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Self-Organized Criticality (SOC) Tino Duong Biological Computation
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Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

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Page 1: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Self-Organized Criticality (SOC)

Tino Duong

Biological Computation

Page 2: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Agenda

l Introduction

l Background material

l Self-Organized Criticality Defined

l Examples in Nature

l Experiments

l Conclusion

Page 3: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

SOC in a Nutshell

l Is the attempt to explain the occurrence ofcomplex phenomena

Page 4: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Background Material

Page 5: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

What is a System?

l A group of components functioning as a whole

Page 6: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Obey the Law!

l Single components in a system are governedby rules that dictate how the componentinteracts with others

Page 7: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

System in Balance

l Predictable

l States of equilibrium– Stable, small disturbances in system have only local

impact

Page 8: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Systems in Chaos

l Unpredictable

l Boring

Page 9: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Example Chaos: White Noise

Page 10: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Edge of Chaos

Page 11: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Emergent Complexity

Page 12: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Self-Organized Criticality

Page 13: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Self-Organized Criticality: Defined

l Self-Organized Criticality can be considered asa characteristic state of criticality which isformed by self-organization in a long transientperiod at the border of stability and chaos

Page 14: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Characteristics

l Open dissipative systems

l The components in the system are governedby simple rules

Page 15: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Characteristics (continued)

l Thresholds exists within the system

l Pressure builds in the system until it exceedsthreshold

Page 16: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Characteristics (Continued)

l Naturally Progresses towards critical statel Small agitations in system can lead to system effects called

avalanches

l This happens regardless of the initial state of the system

Page 17: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Domino Effect: System wide events

l The same perturbation may lead to smallavalanches up to system wide avalanches

Page 18: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Example: Domino Effect

By: Bak [1]

Page 19: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Characteristics (continued)

l Power Lawl Events in the system follow a simple power law

Page 20: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Power Law: graphed

i)

ii)

Page 21: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Characteristics (continued)

l Most changes occurs through catastrophicevent rather than a gradual change

l Punctuations, large catastrophic events that effect theentire system

Page 22: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

How did they come up with this?

Page 23: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Nature can be viewed as a system

l It has many individual components workingtogether

l Each component is governed by lawsl e.g, basic laws of physics

Page 24: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Nature is full of complexity

l Gutenberg-Richter Law

l Fractals

l 1-over-f noise

Page 25: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Earthquake distribution

By: Bak [1]

Page 26: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Gutenberg-Richter Law

By: Bak [1]

Page 27: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Fractals:

l Geometric structures with features of all lengthscales (e.g. scale free)

l Ubiquitous in nature

l Snowflakes

l Coast lines

Page 28: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Fractal: Coast of Norway

By: Bak [1]

Page 29: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Log (Length) Vs. Log (box size)

By: Bak [1]

Page 30: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

1/F Noise

By: Bak [1]

Page 31: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

1/f noise has interesting patterns

1/f Noise White Noise

Page 32: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Can SOC be the common link?

l Ubiquitous phenomenal No self-tuning

l Must be self-organized

l Is there some underlying link

Page 33: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Experimental Models

Page 34: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Sand Pile Model

l An MxN grid Z

l Energy enters the model by randomly addingsand to the model

l We want to measure the avalanches causedby adding sand to the model

Page 35: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Example Sand pile grid

l Grey border representsthe edge of the pile

l Each cell, represents acolumn of sand

Page 36: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Model Rules

l Drop a single grain of sand at a randomlocation on the grid

l Random (x,y)

l Update model at that point: Z(x,y) ‡ Z(x,y)+1

l If Z(x,y) > Threshold, spark an avalanchel Threshold = 3

Page 37: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Adding Sand to pile

l Chose Random (x,y)position on grid

l Increment that celll Z(x,y) ‡ Z(x,y)+1

l Number of sand grainsindicated by colour code

By: Maslov [6]

Page 38: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Avalanches

l When threshold hasbeen exceeded, anavalanche occurs

l If Z(x,y) > 3l Z(x,y) ‡ Z(x,y) – 4

l Z(x+-1,y) ‡ Z(x+-1,y) +1

l Z(x,y) ‡ Z(x,y+-1) +1

By: Maslov [6]

Page 39: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Before and After

Before After

Page 40: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Domino Effect

l Avalanches maypropagate

By: Bak [1]

Page 41: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

DEMO: By Sergei Maslov

http://cmth.phy.bnl.gov/~maslov/Sandpile.htm

Sandpile Applet

Page 42: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Observances

l Transient/stable phase

l Progresses towards Critical phasel At which avalanches of all sizes and durations

l Critical state was robustl Various initial states. Random, not random

l Measured events follow the desired Power Law

Page 43: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Size Distribution of Avalanches

By: Bak [1]

Page 44: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Sandpile: Model Variations

l Rotating Drum

l Done by Heinrich Jaeger

l Sand pile forms alongthe outside of the drum

Rotating Drum

Page 45: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Other applications

l Evolution

l Mass Extinction

l Stock Market Prices

l The Brain

Page 46: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

Conclusion

l Shortfallsl Does not explain why or how things self-organize into the

critical state

l Cannot mathematically prove that systems follow thepower law

l Benefitsl Gives us a new way of looking at old problems

Page 47: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

References:

l [1] P. Bak, How Nature Works. Springer -Verlag, NY,1986.

l [2] H.J.Jensen. Self-Organized Criticality – EmergentComplex Behavior in Physical and Biological Systems.Cambridge University Press, NY, 1998.

l [3] T. Krink, R. Tomsen. Self-Organized Criticality andMass Extinction in Evolutionary Algorithms. Proc. IEEEint. Conf, on Evolutionary Computing 2001: 1155-1161.

l [4] P.Bak, C. Tang, K. WiesenFeld. Self-OrganizedCriticality: An Explanation of 1/f Noise. PhysicalReview Letters. Volume 59, Number 4, July 1987.

Page 48: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

References Continued

l [5] P.Bak. C. Tang. Kurt Wiesenfeld. Self-OrganizedCriticality. A Physical Review. Volume 38, Number 1.July 1988.

l [6]S. Maslov. Simple Model of a limit order-drivenmarket. Physica A. Volume 278, pg 571-578. 2000.

l [7] P.Bak. Website: http://cmth.phy.bnl.gov/~maslov/Sandpile.htm.Downloaded on March 15th 2003.

l [8] Website:http://platon.ee.duth.gr/~soeist7t/Lessons/lessons4.htm.Downloaded March 3rd 2003.

Page 49: Self-Organized Criticality (SOC) · References: l[1] P. Bak, How Nature Works.Springer -Verlag, NY, 1986. l[2] H.J.Jensen.Self-Organized Criticality – Emergent Complex Behavior

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