Complex systems 3
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Transcript of Complex systems 3
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Complex Systems
Session 3.
Self-organization and criticality
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The Matthew Effect
Robert K. Merton (1968)
Herbert A. Simon (1955) Nobel M. Prize (1978)
Udny Yule (1925)
Yule-Simon process and distribution
SzEEDSM Complex Systems 2017
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Yule-Simon
SzEEDSM Complex Systems 2017
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Preferential attachment
SzEEDSM Complex Systems 2017
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Scaling in networks
SzEEDSM Complex Systems 2017
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Complex Networks
SzEEDSM Complex Systems 2017
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Degree distribution of Internet
routers
SzEEDSM Complex Systems 2017
α= 1.1
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Degree distribution of WWW pages
α= 1.1
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Degree distribution in Social
Networks
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Gibrat’s law
Robert Gibrat (1931)
Growth rate (x=company- asset- or city size)
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Growth is independent of size
Under this assumption, size distribution is power law again
with α approximately 1.
SzEEDSM Complex Systems 2017
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Firm size distribution
SzEEDSM Complex Systems 2017
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Sales &
Profit
SzEEDSM Complex Systems 2017
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The Black Swan
SzEEDSM Complex Systems 2017 Nassim Taleb
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Fat Tail
SzEEDSM Complex Systems 2017
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Earthquake size distribution
SzEEDSM Complex Systems 2017
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Phase transitions
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Ferromagnetism and the Ising
model
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Cluster size distribution
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Mean field approximation
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Mean field diagram
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Bistability and bifurcation
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Percolation
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SzEEDSM Complex Systems 2017
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Epidemic processes
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SIS model
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Traffic congestion
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Inverse U curve
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Social collapse
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Collective intelligence
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Power outages
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10 4 10 5 10 6 10 710
-2
10-1
100
101
N= # of customers affected by outage
Frequency
(per year) of
outages > N
1984-1997
August 10, 1996
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Square
site
percolation
or
simplified
“forest
fire”model.
The simplest possible toy model of cascading
failure.
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connected
not
connected
Connected clusters
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A “spark” that hits
a cluster causes
loss of that
cluster.
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yield
=
density
- loss
Assume: one randomly
located spark
(average)
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yield
=
density
- loss
Think of (toy) forest fires.
(average)
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(avg.)
yield
density
“critical point”
N=100
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Critical point
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criticality
This picture is very generic.
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100
101
102
103
104
10-1
100
101
102
Power laws Criticality
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100
101
102
103
104
10-1
100
101
102
Power
laws: only
at the
critical point
low density
high density
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Life, networks, the brain, the universe and
everything are at “criticality” or the “edge of
chaos.”
Does anyone really believe
this?
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Self-organized criticality:
dynamics have critical point as global attractor
Simpler explanation: systems that
reward yield will naturally evolve to
critical point.
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Would you
design a
system this
way?
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Maybe random
networks
aren’t so great
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High yields
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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isolated
critical
tolerant
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Why power laws?
Almost any
distribution
of sparks
Optimize
Yield
Power law
distribution
of events
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
random
“optimized”
density
yield
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5 10 15 20 25 30
5
10
15
20
25
30
Probability distribution (tail of normal)
High probability region
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Optimal “evolved”
“Evolved” = add one site at
a time to maximize
incremental (local) yield
Very local and limited optimization, yet still
gives very high yields.
Small events likely
large events
are unlikely
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
random
“optimized”
density
High yields.
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Optimized grid
Small events likely
large events
are unlikely
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Optimized grid
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
random
grid
High yields.
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This source of power
laws is quite universal.
Almost any
distribution
of sparks
Optimize
Yield
Power law
distribution
of events
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Tolerance is very different
from criticality.
• Mechanism generating power laws.
• Higher densities.
• Higher yields, more robust to sparks.
• Nongeneric, won’t arise due to random
fluctuations.
• Not fractal, not self-similar.
• Extremely sensitive to small perturbations that were
not designed for, “changes in the rules.”
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Extreme robustness and extreme hypersensitivity.
Small
flaws
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
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0.4
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1