Transcript Diapositiva 1

DYNAMICS OF COMPLEX SYSTEMS
Self-similar phenomena and Networks
Guido Caldarelli
CNR-INFM Istituto dei Sistemi Complessi
[email protected]
2/6
•STRUCTURE OF THE COURSE
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.
BIOLOGICAL NETWORKS
•STRUCTURE OF THE LECTURE
2.1 FREQUENCY DISTRIBUTIONS
2.2 ASSORTATIVITY
2.3 BETWEENNESS
2.4
COMMUNITIES (GIRVAN AND NEWMAN)
2.5
COMMUNITIES (SPECTRAL ANALYSIS)
2.6
•2.1 BASIC OF GRAPH THEORY
·1 Degree frequency density P(k) = how many times you find a vertex with degree k
P(k )  e
P(k)
 pN
( pN ) k
k!
P(k )  k 
k
·2 Degree Corr. Knn (k) = average degree of a neighbour of a vertex with degree k
·3 Clustering Coefficient (k) = the average value of c for a vertex whose degree is k
•2.2 BASIC OF GRAPH THEORY
Assortative networks
Disassortative networks
Techological,
Biological networks
• Real networks always display one of these two tendencies,
Social networks
• “similar” networks display “similar” behaviours.
Assortativity coefficient
r  ij  i j
>0
:
Assortative
=0
:
Non assortative
<0
:
Disassortative
•2.2 BASIC OF GRAPH THEORY
Consequences of assortativity: - Resistence to attacks
- Percolation
- Epidemic spreading
M.E.J. Newman, Physical Review E, 67 026126 , (2003).
•2.3 BASIC OF GRAPH THEORY
·4 Centrality betweenness b(k) = The probability that a vertex whose degree
is
k has betweenness b
betweenness of I is the number of distances
between any pair of vertices passing
through I
·5 TREES ONLY!!! P(A) = Probability Density for subbranches of size A
11
1 1
Size distribution:
Allometric relations:
35
33
11
35
1 1 0,6 0,5 P(A
C(A
0,5
11 5
2
3
30
)
0,4
)
25
22
20
0,3
15
1 1
22 8
0,2
0,1
0,1
0,1
0,1
0,1
A
0
10
33
11
10
0,1
1
2
3
4
5
6
7
8
9
10
5
5
A
3
1
0
0
2
4
6
8
10
12
•2.4 COMMUNITIES: GN Algorithm
•2.4 COMMUNITIES: GN Algorithm
•2.4 COMMUNITIES: GN Algorithm
Network of e-mails in the University of Tarragona (Spain).
On the left the total network, different vertices represent persons and the colour
of vertices the various Department.
On the right the tree of communities.
•2,5 COMMUNITIES: Spectral Analysis
 a11 a12

a22
a
A   21
... ...

a
 n1 an 2
•
•
... a1n 

... a2 n 
... ... 

... ann 
Undirected graphs →
aii = 0
aij= aji
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
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
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
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
•COMMUNITIES: Spectral Analysis
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
 k1 0

 0 k2
K 
... ...

0 0

0

0 0
... ... 

0 k n 
0
Note that by definition for every node i
ki 
a
j 1, N
ij
•2.5 COMMUNITIES: Spectral Analysis
   a1 j
a12
 j 1, N

  a2 j
 a21
j 1, N
L
...
 ...
 an1
an 2


a12 / k1
 0

0
 a21 / k 2
N 
...
...

a / k a / k
 n1 n n 2 n



... a2 n 

...
... 
...   anj 
j 1, N

...
a1n
... a1n / k1 

... a2 n / k 2 
...
... 

...
0 
 f1 
 
 f2 
 ... 
 
f 
 n
If f’ = Lf
f 'i 
2
a
f

k
f


 ij j i i fi
j 1, n
• Laplacian Matrix
The elements of matrix N give
the probability with which one
field f passes from a vertex i to
the neighbours.
• Normal Matrix
•2.5 COMMUNITIES: Spectral Analysis
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!)
•2.5 COMMUNITIES: Spectral Analysis
In general, the probability to pass from one vertex to a neighbour
depend upon the nature of the edge.
In the case of Internet different cables have different capacity,
speed or cost.
In social networks, the edge has the strength of the connection.
Therefore it is customary to generalize this approach to a case
where instead of the aij we have real numbers (weights) wij.
kiw 
 0

 w21
w
A 
...

w
 n1
w12
...
0
...
...
...
wn 2
...
w1n 

w2 n 
... 

0 
w
j 1, N
ij
 k1w

0
Kw 
 ...
0

0
k 2w
0
0
...
0
...
0
0

0
... 
k nw 
•2.5 COMMUNITIES: Lagrange Multipliers
To formalize the role between eigenvectors and communities
we express the eigen-problem as a research of a minimum under constraint
This is a three step procedure
1. Define a ficticious quantity x for the sites of the graph
2. Define a suitable function z on these x’s (a “distance”)
3. Define a suitable constraint on these x’s (to avoid having all equal or all 0)
For example
z ( x) 
2
(
x

x
)
 i j wij
i , j 1, N
where the xi are values assigned to the nodes, with the constraint expressed by
x x m
i , j 1, N
i
j
ij
1
(A)
Looking for stationary points of z(x) + constraint (A) →
Lagrange multiplier
•2.5 COMMUNITIES: Lagrange Multipliers
 z ( x)   ( xi  x j ) 2 wij

i , j 1, N

xi x j mij  1


i , j 1, N

xi
 wij 
j 1, N
 z ( x)


i 

xi
 xi
 wij x j 
j 1, N


( K  A ) x  Mx
w
w

2
x m
j 1, N
j
ij

xi x j mij   0

i , j 1, N

0
(   / 2)
•2.5 COMMUNITIES: Lagrange Multipliers

M  K w  K w 1 Aw x
 (1  2 ) x



 M  1  ( K w  Aw ) x  x
M

M
 Kw 
 1 
Lagrange Multiplier = Normal Eigenvalue problem
Lagrange Multiplier = Laplacian Eigenvalue problem
•2.6 COMMUNITIES: Application
In order to test if the model works properly we choose a suitable network
Where communities can be easily spotted.
The data are collected through a psychological experiment:
Persons (about 100) are given as a stimulus a single word i.e. “House”
They must answer with the first word that comes on their mind i.e.“Family”.
Answer are given as new stimula, so that a network of average associations forms.
Dog
Road
Car
Job
House
Mortgage
Family
•2.6 COMMUNITIES: Application
Therefore we expect similar words to be on the same plateau.
We can measure the correlation between the values of various vertices averaged
over 10 different eigenvectors.
science
1
literature
1
piano
1
scientific
0.994
dictionary
0.994
cello
0.993
chemistry
0.990
editorial
0.990
fiddle
0.992
physics
0.988
synopsis
0.988
viola
0.990
concentrat
e
0.973
words
0.987
banjo
0.988
thinking
0.973
grammar
0.986
saxophone
0.985
test
0.973
adjective
0.983
director
0.984
lab
0.969
chapter
0.982
violin
0.983
brain
0.965
prose
0.979
clarinet
0.983
equation
0.963
topic
0.976
oboe
0.983
examine
0.962
English
0.975
theater
0.982
•2.7 DATA
With Graph Theory we can describe and analyse a series of different systems
FINANCIAL SYSTEMS
Portfolio
Board of Directors
Price correlations
SOCIAL INTERACTIONS
Actors, Scientists
Sex
TECHNOLOGICAL NETWORKS
WWW, Internet
e-mail
LINGUISTIC NETWORKS
Syntactic Networks
Word
Associazioni di parole.
BIOLOGICAL NETWORKS
Protein and Metabolic networks
Food Webs
Taxonomies
•2.7 DATA: PORTFOLIOS
Vertices = Companies
Edges = ownership
•2.7 DATA: BOARD OF DIRECTORS
Vertices = companies
Edges = to be in the same
board
Vertices = Boards
Edges = share a Director
•2.7 DATA: STOCK CORRELATIONS
Vertices = Companies
Edges = High correlation in
stock prices
•2.7 DATA: SOCIAL NETWORKS
MOVIES
Days of Thunder (1990)
Far and Away (1992)
Eyes Wide Shut (1999)
CITATIONS
SEXUAL NETWORKS
•2.7 DATA: LINGUISTIC NETWORKS
ΜΗΝΙΝ ΑΕΙΔΕ ΘΕΑ ΠΗΛΙΑΔΕΩ ΑΧΙΛΛΗΟΣ,
ΟΥΛΟΜΕΝΗΝ Ή ΜΥΡΙΑ ΑΧΑΙΟΙΣ ΑΛΓΕ' ΕΘΗΚΕ,
ΠΟΛΛΑΣ Δ' ΙΦΘΙΜΟΥΣ ΨΥΧΑΣ ΑΙΔΙ ΠΡΟΙΑΨΕΝ ΗΡΩΩΝ,
ΑΥΤΟΥΣ ΔΕ ΕΛΩΡΙΑ ΤΕΥΧΕ ΚΥΝΕΣΙΝ ΟΙΩΝΟΙΣΙ ΤΕ ΠΑΣΙ.
ΔΙΟΣ ΔΕ ΕΤΕΛΕΙΕΤΟ ΒΟΥΛΗ.
ΕΞ ΟΥ ΔΗ ΤΑ ΠΡΩΤΑ ΔΙΑΣΤΗΤΗΝ ΕΡΙΣΑΝΤΕ
ΑΤΡΕΙΔΗΣ ΤΕ ΑΝΑΞ ΑΝΔΡΩΝ ΚΑΙ ΔΙΟΣ ΑΧΙΛΛΕΥΣ
......
•2.7 DATA: TECHNOLOGICAL NETWORKS
•2.7 DATA: FOOD WEBS
•2.7 DATA: TAXONOMIES
phylum
subphylum
class
subclass
order
family
genus
species
•2.7 DATA: PROTEIN INTERACTION NETWORKS
•2.7 DATA: PROTEIN INTERACTION NETWORKS
•2.7 DATA: NEURAL NETWORKS
•2.7 DATA: NEURAL NETWORKS
Example of a network
extracted from a correlation
matrix of brain activity
Above (1/8 of the total.
Colour corresponds to degree
yellow = 1, green = 2, red =
3, blue = 4, violet = 5).
The plot of the degree
distribution for three different
values of threshold.
•2.8 EPIDEMICS
EPIDEMICS IS A
FUNDAMENTAL
APPLICATION
The web of Human sexual contacts
[Lilijeros et al., Nature (2001)]
•2.8 EPIDEMICS
Individuals are present in different stages
• HEALTY
• INFECT
• IMMUNE
• DEAD..
Again the network is the most
immediate representation
•Every node is an individual
•Every edge is a connection
•2.8 EPIDEMICS
r
Absorbing phase
Dead of
virus
Active phase
Infection
c
In any model
WE HAVE A CRITICAL
THRESHOLD

•2.8 EPIDEMICS
Actually all the models assume a regular grid
IF THE NETWORK IS SELFSIMILAR
• There is no activation!!!
• Mass vaccination is useless
WE MUST ACT ON HUBS!!!