Criticality and power laws

1
Criticality and power laws

• Tuning 1-2 parameters ? critical point
• In certain model systems (percolation, Ising, )
  power laws and universality iff at criticality.
• Physics power laws are suggestive of criticality
• Engineers/mathematicians have opposite
  interpretation
• Power laws arise from tuning and optimization.
• Criticality is a very rare and extreme special
  case.
• What if many parameters are optimized?
• Are evolution and engineering design different?
  How?
• Which perspective has greater explanatory power
  for power laws in natural and man-made systems?

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6
Data compression (Huffman)
WWW files Mbytes (Crovella)
5
4
Cumulative
3
Frequency
Forest fires 1000 km2 (Malamud)
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
Decimated data Log (base 10)
Size of events
3
Size of events x vs. frequency
log(Prob gt size)
log(rank)
log(size)
4
6
Web files
5
Codewords
4
Cumulative
3
Frequency
Fires
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
Size of events
Log (base 10)
5
Forest fires dynamics
Intensity Frequency Extent
6
A HOT forest fire abstraction
Fire suppression mechanisms must stop a 1-d front.
Optimal strategies must tradeoff resources with
risk.
7
Generalized coding problems

• Optimizing d-1 dimensional cuts in d dimensional
  spaces
• To minimize average size of files or fires,
  subject to resource constraint.
• Models of greatly varying detail all give a
  consistent story.
• Power laws have ? ? 1/d.
• Completely unlike criticality.

Data compression
Web
8
Theory
d 0 data compression d 1 web layout d
2 forest fires
9
Data
6
DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
10
Data Model/Theory
6
DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
11
Forest fires?
Fire suppression mechanisms must stop a 1-d front.
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Forest fires?
Geography could make d lt2.
13
California geographyfurther irresponsible
speculation

• Rugged terrain, mountains, deserts
• Fractal dimension d ? 1?
• Dry Santa Ana winds drive large (? 1-d) fires

14
Data HOT Model/Theory
6
5
California brushfires
4
3
FF (national) d 2
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
15
Data HOTSOC
6
5
4
3
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
16
Critical/SOC exponents are way off
Data ? gt .5
SOC ? lt .15
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18 Sep 1998
Forest Fires An Example of Self-Organized
Critical Behavior Bruce D. Malamud, Gleb Morein,
Donald L. Turcotte
4 data sets
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HOT FF d 2
2
10
1
10
0
10
-2
-1
0
1
2
3
4
10
10
10
10
10
10
10
Additional 3 data sets
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(No Transcript)
20
Fires 1930-1990
Fires 1991-1995
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HOT
SOC and HOT have very different power laws.
d1
SOC
d1

• HOT ? decreases with dimension.
• SOC?? increases with dimension.

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• HOT yields compact events of nontrivial size.
• SOC has infinitesimal, fractal events.

HOT
SOC
large
infinitesimal
size
23
SOC and HOT are extremely different.
HOT
SOC
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SOC and HOT are extremely different.
HOT
SOC
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Robust
Log(freq.) cumulative
yet fragile
Log(event sizes)
26
Power laws are inevitable.
Gaussian
log(probgtsize)
log(size)
27
Power laws summary

• Power laws are ubiquitous
• HOT may be a unifying perspective for many
• Criticality, SOC is an interesting and extreme
  special case
• but very rare in the lab, and even much rarer
  still outside it.
• Viewing a complex system as HOT is just the
  beginning of study.
• The real work is in new Internet protocol design,
  forest fire suppression strategies, etc

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Universal network behavior?
Congestion induced phase transition.
throughput

• Similar for
• Power grid?
• Freeway traffic?
• Gene regulation?
• Ecosystems?
• Finance?

demand
29
Web/Internet?
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Networks

• Making a random network
• Remove protocols
• No IP routing
• No TCP congestion control
• Broadcast everything
• ? Many orders of magnitude slower

log(thru-put)
log(demand)
31
Networks
HOT
log(thru-put)
log(demand)
32
Complexity, chaos and criticality

• The orthodox view
• Power laws suggest criticality
• Turbulence is chaos
• HOT view
• Robust design often leads to power laws
• Just one symptom of robust, yet fragile
• Shear flow turbulence is noise amplification
• Other orthodoxies
• Dissipation, time irreversibility, ergodicity and
  mixing
• Quantum to classical transitions
• Quantum measurement and decoherence

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Epilogue

• HOT may make little difference for explaining
  much of traditional physics lab experiments,
• So if youre happy with orthodox treatments of
  power laws, turbulence, dissipation, quantum
  measurement, etc then you can ignore HOT.
• Otherwise, the differences between the orthodox
  and HOT views are large and profound,
  particularly for
• Forward or reverse (eg biology) engineering
  complex, highly designed or evolved systems,
• But perhaps also, surprisingly, for some
  foundational problems in physics

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Data Model/Theory
6
DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2