Power-law degree distribution
in a coevolving network of social
interactions
Tomasz Raducha, Tomasz Gubiec
Faculty of Physics University of Warsaw
[email protected]
Physica A 471 2017
Econophysics Workshop 07.03.2017 Leicester
Modeling social interactions
• R. Axelrod The Dissemination of Culture
• Individuals located on a static square lattice
• Agents can interact becoming more similar
• Final state not always homogeneous
Solomon Islands
over 100 separate languages
# of languages ∼ size of an island
Coevolutionary model
• Network with N nodes and average degree hki
• Every node has F traits
• Every trait can adopt one of q values
5, 4, 2, 2, 8
4, 3, 4, 9, 2
3, 6, 2, 1, 6
3, 7, 2, 9, 2
F = 2, q = 3
1, 2
2, 2
1, 3
2, 3
3, 1
3, 3
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 3
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 3
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 3
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 3
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 3
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
1, 2
2, 2
1, 3
2, 3
3, 1
3, 1
3, 2
1, 1
How to draw a new neighbour?
A. PA (i) ∼ ki
B. PB (i) ∼ ki + 1
C. PC (i) ∼ (ki + 1)2
D. switching to nodes distant by two edges
Phase Diagram
0.8
clustering coefficient
largest component and domain
1.0
0.6
0.4
0.2
0.0 0
10
component
domain
101
102
q
103
104
Phase Diagram
0.8
clustering coefficient
largest component and domain
1.0
0.6
0.4
0.2
0.0 0
10
model A
model B
101
102
q
103
104
Phase Diagram
0.8
clustering coefficient
largest component and domain
1.0
0.6
0.4
0.2
0.0 0
10
model C
model B
101
102
q
103
104
Phase Diagram
0.8
clustering coefficient
largest component and domain
1.0
0.6
0.4
0.2
0.0 0
10
model D
model B
101
102
q
103
104
Phase Diagram
0.8
clustering coefficient
largest component and domain
1.0
0.6
0.4
0.2
0.0 0
10
local c. c.
global c. c.
component
domain
101
102
q
103
104
First transition point
100
probability
10-1
slope = -1.14
10-2
10-3
10-4
100
101
102
size of a component
103
Model A (ki )
Model B (ki + 1)
Model C (ki + 1)2, Phase I
10-1
probability
slope = -3.3
10-3
10
N = 500
N = 1000
N = 2000
-5
100
101
degree
102
103
Model C (ki + 1)2, Phase II
probability
10-1
slope = -1.96
10-3
10
N = 500
N = 1000
N = 2000
-5
100
101
degree
102
Model C (ki + 1)2, Phase III
probability
10-1
slope = -1.86
10-3
10
N = 500
N = 1000
N = 2000
-5
100
101
degree
102
Model D (local switching) - scaling
1.0
N = 500
N = 1000
N = 1500
N = 2000
N = 4000
largest component
0.8
0.6
0.4
0.2
0.0 0
10
101
102
q
103
104
Model D (local switching) - # of domains
500
number of domains
400
model D, q = 50
model D, q = 100
random, q = 50
random, q = 100
300
200
100
0
200
400
N
600
800
1000
Summary
• High value of the clustering coefficient
• Small-world effect (except model D)
• Various degree distributions including power law
• Different levels of recombination
• Consistent with Solomon Islands case
Thank you for your attention
aaa
www.ec2017.org
References
• Terrell (1977) Fieldiana. Anthropology 68
• Axelrod (1997) J. Conflict Res. 41
• Castellano, Marsili, Vespignani (2000) Phys. Rev. Lett. 85
• Vazquez, González-Avella, Eguı́luz, San Miguel (2007) Phys. Rev. E 76
• Raducha, Gubiec (2017) Physica A 471
Summary
• High value of the clustering coefficient
• Small-world effect (except model D)
• Various degree distributions including power law
• Different levels of recombination
• Consistent with Solomon Islands case
Thank you for your attention
aaa
www.ec2017.org
References
• Terrell (1977) Fieldiana. Anthropology 68
• Axelrod (1997) J. Conflict Res. 41
• Castellano, Marsili, Vespignani (2000) Phys. Rev. Lett. 85
• Vazquez, González-Avella, Eguı́luz, San Miguel (2007) Phys. Rev. E 76
• Raducha, Gubiec (2017) Physica A 471
largest component and domain
1.0
component
domain
local c. c.
global c. c.
0.8
0.6
0.4
0.2
0.0 0
10
101
102
q
103
clustering coefficient
Model A
104
Model B
0.8
clustering coefficient
largest component and domain
1.0
0.6
0.4
0.2
0.0 0
10
local c. c.
global c. c.
component
domain
101
102
q
103
104
largest component and domain
1.0
component
domain
local c. c.
global c. c.
0.8
0.6
0.4
0.2
0.0 0
10
101
102
q
103
clustering coefficient
Model C
104
largest component and domain
1.0
component
domain
local c. c.
global c. c.
0.8
0.6
0.4
0.2
0.0 0
10
101
102
q
103
clustering coefficient
Model D
104
Model A
1.0
N = 500
N = 1000
N = 1500
N = 2000
N = 3000
largest component
0.8
0.6
0.4
0.2
0.0 0
10
101
102
q
103
104
Model B
1.0
largest component
0.8
0.6
0.4
0.2
0.0 0
10
N = 500
N = 1000
N = 1500
N = 2000
N = 3000
101
102
q
103
104
Model D
1.0
N = 500
N = 1000
N = 1500
N = 2000
N = 4000
largest component
0.8
0.6
0.4
0.2
0.0 0
10
101
102
q
103
104
Model A, Phase I
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
5
10
15
20
degree
25
30
35
Model A, Phase II
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
10
20
30
40
degree
50
60
70
Model A, Phase III
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
10
20
30
40
degree
50
60
70
Model B, Phase I
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
5
10
15
degree
20
25
30
Model B, Phase II
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
10
20
degree
30
40
50
Model B, Phase III
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
5
10
15
20
degree
25
30
35
40
Model C, Phase I
10-1
probability
slope = -3.3
10-3
10
N = 500
N = 1000
N = 2000
-5
100
101
degree
102
103
Model C, Phase II
probability
10-1
slope = -1.96
10-3
10
N = 500
N = 1000
N = 2000
-5
100
101
degree
102
Model C, Phase III
probability
10-1
slope = -1.86
10-3
10
N = 500
N = 1000
N = 2000
-5
100
101
degree
102
Model D, Phase I
0.25
N = 500
N = 1000
N = 2000
probability
0.20
0.15
0.10
0.05
0.000
5
10
degree
15
20
25
Model D, Phase II
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
5
10
15
20
degree
25
30
35
Model D, Phase III
probability
10-1
10-3
10
N = 500
N = 1000
N = 2000
-5
0
5
10
15
20
degree
25
30
35
40
Model A
Model B
Model C
Model D
Phase I
A. Poisson / Exponential
B. Poisson / Exponential
C. Power law
D. Poisson
Phase II
A. Exponential
B. Exponential
C. Power law
D. Unclassified
Phase III
A. Poisson
B. Exponential
C. Power law
D. Poisson