Workshop on Self-Organization

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Workshop on Self-Organization

• Clint Sprott (workshop leader)
• Department of Physics
• University of Wisconsin - Madison
• Presented at the Annual Meeting of the
• Society for Chaos Theory in Psychology and Life
  Sciences
• at Marquette University
• in Milwaukee, WI
• on July 23, 2009

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Agenda

• Introductions
• Introductory lecture
• Challenge to participants
• Choose discussion groups
• Break
• Group discussions
• Presentation of results
• Summary

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Self-OrganizationNatures Intelligent Design

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented to
• Society for Chaos Theory in Psychology and Life
  Sciences
• at Marquette University
• in Milwaukee, WI
• on July 23, 2009

1 Diffusion-limited aggregation
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Self-organized Structures in Nature
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What is a complex system?

• Complex ? complicated
• Not real and imaginary parts
• Not very well defined
• Contains many interacting parts
• Interactions are nonlinear
• Contains feedback loops ( and -)
• Cause and effect intermingled
• Driven out of equilibrium
• Evolves in time (not static)
• Usually chaotic (perhaps weakly)
• Can self-organize and adapt (CAS)

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Landscape of Early Southern Wisconsin (USA)
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2 Stochastic Cellular Automaton Model
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Cellular Automaton

• Cellular automaton Square array of cells where
  each cell takes one of the 6 values representing
  the landscape on a 1-square mile resolution
• Evolving single-parameter model A cell dies
  out at random times and is replaced by a cell
  chosen randomly within a circular radius r (1 lt r
  lt 10)

• Boundary conditions periodic and reflecting
• Initial conditions random and ordered
• Constraint The proportions of land types are
  kept equal to the proportions of the experimental
  data

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Initial Conditions
Ordered
Random
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Bush-Kerry 2004 Election
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3Deterministic Cellular Automaton Model
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Why a deterministic model?

• Randomness conceals our ignorance
• Simplicity can produce complexity
• Chaos requires determinism
• The rules provide insight

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Deterministic CA
Truth Table
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4
4
2
1
2
4
4
0
1
1
3
3
2
1
2
4
4
3
4
4
210 1024 possible rules for 4 nearest neighbors
22250 10677 possible rules for 20 nearest
neighbors
Totalistic rule
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Is it Fractal?
Deterministic Model
Stochastic Model
D 1.666
D 1.685
0
0
e
e
log C( )
log C( )
-3
-3
e
log
e
0
0
3
3
log
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Power Laws (Zipf)
Size of Power Outages
Words in English Text
Earthquake Magnitudes
Internet Document Accesses
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Other Examples of Power Laws

• Populations of cities
• Size of moon craters
• Size of solar flares
• Size of computer files
• Casualties in wars
• Occurrence of personal names
• Number of papers scientists write
• Number of citations received
• Sales of books, music,
• Individual wealth, personal income
• Many others

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4Lotka-Volterra Models
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Multispecies Lotka-Volterra Model

• Let xi be population of the ith species
  (rabbits, trees, people, stocks, )
• dxi / dt rixi (1 - S aijxj )
• Parameters of the model
• Vector of growth rates ri
• Matrix of interactions aij
• Number of species N

N
j1
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Evolution to the Edge of Chaos
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Minimal High-D Chaotic L-V Model
dxi /dt xi(1 xi 2 xi xi1)
1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
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Space
Time
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Chaos

• Chaos is the unpredictable behavior of
  deterministic systems
• It is sensitive to initial conditions (the
  butterfly effect)
• It produces erratic fluctuations and never
  repeats
• Systems that produce fractal spatial patterns
  usually exhibit temporal chaos

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5Social Network Model
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Social Networks
Rules

1. You tend to befriend friends of your friends
2. You tend to mirror others friendliness toward
   you
3. You have a limited capacity for maintaining
   friendships

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6Strange Attractors
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Aesthetic Evaluation
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Conclusions

• Many simple models exhibit self-organization (the
  spontaneous development of complex structures).
• Some effects may not have easily identifiable
  causes.
• The 2nd law of thermodynamics (increasing
  disorder) is not violated since these systems are
  far from equilibrium (driven by energy flow).
• If there is intelligent design in nature, it is
  at a more fundamental level (the underlying laws
  of nature) than its proponents commonly suppose.

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References

• http//sprott.physics.wisc.edu/
  lectures/sctpls.ppt (this talk)
• http//sprott.physics.wisc.edu/
• chaostsa/ (my book on Chaos)
• sprott_at_physics.wisc.edu (contact me)

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Challenge to Participants

• Break into 3 groups
• Physical science
• Biological science
• Social science
• Develop a model in your field that could
  self-organize
• Reconvene and present your results to the whole
  group

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Groups

• Physical Science
• Physics
• Mathematics
• Meteorology
• Engineering
• Biological Science
• Neurology
• Medicine
• Ecology
• Social Science
• Economics
• Political Science
• Sociology

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Tasks

• Decide on a problem
• Identify the agents
• Decide on the rules
• Identify the resource
• Develop tests for self-organization
• Choose a spokesman to present to whole group

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Decide on a Problem

• In your chosen field
• Interesting and novel
• Not trivial or too complex
• Potentially publishable

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Identify the Agents

• Examples Dust grains, spatial cells, species,
  persons,
• Quantify spatial position, landscape type,
  species population, friendliness,
• Interactions stickiness, replacement,
  competition, conversation,

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Decide on the Rules

• Dust grains
• Move randomly in 2-D
• Stick when collide
• Tree model
• Random death and replacement
• Lotka-Volterra model
• Logistic growth
• Competition matrix
• Friendship model
• 3 rules previously explained

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Identify the Resource

• Dust grains (DLA)
• Grains
• Enter at edge ? stick
• Tree model
• Sunlight - photosynthesis
• Growth ? death
• Lotka-Volterra model
• Food - metabolism
• Production ? consumption
• Friendship model
• Information

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Tests for Self-organization

• Spatio-temporal plots
• Fractal structure
• Power laws
• Chaotic dynamics

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Other Questions to Consider

• What is internal/external to the model?
• What are the adjustable parameters?
• Is the model deterministic or stochastic?
• What are reasonable initial conditions?
• How many spatial dimensions required?
• How many agents are required?
• How many generations are required?
• How would you implement the model?
• How do you test the model?