Thermohaline Circulation PIK Potsdam www'pikpotsdam'de Stefan Rahmstorff

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Thermohaline CirculationPIK Potsdam
(www.pik-potsdam.de)Stefan Rahmstorff
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LectureComplex Adaptive SystemsCellular
Automata
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Contents

• Characteristics of cellular automata
• Examples of CAs
• Self-organized criticality
• Application of CAs to represent social systems
  (potential and limitations)

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CA Definition

• A cellular automaton (CA) is a discrete dynamical
  system, where space, time, and the states of the
  system are all discrete.
• It is grid or lattice of a large number of
  identical cells in a regular lattice in one, two
  or three dimensions.
• Simulated time proceeds in discrete units often
  called steps, cycles or generations.

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CA Definition cont.

• Each site or cell can be in one of a finite set
  of states. The state may be represented by
  integer values. They may also be boolean (eg.
  dead/TRUE dead/FALSE alive). Changes in a
  cells state are controlled by rules. The cells
  rules depend only on the state of the cell and
  its neighbours.
• At each step, the state of every cell (at time
  t1) is calculated using the states of
  neighbouring cells at time t.
• The type of neighbourhood chosen determines the
  cells included as neighbors.

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Most important neighborhoods
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CA Dynamics

• In general, the update of all cells is done
  simultaneously not sequentially. Let us call S
  the set of all cells in a 2-dimensional cellular
  automaton and Sij(tk) the state of a cell Cij at
  a specific location, i,j at the time step tk.
• S -gt T (store states in temporary grid)
• And then

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Boundary Conditions

• Each CA is implemented on a finite grid. The
  boundary conditions chosen depend on the nature
  of the system.
• Periodic Boundary conditions are chosen for
  infinite systems. Border sites have sites on the
  opposite borders as neighbours for example in a
  squared lattice with dimension n, all S1j have as
  the neighbour the cell Snj.
• Absorbing (open) boundary and reflecting (closed)
  boundary conditions act as sinks and sources,
  respectively. A unique and fixed value is chosen
  for boundaries - rules are chosen to specify
  states of cells adjacent to border cells.

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The Game of Life

• A cell is either dead or alive.
• A dead cell becomes alive if exactly three of its
  neighbors are alive, a living cell stays alive if
  two or three of its neighbors are alive, in all
  other cases with less than two or more then three
  neighbors a living cell dies the neighborhood
  is a Moore Neighborhood.

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• Automata Classes
• The various types of CA fall into 4 Classes
  (defined by S. Wolfram)
• Class 1 - point attractors. The system freezes
  into a fixed state after a short time (the
  transient behaviour).
• Class 2 - limit cycles. The system develops
  periodic behaviours, which then repeat
  continuously.
• Class 3 - chaotic. The system becomes aperiodic,
  continuously changing in unpredictable and random
  ways.                      
• Class 4 - structured. System can develop in
  highly patterned but unstable ways.
  Computationally rich, e.g. Game of Life. This
  denotes automata between states 2 and 3.

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Self Organized Criticality

• Per Bak et al introduced probabilistic CAs
  showing the phenomenon of self-organized
  criticality (SOC).
• Systems are able to sustain a limited amount of
  stress. If stress exceeds locally a certain
  threshold, the system relaxes locally to an
  unstressed state and the stress is distributed to
  the neighborhood.

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SOC and sand piles

• The original example was a sandpile - the
  phenomenon of SOC was studied both with CA models
  and experimentally.
• In the experimental case a sandpile is
  constructed by adding single grains of sand to an
  existing heap and by observing the size of the
  avalanches that are generated when the slope
  becomes unstable

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Sand Pile Model

• The sandpile CA uses a 2d square lattice with
  absorbing boundary conditions.
• Each site i is characterized by a discrete
  variable zi with threshold (N1) where a site
  becomes unstable. For the 2d CA, N 4.
• Internal lattice sites have initial random values
  between one and N while the sites at the borders
  have a value of zero.
• When the overall configuration is stable, a site
  is chosen at random. At this site zi is
  increased by 1 unit, zi -gt zi 1.
• When the configuration becomes unstable since
  some zi gt N1 the following rule is applied
• zi -gt zi N, zik -gt zk 1 for all N
  neighbors k of i unless the neighbors are
  bordering sites which remain zero.
• If neighboring sites become unstable the rule is
  repeated until the whole system is relaxed.

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Example for temporal evolution in sand pile CA
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Aerial extent (domain) for several different
avalanches in a SOC model. Each avalanche was
triggered by the addition of a single grain.
Avalanches have orders of magnitude difference in
their sizes (Bak, et. al, 1988).
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Log-log plot of the frequency of occurrence of
given avalanche size D(s) vs. the size of the
avalanche s for 200 avalanches Avalanches exhibit
a power law distribution (Bak, et. al, 1988).
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Schematic plot of the distribution of the size of
avalanches (1/f noise)
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SOC more general

• SOC describes the tendency of large dynamical
  systems to drive themselves to a critical state
  with a wide range of time and space scales.
• It may the origin of 1/f noise observed for many
  phenomena in natural and social systems.

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SOC in other systems

• Movement of tectonic plates. When two plates
  slide, the bulks of the plates drift slowly
  relative to each other but the friction at the
  boundaries increases. Plates get distorted until
  the tension is relaxed by an earthquake. This
  local release of stress may trigger a suite of
  secondary earthquakes. And the size distribution
  of earthquakes follows the pattern expected from
  a system showing SOC.
• Forest fires. Increase in stress is given be the
  accumulation of inflammable material. One has
  observed a size distribution of fires that
  correspond to the patterns of a SOC. Management
  of forests (extinctions of small fires) has lead
  to a decrease in the number of fires but an
  increase in devastating fires since stress could
  not be released.
• Size of sales in a market?

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Gutenberg-Richter-Relationship shown by Log-log
plot of global occurrence of earthquakes from
1977- 1995. Seismic moment (used to determine
magnitude) is plotted vs. number of events. The
distribution follows a power law relationship
(Lay and Wallace, 1995).
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Earthquake Model

• Consider a two dimensional array of blocks. Each
  block is connected by springs to its nearest
  neighbor, as well as to a driving plate (see
  figure 7). This system is analogous to a "sand
  pile" in that the local addition of slope to the
  sand pile
• z(x,y) -gt z(x,y)1
• is like adding stress to the "fault plane". The
  addition of stress eventually causes a block to
  fail and an earthquake (avalanche) occurs
• z(x,y) -gt z(x,y)-4
• z(x/-1,y) -gt z(x/-1,y)1
• z(x,y/-1) -gt z(x,y/-1)1.
• If subsequent redistribution of stress or "stress
  drop" causes a nearby block to exceed the
  critical stress value, then it too will fail.
  Thus, earthquakes can range in size from the
  failure of a single block to failures which
  encompass nearly the entire system.
• As with computer models of sand piles, this
  simple set of rules produces computer-earthquakes
  which follow a power law distribution.

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Conceptual drawing of a two-dimensional
spring-slider block model. Leaf springs (K1)
connect a moving plate to an array of smaller
sliding blocks. These blocks are in turn
connected to their nearest neighbors via coil
springs (K2, K3). Sliding blocks also have a
frictional contact with a fixed plate (Bak,
1996).
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Implications for predictability and management of
systems and convergence of averages from time
series

• - random fluctuations and disturbance are better
  represented by red than white noise if the source
  of the noise is a system in a SOC (not the only
  source of 1/f) noise.
• - extreme values are very difficult to predict
  and it is very difficult to extract the
  parameters for extreme value statistics from data
• - management of such systems may have detrimental
  effects if the system cannot relax anymore on
  many scales but tension still builds up (e.g.
  forest fires) -gt management may lead to less
  extreme events but more events of extreme
  magnitude.

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CA and social systems

• Gradual transition to agent based models

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Spread of gossip

• A very simple example for the spread of an
  innovation or gossip can be based on a cellular
  automaton. Individuals are modelled as cells and
  the interaction between people is modelled using
  the cells rules.
• The spread of information from a single origin is
  like a diffusion process. Each cell has two
  states ignorance and knowing the gossip.

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Rumor Mill (Netlogo)

• This program models the spread of a rumor. The
  rumor spreads when a person who knows the rumor
  tells one of their neighbors. In other words,
  spatial proximity is a determining factor as to
  how soon (and perhaps how often) a given
  individual will hear the rumor.
• The neighbors can be defined as either the four
  adjacent people or the eight adjacent people. At
  each time step, every person who knows the rumor
  randomly chooses a neighbor to tell the rumor to.
  The simulation keeps track of who knows the
  rumor, how many people know the rumor, and how
  many "repeated tellings" of the rumor occur.

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Options Boundary Conditions and NeighborHood

• WRAP? is a switch which when set on allows the
  rumor to wrap top and bottom and left and right,
  as if the grid were on a torus. When set off, the
  rumor spreads as if the grid is bounded, without
  wrapping.
• EIGHT-MODE? is a switch that determines whether
  at each time step the rumor spreads to one of
  four randomly chosen neighbors, or one of eight
  such neighbors.

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Options Initital Conditions

• 1) Single source Press the SETUP-ONE button.
  This starts the rumor at one point in the center
  of the screen.
• 2) Random source Press the SETUP-RANDOM button
  with the INIT-CLIQUE slider set greater than 0.
  This "seeds" the rumor randomly by choosing a
  percentage of the population that knows the rumor
  initially. This percentage is set using the
  INIT-CLIQUE slider.
• 3) Choose source with mouse Press either
  SETUP-ONE or SETUP-RANDOM, then press the
  SPREAD-RUMOR-WITH-MOUSE button. While this
  button is down, clicking the mouse button on a
  patch in the graphics window will tell the rumor
  to that patch.

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XY Plots Presented

• RUMOR SPREAD - plots the percentage of people who
  know the rumor at each time step.
• SUCCESSIVE DIFFERENCES - plots the number of new
  people who are hearing the rumor at each time
  step.
• SUCCESSIVE RATIOS - plots the percentage of new
  people who are hearing the rumor at each time
  step.

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Spatial prisoners dilemma model

• The prisoners dilemma is a game theoretic
  analysis of the cooperation between two
  individuals.
• It deals with the situation in a single encounter
  when each individual must choose to cooperate or
  not to cooperate.
• In an isolated interaction between two
  individuals there exist four possible outcomes of
  the interaction
• Win-Win Win-Lose Lose-Win Lose-Lose

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Outcomes prisoners dilemma model

• WW - both cooperate (each receives payoff R)
• WL and LW one cooperates whereas the other
  defects (non-cooperative receives a payoff T and
  the cooperative player receives a payoff, D),
• LL none is cooperative (each receives payoff P).
• If the payoffs are ordered according to
  T gt R gt P
  gt Sit is always in each players self-interest
  not to cooperate

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PayOffMatrix prisoners dilemma model
In the original version of the game the players
were assumed to be prisoners. The payoff was
related to the years to be spend in prison. The
value in the left of each column in the payoff
of player A, in the right of player B.
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Iterated prisoners dilemma

• The problem is made more interesting by playing
  it repeatedly with the same group of players,
  thereby permitting partial time histories of
  behaviour to guide future decisions.
• This so-called iterated prisoner's dilemma has
  drawn interest from game theorists for a while.
• Computer tournaments have pitted different
  computer procedures against one another in two
  contests.
• In both contests, a very simple strategy called
  "tit for tat" was the overall winner among 76
  entrants. It simply cooperates on the first move
  and then does whatever the opponent does on the
  previous move 1st move cooperate
  subsequent move copy opponents previous move

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CA with evolving strategies

• CA with 10002 cells (Gilbert Troitzsch, 1996)
• Each cell remembers the last three moves of its
  IPD opponent and follows one of the 215 possible
  different strategies
• At the end of each step, a cell randomly finds
  another player and switches to the other cells
  strategy if that is better (higher total payoff)
• Calculation of difference between payoff is
  subject to noise
• New strategies mutate (noisy transmission)

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Results

• Even in the presence of noise, cooperation is
  likely and stable
• There is no one best strategy rather there are
  bundles of strategies which give a high payoff
  together
• The mix of strategies within the bundle changes
  over time.

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A socio-economic model of regional structural
change

• The cellular automaton approach was used to
  analyse structural change and path dependence in
  agriculture
• (Farm-based modelling of regional structural
  change A cellular automata approach -
  http//www.alfons-balmann.de/).
• Like a cellular automaton, the region is
  subdivided in a number of spatially ordered
  plots.
• Albeit rule based, farms are characterized by
  quite complex, optimizing behaviour.

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Difference between CAs and Agent Based Models

• The last model could already be labeled an agent
  based model.
• It has the properties of a cellular automaton
  regarding the discrete spatial location which is
  fixed, rule-based behaviour and discrete states.
• Behaviour is already quite complex and rules use
  historical information.
• The distinction between CAs and Agents is here a
  bit fluid.
• -gt Clearly agents when movement is involved.