.
Different centrality
properties arise in such
structure.
6.2
6.3
7
8
9
9.1
9.2
10
11
A. Broder et al. Computer Network, 33, 309 (2000).
D. Donato et al. Journal of Phys A, 41, 224017 (2008).
Lectures on Complex Networks
Social Networks
6
Wikipedia
6.1
6.2
6.3
7
8
9
9.1
9.2
10
11
Lectures on Complex Networks
Social Networks
6
Wikipedia
The degree shows fat tails
that can be approximated
by a power-law function of
the kind P(k) ~ k-γ
Where the exponent is
the same both for
in-degree and out-degree.
6.1
6.2
6.3
7
8
9
9.1
9.2
10
11
in–degree(empty) and out–degree(filled)
Occurrency distributions for the Wikgraph
in English (o) and Portuguese (∆).
V. Zlatic et al. Physical Review E, 74, 016115 (2006).
A. Capocci et al. Physical Review E, 74, 036116 (2006).
Lectures on Complex Networks
Financial Networks
6
Stock Correlation Network
6.1
6.2
6.3
7
8
9
9.1
9.2
10
11
Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-year
period 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 manufacturing
green for transportation,,
light blue for public
electric,gas and sanitary services
administration
magenta wholesale trade
black for retail trade
purple for finance and insurance
orange for service industries
Bonanno et al. Physical Review E 68
046130 (2003).
Lectures on Complex Networks
Financial Networks
6
Stock Ownership Network
6.1
6.2
6.3
7
8
9
9.1
9.2
10
11
Garlaschelli et al. Physica A, 350 491 (2005).
Lectures on Complex Networks
Financial Networks
6
Trading Webs
6.1
6.2
6.3
7
8
9
9.1
9.2
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11
M. Angeles Serrano, M. Boguna Phys Rev. E 68 015101 (2003)
D. Garlaschelli et al. Physica A, 350 491 (2005).
C.A. Hidalgo, et al. Science 317 482 (2007)
Lectures on Complex Networks
Financial Networks
6
6.1
6.2
6.3
7
8
9
Bank’s Credit Network
Banks exchange money
overnightly, in order to
meet the customer needs
of liquidity as well as
ECB requirements
9.1
9.2
10
11
The network shows a rather peculiar architecture
The banks form a disassortative network where large
banks interact mostly with small ones.
G. De Masi et al. PRE, 98 208701 (2007)
Lectures on Complex Networks