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Self-Organized Criticality (SOC) Tino Duong Biological Computation - - PowerPoint PPT Presentation
Self-Organized Criticality (SOC) Tino Duong Biological Computation - - PowerPoint PPT Presentation
Self-Organized Criticality (SOC) Tino Duong Biological Computation Agenda l Introduction l Background material l Self-Organized Criticality Defined l Examples in Nature l Experiments l Conclusion SOC in a Nutshell l Is the attempt to explain the
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SOC in a Nutshell
l Is the attempt to explain the occurrence of
complex phenomena
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Background Material
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What is a System?
l A group of components functioning as a whole
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Obey the Law!
l Single components in a system are governed
by rules that dictate how the component interacts with others
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System in Balance
l Predictable l States of equilibrium
– Stable, small disturbances in system have only local
impact
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Systems in Chaos
l Unpredictable l Boring
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Example Chaos: White Noise
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Edge of Chaos
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Emergent Complexity
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Self-Organized Criticality
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Self-Organized Criticality: Defined
l Self-Organized Criticality can be considered as
a characteristic state of criticality which is formed by self-organization in a long transient period at the border of stability and chaos
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Characteristics
l Open dissipative systems l The components in the system are governed
by simple rules
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Characteristics (continued)
l Thresholds exists within the system l Pressure builds in the system until it exceeds
threshold
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Characteristics (Continued)
l Naturally Progresses towards critical state
l Small agitations in system can lead to system effects called
avalanches
l This happens regardless of the initial state of the system
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Domino Effect: System wide events
l The same perturbation may lead to small
avalanches up to system wide avalanches
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Example: Domino Effect
By: Bak [1]
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Characteristics (continued)
l Power Law
l Events in the system follow a simple power law
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Power Law: graphed
i) ii)
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Characteristics (continued)
l Most changes occurs through catastrophic
event rather than a gradual change
l Punctuations, large catastrophic events that effect the
entire system
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How did they come up with this?
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Nature can be viewed as a system
l It has many individual components working
together
l Each component is governed by laws
l e.g, basic laws of physics
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Nature is full of complexity
l Gutenberg-Richter Law l Fractals l 1-over-f noise
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Earthquake distribution
By: Bak [1]
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Gutenberg-Richter Law
By: Bak [1]
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Fractals:
l Geometric structures with features of all length
scales (e.g. scale free)
l Ubiquitous in nature
l Snowflakes l Coast lines
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Fractal: Coast of Norway
By: Bak [1]
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Log (Length) Vs. Log (box size)
By: Bak [1]
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1/F Noise
By: Bak [1]
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1/f noise has interesting patterns
1/f Noise White Noise
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Can SOC be the common link?
l Ubiquitous phenomena
l No self-tuning l Must be self-organized
l Is there some underlying link
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Experimental Models
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Sand Pile Model
l An MxN grid Z l Energy enters the model by randomly adding
sand to the model
l We want to measure the avalanches caused
by adding sand to the model
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Example Sand pile grid
l Grey border represents
the edge of the pile
l Each cell, represents a
column of sand
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Model Rules
l Drop a single grain of sand at a random
location 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 avalanche
l Threshold = 3
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Adding Sand to pile
l Chose Random (x,y)
position on grid
l Increment that cell
l Z(x,y) ‡ Z(x,y)+1
l Number of sand grains
indicated by colour code
By: Maslov [6]
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Avalanches
l When threshold has
been exceeded, an avalanche occurs
l If Z(x,y) > 3
l 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]
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Before and After
Before After
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Domino Effect
l Avalanches may
propagate
By: Bak [1]
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DEMO: By Sergei Maslov
http://cmth.phy.bnl.gov/~maslov/Sandpile.htm
Sandpile Applet
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Observances
l Transient/stable phase l Progresses towards Critical phase
l At which avalanches of all sizes and durations
l Critical state was robust
l Various initial states. Random, not random
l Measured events follow the desired Power Law
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Size Distribution of Avalanches
By: Bak [1]
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Sandpile: Model Variations
l Rotating Drum l Done by Heinrich Jaeger l Sand pile forms along
the outside of the drum Rotating Drum
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Other applications
l Evolution l Mass Extinction l Stock Market Prices l The Brain
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Conclusion
l Shortfalls
l Does not explain why or how things self-organize into the
critical state
l Cannot mathematically prove that systems follow the
power law
l Benefits
l Gives us a new way of looking at old problems
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References:
l [1] P. Bak, How Nature Works. Springer -Verlag, NY,
1986.
l [2] H.J.Jensen. Self-Organized Criticality – Emergent
Complex Behavior in Physical and Biological Systems. Cambridge University Press, NY, 1998.
l [3] T. Krink, R. Tomsen. Self-Organized Criticality and
Mass Extinction in Evolutionary Algorithms. Proc. IEEE
- int. Conf, on Evolutionary Computing 2001: 1155-1161.
l [4] P.Bak, C. Tang, K. WiesenFeld. Self-Organized
Criticality: An Explanation of 1/f Noise. Physical Review Letters. Volume 59, Number 4, July 1987.
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References Continued
l [5] P.Bak. C. Tang. Kurt Wiesenfeld. Self-Organized
- Criticality. A Physical Review. Volume 38, Number 1.
July 1988.
l [6]S. Maslov. Simple Model of a limit order-driven
- market. 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.
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