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