The Dynamics of Cooperation
in Small World Networks
Dan FitzGerald
Dr. Gregg Hartvigsen
SUNY Geneseo Biomath Initiative
The Dynamics of Cooperation in Small World Networks
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Introduction
So, lets say that I have a piano…
What is a Small World Network?
Results
Acknowledgements
1
So, lets say that I have a piano…
(How the model works)
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Introduction
So, lets say that I have a piano…
What is a Small World Network?
Results
Acknowledgements
Our Simulated “Individual”
p = 0.500
k = 2 neighbors
p value; 50% chance
of cooperating with
another individual if
asked
2
Rules of Engagement
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Model runs in discrete time
For each time step, we visit every
node on the network
Rules of Engagement
Choose individual
If rnd <= pi
If rnd > pi
Choose player from neighborhood
If rnd <= pk
If rnd > pk
C-C
C-D
3
Rules of Engagement
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After an interaction, an individual’s p
will go either up or down depending
on the whether their neighbour
cooperated with them or not.
This change will be ±ε, where ε is a
user-defined variable into the model
What is a Small World Network?
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Introduction
So, lets say that I have a piano…
What is a Small World Network?
Results
Acknowledgements
4
What is a Small World Network?
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Think of this as “Six Degrees of Kevin
Bacon”; strangers are all connected
through mutual acquaintances
swnP – “Small World Network p”; the
probability that an edge from any
given vertex in a circulant will be
rewired
What is a Small World Network?
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L(p) – “Characteristic path length”; the
average of the average path length of
every vertex in the network. This is
small on a SWN
C(p) – “Clustering coefficient”; the
proportion of the host neighbors that
are connected to each other relative to
the number of possible connections.
This is high on a SWN.
5
SWN-P: 0.00
n Vertices: 10
k Neighbors: 4
SWN-P: 0.10
n Vertices: 10
k Neighbors: 4
6
SWN-P: 1.00
n Vertices: 10
k Neighbors: 4
Results
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„
Introduction
So, lets say that I have a piano…
What is a Small World Network?
Results
Acknowledgements
7
Results
On a lattice, a single defector spreads; simulations go either to global
cooperation or defection based on initial p
Results
On the SWN, there exists a small area of parameter space where simulations
never go to either cooperation or defection. This is around initial p = 0.50.
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Results
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All simulations had…
1.
2.
3.
4.
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k=8
constant initial p distribution
10000 nodes in community
Epsilon of 0.01
Simulations without mutation had…
1. Initial p of 0.500
2. SWNSWN-P values of 0.0, 0.01, 0.05, 0.10, 0.25,
0.50, 1.00
3. Five replicate simulations
Results
So what exactly are “components”?
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Components are the groups formed by the
presence of one or more of the same type
of node (e.g. cooperator or defector)
Larger components (size > 1) are
structures of linked individuals of the same
type. For example, a series of defecting
individuals that are all connected to each
other.
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Results
The number of defector components steadily decrease until SWN-P = 0.25,
after which they experience a sharp increase
Main Effects Plot (fitted means) for Defector Components
180
Mean of Defector Components
160
140
120
100
80
60
40
20
0
0.00
0.01
0.05
0.10
SWN-P
0.25
0.50
1.00
Results
On a circulant, the distribution of component sizes covers a diverse area;
there are many components of varying sizes
Degree Distribuiton for SWN-P = 0.0
25
Frequency
20
15
10
5
0
0.3
0.6
0.9
1.2
1.5
Size of Component (log)
1.8
SWN-P = 0.00
Mean k = 8
n = 10000 vertices
10
Results
When SWN-P = 0.01, the distribution of component sizes is more limited;
medium-sized components have merged into a single large component
Degree Distribuiton for SWN-P = 0.01
25
Frequency
20
15
10
5
0
0.5
1.0
1.5
2.0
2.5
Size of Component (log)
3.0
3.5
SWN-P = 0.01
Mean k = 8
n = 10000 vertices
Results
When SWN-P = 0.05, the distribution of component sizes is even more
extreme; the large component is a few thousand nodes large
Degree Distribution for SWN-P = 0.05
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Frequency
4
3
2
1
0
0.0
0.5
1.0
1.5
2.0
2.5
Size of Component (log)
3.0
3.5
SWN-P = 0.05
Mean k = 8
n = 10000 vertices
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Results
At SWN-P = 0.50, most defecting individuals have less than half of their
neighbors as other defectors
Distribution of the Percentage of Defecting Neighbors
SWN-P = 0.5
A nderson-D arling Normality Test
0.0
12.5
25.0
37.5
50.0
62.5
75.0
87.5
A -S quared
P -V alue <
8.21
0.005
M ean
S tD ev
V ariance
S kew ness
Kurtosis
N
37.373
21.348
455.744
-0.073985
-0.757065
1192
M inimum
1st Q uartile
M edian
3rd Q uartile
M aximum
0.000
20.000
37.500
54.371
87.500
95% C onfidence Interv al for M ean
36.159
38.586
95% C onfidence Interv al for M edian
37.500
9 5 % C onfidence Inter vals
40.000
95% C onfidence Interv al for S tD ev
20.524
Mean
22.241
Median
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37
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39
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Results
On a random network, most defecting individuals are connected to no other
defecting individuals. They are surrounded by cooperators.
Distribution of the Percentage of Defecting Neighbors
SWN-P = 1.0
A nderson-D arling N ormality Test
0.0
12.5
25.0
37.5
50.0
62.5
75.0
87.5
A -S quared
P -V alue <
16.87
0.005
M ean
S tDev
V ariance
S kew ness
Kurtosis
N
1.8747
4.7835
22.8818
2.26464
3.47530
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M inimum
1st Q uartile
M edian
3rd Q uartile
M aximum
0.0000
0.0000
0.0000
0.0000
16.6667
95% C onfidence Interv al for M ean
0.6169
3.1324
95% C onfidence Interv al for M edian
0.0000
9 5 % C onfidence Inter vals
0.0000
95% C onfidence Interv al for S tD ev
4.0440
Mean
5.8565
Median
0.0
0.5
1.0
1.5
2.0
2.5
3.0
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Results
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Simulations with mutation had…
1. Initial p of 0.500
2. SWN-P values of 0.0, 0.01, 0.05, 0.10,
0.25, 0.50, 1.00
3. Five replicate simulations
4. Mutation occurs with a probability of
1/100
5. In a mutation, an individual’s p can
change by ±0.001.
Results
Mutation has an effect on the number of defector components; many of the
isolated defector components are pulled to cooperation
Interaction Plot (fitted means) for Number of Defector Components
180
Mutation?
0
1
160
140
Mean
120
100
80
60
40
20
0
0.00
0.01
0.05
0.10
SWN-P
0.25
0.50
1.00
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Conclusions
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There are four conclusions from my
research to-date.
Let’s briefly summarize them…
Conclusions
1. Cooperation is more likely on more random networks.
Effect of SWN-P on Final Average p
1.0
0.9
Mean of Average p
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0.8
0.7
0.6
0.5
0.00
0.01
0.05
0.10
SWN-P
0.25
0.50
1.00
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Conclusions
2. In simulations with initial p near 0.5, both cooperators and
defectors coexist
Conclusions
3. Increasing SWNSWN-P, we find that our components merge into
two large components of cooperators and defectors
SWN-P = 0.0
SWN-P = 0.01
SWN-P = 0.05
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Conclusions
4. In random networks, there is a negligible amount of defectors
in the system
Boxplot of Number of Defectors vs SWN-P
5000
Number of Defectors
4000
3000
2000
1000
0
0.00
0.01
0.05
0.10
SWN-P
0.25
0.50
1.00
Acknowledgements
„
„
„
„
„
Introduction
So, lets say that I have a piano…
What is a Small World Network?
Results
Acknowledgements
16
Muchas Gracias
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Dr. Hartvigsen
SUNY Geneseo Biomath Program and fellow
biomath researchers
Professor Homa Farian and Matt Haas, of
the SUNY Geneseo Distributed Lab
Rob, for telling me to go into research
Stacy, for saving me from pointer-error
purgatory
Chris, just because
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