Cellular Automata
Spatio-Temporal Information for Society
Münster, 2014
System Theory
AdvantagesSimple representation of the worldVisual representationModular and hierarchical
DisadvantagesNo heterogeneityImplicit spatial representationFixed connections between stocks
Where does this image come from?
Where does this image come from?
Map of the web (Barabasi) (could be brain connections)
Information flows in Nature
Ant colonies live in a chemical world
Information flows generate cooperation
White cells attact a cancer cell (cooperative activity)
Foto: National Cancer Institute, EUA http://visualsonline.cancer.gov/
Agents moving
Agents moving
Agents moving
Segregation
Some studies show that most people prefer to live in a non-segregated society. Why there is so much segregation?
SegregationSegregation is an outcome of individual choices
But high levels of segregation indicate mean that people are prejudiced?
Schelling’s Model of Segregation
Schelling (1971) demonstrates a theory to explain the persistence of racial segregation in an environment of growing tolerance
If individuals will tolerate racial diversity, but will not tolerate being in a minority in their locality, segregation will still be the equilibrium situation
Cellular Automata
Firstly developed by Hungarian mathematician John von Neumann, who proposed a model based on the idea of logical systems that were self-replicating.
Self-replicating Automata
Basic Cellular Automaton
Grid of cells Neighbourhood Finite set of discrete states Finite set of transition rules Initial state Discrete time
2-Dimensional Automaton
A 2-dimensional cellular automaton consists of an infinite (or finite) grid of cells, each in one of a finite number of states. Time is discrete and the state of a cell at time t is a function of the states of its neighbors at time t-1.
Neighborhood and Rules
RulesNeighbourhood
States
Space and Time
t
t1
Each cell is autonomous and change its state according to its current state and the state of its neighborhood.
www.terrame.org
“CAs contain enough complexity to simulate surprising and novel change as reflected in emergent phenomena”(Mike Batty)
19
Source: Rita Zorzenon
Game of life
CellularSpace
A CellularSpace is a set of Cells. It consists of an area of interest, divided into a regular grid.
world = CellularSpace{xdim = 5,ydim = 5
}
forEachCell(world, function(cell)cell.value = 3
end)
Neighborhood A Neighborhood represents the proximity relations
of a cell.
world:createNeighborhood{
strategy = "moore",self = false
}
Von Neumann Moore
Legend
Defines colors to draw the Cells of a CellularSpace. Can be used with map observers.
coverLeg = Legend {grouping = "uniquevalue",colorBar = {
{value = 0, color = "white"},{value = 1, color = "red"},{value = 2, color = "green”}
}}
Synchronizing a CellularSpace
TerraME can keep two copies of a CellularSpace in memory: one stores the past values of the cells, and another stores the current (present) values of the cells.
The model equations must read the past copy and write the values to the present copy of the cellular space.
At the correct moment, it will be necessary to synchronize the past copy with the current values of the cellular space.
Characteristics of CA models
Self-organising systems with emergent properties: locally defined rules resulting in macroscopic ordered structures. Massive amounts of individual actions result in the spatial structures that we know and recognise;
Which Cellular Automata?
For realistic geographical modelsthe basic CA principles too constrained to be useful
Extending the basic CA paradigm From binary (active/inactive) values to a set of
inhomogeneous local statesFrom discrete to continuous values (30% cultivated land, 40%
grassland and 30% forest)Transition rules: diverse combinations Neighborhood definitions from a stationary 8-cell to
generalized neighbourhoodFrom system closure to external events to external output
during transitions
Game of Life
Static Life
Oscillating Life
Migrating Life
Conway’s Game of Life
The universe of the Game of Life is an infinite two-dimensional grid of cells, each of which is either alive or dead. Cells interact with their eight neighbors.
Schelling Model for Segregation
Start with a CA with “white” and “black” cells (random)The new cell state is the state of the majority of the
cell’s Moore neighboursWhite cells change to black if there are X or more black
neighboursBlack cells change to white if there are X or more white
neighbours
How long will it take for a stable state to occur?
Schelling’s Model of Segregation
< 1/3
Micro-level rules of the game
Stay if at least a third of neighbors are “kin”
Move to random location otherwise
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