„small-world” networks

Transcription

International Workshop EDEN V –Exploratory Domains of Econophysics. News - UPitESW
2013 June 29th, 2013
UNIVERSITY OF PITESTI ECONOPHYSICS AND SOCIOPHYSICS WORKSHOP - UPitESW
FROM THE „SIX DEGREES OF
SEPARATION” TO THE WEIGHTED
„SMALL-WORLD” NETWORKS
Mircea Gligor
National College Roman Voda Roman
The Stanley Milgram’s experiment (1967)
The letters path: Nebraska-Boston
Criteria: name, residence, field of
activity
S. Milgram, University of Yale
The letters path: Nebraska-Boston
Criteria: name, residence, field of
activity
CONCLUSION
• A reasonable number of
letters did reach their
destination
• The letters covered an
average of six steps to get
from Nebraska to Boston.
• Two randomly chosen
individuals can be connected
by only a short chain of
intermediate acquaintances
(„small world effect”)
Milgram, S. The small world
problem, Psychology Today,
2, 60-67 (1967)
„The six degrees of separation” in games
and talk-show industry
During an interview for Premiere in jan. 1994 about
the movie Wild River, the actor Kevin Bacon claimed
that he knows everybody at Hollywood or somebody
who worked with him/her. In april 1994 appeared on
a blog a new game: "Kevin Bacon is the Centre of
Universe"
•
•
•
•
•
•
Jaye Davidson was in THE CRYING
GAME with
Stephen Rea who was in ANGIE with
Geena Davis who was in A LEAGUE
OF THEIR OWN with
Rosie O'Donnell who will be in THE
FLINTSTONES with
Elizabeth Perkins who was in HE
SAID, SHE SAID
with Kevin Bacon
Kevin Bacon
Universe"
•
•
•
•
•
Stockard Channing was in GREASE
with John Travolta who was in LOOK
WHO"S TALKING
with Bruce Willis who was in THE
PLAYER
with Julia Roberts who was in
FLATLINERS
with Kevin Bacon.
Kevin Bacon
Universe"
•
•
•
•
•
•
Omar Sharif who was in FUNNY GIRL
with Barbra Streisand who was in
NUTS
with Richard Dreyfuss who was in
MOON OVER PARADOR
with Sonia Braga who was in THE
MILAGRO BEANFIELD WAR
with Daniel Stern who was in DINER
with Kevin Bacon.
Kevin Bacon
2013 June 29th, 2013
Passing over the fashionable account of the
phenomenon, we point out that the problem is
crucially important for communications. Most
human interactions take place directly between
individuals. The spread of news, rumours and
fashions, as well as various diseases (from simple
flues to the HIV virus) can spread far faster in a small
world network than in a network where the average
degree of separation is, say, 10.000
2013 June 29th, 2013
The Watts-Strogats model
Duncan Watts, Univ. Colombia
Steven Strogatz, Univ. Cornell
2013 June 29th, 2013
The terminology of the complex networks
• NETWORK = NODES (VERTICES) + EDGES (LINKS)
The edges are usually considered of unit length.
• The vertex DEGREE = the number of the vertex links
<k> = the average degree of the network
• MINIMAL PATH LENGTH = the length of the shortest
trajectory joining two vertices.
<ℓ> = the average minimal path length of the network
• CLUSTERING COEFFICIENT Ci = number of actual links /
number of the first-order neighbors
<C> = the average clustering coefficient
2013 June 29th, 2013
OPORTUNITIES:
The development of
the techniques for
numerical simulations
(1970-1985)
The possibility of
testing the results on
new networks www.
and Internet.
2013 June 29th, 2013
REWIRING LINKS
 We consider firstly a 1-dimensional network with periodical
boundary conditions.
 One takes all the edges of the lattice in turn and with
probability p rewires to randomly chosen vertices. In such a
way, a number of far connections appears.
2013 June 29th, 2013
The transition from a regular lattice to a SW network
< l >(p)/l(0)
C(p)/C(0)
-2
-1.5
1.2
1
0.8
0.6
0.4
0.2
0
-4
-3.5
-3
-2.5
lg(p)
-1
-0.5
0
2013 June 29th, 2013
The spread of disease in regular and “small world” networks
1
0.9
0.8
0.7
ni/L
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
t
6
7
8
9
10
2013 June 29th, 2013
The small-world network of
pupils in school
2013 June 29th, 2013
network
N
ℓ
C
Crandom
pupils
40
2.7
0.21
0.05
2013 June 29th, 2013
Simulation of the pupils small-world network by Watts-Strogatz method
network
N
ℓ
C
p
pupils
40
2.7
0.21
0.33
2013 June 29th, 2013
The overlapping index:
 N

N
N


1


Oij 
(
w
w
)
w
w
 il jl  

ip
jq 
2( N  1)( N  2) l 1
p 1
q 1


l i , j
q j
 p i

ij
N
Oi   Oij
j 1
v01
v02
v03
v04
v05
v06
v07
v08
v09
v10
v11
v12
v13
v14
12.56
12.23
11.97
11.79
10.56
9.55
9.47
9.42
9.34
9.23
9.15
9.10
8.94
8.76
v15
v16
v17
v18
v19
v20
v21
v22
v23
v24
v25
v26
v27
v28
8.71
8.65
8.65
8.60
8.55
8.52
8.52
8.47
8.34
8.23
8.13
8.10
8.10
8.07
v29
v30
v31
v32
v33
v34
v35
v36
v37
v38
v39
v40
8.07
8.05
7.94
7.84
7.73
7.73
7.59
7.42
7.11
6.79
6.79
6.29
2013 June 29th, 2013
Phase transitions in small-world
networks during their deconstruction
2013 June 29th, 2013
The statistical physics of weighted networks







For the weighted small-world network we can define:
The partition function;
The entropy;
The free energy;
The average energy / link
The “thermal” susceptibility
Some of these physical quantities are expected to exhibit
critical points during a thinking process of network
deconstruction by removing vertices. The changes of the
structure may be studied by the phase transition formalism.
2013 June 29th, 2013
Small world network deconstruction
1.E+00
Removing the weak connected first
Heat Capacity
1.E-01
1.E-02
1.E-03
1.E-01
1 / Beta
1.E+00
2013 June 29th, 2013
0.4
Removing the weak connected first
0.3
0.2
VL
0.1
0.0
-0.1
-0.2
-0.3
0.15
0.20
0.25
0.30
1 / Beta
0.35
0.40
0.45
2013 June 29th, 2013
1.E+00
Removing the strong connected first
Heat Capacity
1.E-01
1.E-02
1.E-03
0.10
1.00
1 / Beta
2013 June 29th, 2013
0.2
Removing the strong connected first
0.2
0.1
0.1
0.0
VL
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
0.15
0.20
0.25
0.30
1 / Beta
0.35
0.40
0.45
2013 June 29th, 2013
Conclusion
 As long as the most human interactions take place directly
between individuals, we may expect that a large number of social
networks to display the small-world effect.
 The pupils network in school may be easily simulated by
Watts-Strogatz method of links rewiring
 One alternative to the Watts-Strogatz method is to explain the
small-world effect by existence of few nodes in the network which
have unusually high coordination numbers or which are linked to
a widely distributed set of neighbours.
 In the network there are several nodes whose removal leads to
qualitative changes in the network structure. On this way we may
conclude that, to a large extent, the small-world effect is due to a
few people who are particularly well connected.
2013 June 29th, 2013
How small is the world!
Thank you!

„small-world” networks

Transcription

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