Zelf Organiserende Systemen ZOS

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Zelf Organiserende Systemen (ZOS) - Self
Organising Systems - Vrij Universiteit
Amsterdam Lecture 5 Self Organisation to
Criticality http//www.cs.vu.nl/zos Martijn
Schut schut_at_cs.vu.nl
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SOS - Lecture 5
Overview

• Criticality in theory
• Sandpiles
• and in practice
• Forest fires
• Ecosystems (Lotka-Volterra)
• Panic
• Chaos
• Exercises
• Turing Machines
• This week
• Panic

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SOS - Lecture 5
Criticality

• What came first, chicken or egg?
• Biological systems are organised by the
  information present in them, and the information
  in turn originates in the sel-organised state by
  means of selection. (Adami, 1998)
• self-organisation is one of the hallmarks of
  living systems
• selection relates criticality to evolution (week
  7)

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SOS - Lecture 5
Critical Values
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SOS - Lecture 5
Criticality

• Some important things to remember
• self organised means that critical values are
  included with the system
• in chaotic systems, critical values can make the
  system abdruptly change state
• These two characteristics are illustrated in
  this Lectures examples

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SOS - Lecture 5
Sandpiles

• What is the problem?
• Very simple
• Imagine a chess board with miniscule squares.
  Let grains of sand drop on this board at random
  locations for an extended amount of time.
• What do you see?
• Many events where only one grain topples
• Some events where a few grains are involved in
  an avalanche
• Rare events where all grains are involved in an
  avalanche
• We are interested to know what the distribution
  is of these size of events

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SOS - Lecture 5
Sandpiles

• Solution?
• Surprisingly difficult to analyse from first
  principles!
• Quite easy to see by simple performing the
  experiment
• Experiments result in very nice graphs
• Can we use these experiments for analysis?

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SOS - Lecture 5
Sandpiles
A one-dimensional sandpile
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SOS - Lecture 5
Sandpiles

• Rules
• if difference in sizes between locations is gt
  2, then one grain will tumble to the next
  location
• Even in this extremely simple example, it is very
  hard to
• predict the size of avalanches over time.

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SOS - Lecture 5
Sandpiles
But what turns out to be the case? With
experiments, we find that there is one critical
state towhich the system always returns! /
Note state is defined on the average slope /
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SOS - Lecture 5
Sandpiles

• What does such a state look like?
• Assume building up from an empty row to a state
  just beforea sliding event (avalanche
  involving all locations)
• In this state, the first location is always
  maximally filled and the differences between
  locations is critical

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SOS - Lecture 5
Sandpiles

• Take home message
• critical state itself is very unstable
• but it makes the whole system very stable

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SOS - Lecture 5
Forest Fires
Example State

• Example Rules
• B ? A
• A ? T
• BT ? AB
• T ? B
• B ? A

• Legend
• T Tree
• B Burning
• A Ashes

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SOS - Lecture 5
Forest Fires

• With these states and rules, the system is not
  critical!
• Instead it is periodic, like disease epidemics
• Over time there will be waves of live and dead
  trees
• What to do to make it critical, what is the
  problem here?

• The system is not driven!
• A driving force is an infinitely small external
  cause of fluctuation
• Thus there is nothing to make it revert to a
  single state

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SOS - Lecture 5
Forest Fires

• Solution light up a cigarette!
• Or have an occasional lightning strike
• This changes the system into a critical one
• Now trees start burning spontaneously
• Thus T ? B with some very small probability
• The critical state is then one in which there is
  some constantnumber of burning trees

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SOS - Lecture 5
Properties of self criticality

• The system must
• resemble or be a self organising system, I.e.
  many interactingsmall components
• carry information throughout the whole system
• carry noise (random elements) throughout the
  whole system
• an infinitesimal driving rate

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SOS - Lecture 5
Chaos Theory

• This driving rate reminds a bit of chaos theory
• Are chaotic systems and critical systems related?

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SOS - Lecture 5
Chaos Theory

• Chaos happens in dynamic systems
• These are basically systems in which the
  components are in motion
• Order in such a system occurs when it is in
  balance
• In balance means
• little disturbances have no consequences
• action reaction
• only dramatic disturbances can cause state
  transactions

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SOS - Lecture 5
Chaos Theory
Examples of systems in balance
ice
sand
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SOS - Lecture 5
Chaos Theory
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SOS - Lecture 5
Chaos Theory

• Lorenz Curve
• Discovered by Lorenz in 60s
• Meteorologist
• How predictable is the weather?

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SOS - Lecture 5
Chaos Theory
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SOS - Lecture 5
Chaos Theory

• Is this not very exceptional?
• No, we can see chaos in
• turbulence of water and air
• wobbling of the planets
• global weather patterns
• electric-chemical activity in the human brain

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SOS - Lecture 5
Chaos Theory

• Then how is it different from stochastics?
• It is exactly the opposite!
• With probabilities, we are able to predict
  long-term outcomes
• For example, flipping a coin
• (remember the law of large numbers?)

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SOS - Lecture 5
Ecologies
An ecosystem is also a self organising critical
system withemerging properties, interacting
components etc etc...
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SOS - Lecture 5
Ecologies
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SOS - Lecture 5
Ecologies
Click here for to see some major state shifts and
their causes
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SOS - Lecture 5
Patchiness
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SOS - Lecture 5
Patchiness
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SOS - Lecture 5
Patchiness
double nextValue(int i, int j) double
nextValue double neighbourhood new
doublenbhWidthnbhLength neighbourhood
makeNbh(i, j) double nbhValue
computeNbhValue(neighbourhood) nextValue
worldij Math.max(B, Math.min(H,
cnbhValue)) return nextValue
double computeNbhValue(double nbh)
double nbhValue 0 int k, l for(k0
kltnbhWidth k) for(l0 lltnbhLength
l) if(!(knbhCenterX l nbhCenterY))
nbhValue nbhklcMatrixkl
return nbhValue
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SOS - Lecture 5
Patchiness
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SOS - Lecture 5
Panic

• Panic behaviour is also self organising
• Trade off between well being of the individual
  and the group
• People never make this choice rationally
• People have investigated panic theoretically
  with experiments
• Use the results of these experiments to guide
  people in panic situations

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SOS - Lecture 5
Panic - Example Simulations
no panic panic stampede fire front column straigh
t corridor corridor with widening herding individ
ualism mixed behaviour
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SOS - Lecture 5
Panic - 2 exits

• People are searching for an exit they can't see
  because of smoke (grey), the desired velocity of
  individuals is v0 5 m/s.
• Pedestrians recognize the door and the walls
  from a distance of 2m, while the range of
  pedestrian-pedestrian interactions is assumed to
  be 5m.
• For pure individualism (unrealistic), people
  find an exit only by chance.
• For strong herding, people follow the mass which
  may move into the wrong direction.
• Most efficient is a mixture of individualism and
  herding, for which small groups are formed
  Successful strategies that individuals found by
  chance are imitated by a reasonable number of
  people.

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SOS - Lecture 5
Panic - Simulation - 2 exits
Type of Behaviour Escaped until t
20s Individualism (p0.01) 66 Mixed behavior
(p0.4) 71 Herding (p0.8) 16
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SOS - Lecture 5
Conclusions

• Criticality in theory
• Sandpiles
• and in practice
• Forest fires
• Ecosystems (Lotka-Volterra)
• Panic
• Chaos