# Criticality and power laws Tuning 1-2 parameters  critical point In certain model systems (percolation, Ising, …) power laws and universality iff at criticality.

## Presentation on theme: "Criticality and power laws Tuning 1-2 parameters  critical point In certain model systems (percolation, Ising, …) power laws and universality iff at criticality."— Presentation transcript:

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?

-6-5-4-3-2012 0 1 2 3 4 5 6 Size of events Frequency Decimated data Log (base 10) Forest fires 1000 km 2 (Malamud) WWW files Mbytes (Crovella) Data compression (Huffman) Los Alamos fire Cumulative

Size of events x vs. frequency log(size) log(probability) log(Prob > size) log(rank)

-6-5-4-3-2012 0 1 2 3 4 5 6 Size of events Frequency Fires Web files Codewords Cumulative Log (base 10) -1/2

Forest fires dynamics Intensity Frequency Extent Weather Spark sources Flora and fauna Topography Soil type Climate/season

A HOT forest fire abstraction… Burnt regions are 2-d Fire suppression mechanisms must stop a 1-d front. Optimal strategies must tradeoff resources with risk.

Generalized “coding” problems Fires Web Data compression 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.

d = 0data compression d = 1web layout d = 2forest fires Theory

-6-5-4-3-2012 0 1 2 3 4 5 6 FF WWW DC Data

-6-5-4-3-2012 0 1 2 3 4 5 6 FF WWW DC Data + Model/Theory

Forest fires? Burnt regions are 2-d Fire suppression mechanisms must stop a 1-d front.

Forest fires? Geography could make d <2.

California geography: further irresponsible speculation Rugged terrain, mountains, deserts Fractal dimension d  1? Dry Santa Ana winds drive large (  1-d) fires

-6-5-4-3-2012 0 1 2 3 4 5 6 FF (national) d = 2 Data + HOT Model/Theory d = 1 California brushfires

-6-5-4-3-2012 0 1 2 3 4 5 6 Data + HOT+SOC d = 1 SOC FF d = 2  .15

Critical/SOC exponents are way off SOC  <.15 Data:  >.5

Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 18 Sep 1998 4 data sets

10 -2 10 10 0 1 2 3 4 0 1 2 SOC FF HOT FF d = 2 Additional 3 data sets

Fires 1991-1995 Fires 1930-1990

HOT SOC d=1 dd dd HOT  decreases with dimension. SOC  increases with dimension. SOC and HOT have very different power laws.

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

HOT SOC  1/d Large Compact Large Data  1/d  d-1 Dimension d LargeSmall Slope  CompactFractalLarge event shape LargeInfinitesimalMax event size HOTSOC SOC and HOT are extremely different.

 1/d  d-1 Dimension d Large Small Slope  Compact FractalLarge event shape Large InfinitesimalMax event size HOT & Data SOC SOC and HOT are extremely different. HOT SOC

yet fragile Robust Gaussian, Exponential Log(event sizes) Log(freq.) cumulative

Gaussian log(size) log(prob>size) Power laws are inevitable. Improved design, more resources

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…

Universal network behavior? demand throughput Congestion induced “phase transition.” Similar for: Power grid? Freeway traffic? Gene regulation? Ecosystems? Finance?

Web/Internet? demand throughput Congestion induced “phase transition.” Power laws log(file size) log(P>)

random networks log(thru-put) log(demand) Networks Making a “random network:” Remove protocols –No IP routing –No TCP congestion control Broadcast everything  Many orders of magnitude slower Broadcast Network

Networks random networks real networks HOT log(thru-put) log(demand) Broadcast Network

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

Epilogue HOT may make little difference for explaining much of traditional physics lab experiments, So if you’re 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

-6-5-4-3-2012 0 1 2 3 4 5 6 FF WWW DC Data + Model/Theory

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