Cellular Automata Models

of Social Processes

Alexander Makarenko

Institute for applied systems analysis

NTUU „KPI”, Prospect Pobedy 37, 03056,

Kiev-56, Ukraine

makalex@i.com.ua

HYSTORY, IDEAS of CLASSICAL

CELLUILAR AUTOMATA

SOURCES OF ORIGIN:

Theory of automates: J. von Neumann

A. Turing

Ideas of cellular automata implementations and applications:

J.Conway („Life” game, 1970);

S.Wolfram (1984); S.Kauffman (1986); K.Nagel (2002); Nagel K., M.Schreckenberg (1992); Helbing D. (2001); Blue V., Adler J. (1999); M.Stepantcov (1998); P.M.A.Sloot, A.G.Hoekstra (2000); H.Klupfel (2003); S.Bandini (2006); S. El Yakobi (2006)

Etc.

BASIC IDEAS

In classical CA:

Regularity:

Discret space

Discret time

Discret states of elements

Dynamics:

Local neighborhood,

Step-by-step rules,

Deterministic rules of CA

or

Probabilistic rules of CA

Game „Life‟by J.Conway

Cells create a lattice.

Local aspects

Neumann‟s and Moor‟s neighboorhood

Rule of CA „Life‟ (1)

The states of each cell take two values

0 or 1 which correspond to „dead' or

„living‟ cell.

The state of the cell is defined by

conditions of the neighbour cells by

rule:

At a time t let some subset of the cells in the

array are living. The living cells at time t+1 are

determined by those at time t according to the

following evolutionary rules:

Rule of CA „Life‟ (1, continue)

1. If a live cell has either two or three live neighboors, it will survive in the next time step, otherwither it will die.

2. If a dead cell has exactly three live neighboors, there will be a „birth‟ at next time step

All „bírth‟ and „death‟ take place simultaniously.

Example

Oscillator

Example

Some solutions (Gosper glider gun):

CA description I.

DEFINITIONS (S.Wolfram; J.-P. Allouche,

M.Courbage and G.Scordev; G.Hedlund)

Zd = the d-dimensional lattice

S = the finite set of states of single element (cell)

on the lattice

si in S is the state of i-th cell from Zd (i- index of

cell)

A configuration on the lattice Zd is a collection of

states of all cells at the same moment of time

All possible configurations constitute the space

of configurations C on Zd

CA description II.

Let T={0, 1,2, …} is a discretization in time and C(t) –configuration of the system at moment of time t (t=0, 1, 2, ….)

The local rule for cell k on the Zd is the transformation Tk which transforms the state sk(t) in S of cell k at moment t to the state sk(T+1) in S of the same cell at moment (t+1).

sk(t+1)=Tk({sk(t)}, Nk, R) ,

where Nk – some neighboorhood of cell k on the lattice Zd; {sk(t)} is the set of cell‟s states within Nk,

the transformation Tk result depends only on the states of elements within the neighboorhood N

CA description III.

The collection of local transformations Tk define the global transfomation G on the configuration space C

C(t+1)=G(C(t));

The initial data C(0) configuration is defined at initial moment t=0

The set of transformations {Tk} or transformation G define the cellular automata on the lattice Zd

with the cell‟s state space S

CA examples

Example 1. 1D cellular automata (on the line)

S={0, 1}

Zd = Z , Z – integer numbers

C – space of all binary strings

Nk = [k+l, …, k+2, k+1, k, k-1, k-2,…, k-l],

CA examples

Example 2. (Game „Life‟)

S={0,1}

Zd=ZxZ – rectangular grid on the plane

C – two-dimensional matrix constituted from 0 or

1

Nkj = Nk1xNj2,

Nk1=[k+l, …, k+1, k, k-1, …, k-l] ;

Outline of this talk 1.

1. Description of a model of pedestrain

movement as a source for further applications

and new problems extracting.

A) 2D model with probabilistic properties

B) Examples of applications for crowds movement

C) Modeling of migration: example of CA application

Outline of this talk 2.

2. Statement and discussion of new problems:

A) Some ways for mentality accounting in

elements which represent pedestrains

B) Discussion on possible optimization

problems

Outline of this talk 3.

3. The anticipatory property and its

consequences for scenarious analysis and

decision – making

A) Anticipation (R.Rosen; D.Dubois etc.)

B) Game „Life‟with anticipation

C) Multivaluedness and decision-making

End of part A

B. Model description of crowd

movement

The models follows to the approach

from the paper by

K.Nagel and M.Shreckenberg, (1992) A

cellular automation model for freeway

traffic. J.Phys. I France, 2,: 2221 – 2229

(see Helbing D. (2001); Blue V., Adler J.

(1999); M.Stepantcov (1998); H.Klupfel

(2003) etc.).

A. Some models of cars and

pedestrians motion

Differential equations (since M.Lighthill, G.Whithem, 1959)

Master equations and kinetic equations (W.Weidlich; G.Haag, 1983; D.Helbing)

Active Brownian particles (F.Schweitzer, W.Ebeling, B.Tilch,

1999)

Multi-agent systems (M.Wooldridge, N.Jennings, N.Gilbert; K.Troightz, W.Jager etc.)

Cellular automata (CA)

(J.Conway; S.Wolfram; G.Hedlund; S.Kauffman;

Model as CA

The models are from the class of cellular automata above

S={0, 1}

Zd=Z2

N (neighboohood) – Moor‟s or Neumann

The model is probabilistic – that is the rules have probabilities components

The rules correspond to possible movements of single pedestrain in dependence on local environment

Problem description and

assumptions I.

Let's consider movement of people

(particles) on a plane which part is occupied

by impassable obstacles.

The lattice of the cellular automata is the

orthogonal grid which sets four (in case of

Neumann's neighbourhood) or eight (in case

of Moore's neighbourhood) possible

directions of movement (along lines of a

grid).

The state of the cell corresponds to presence

or absence of the particle (pedestrian) in the

given cell.

Problem description and

assumptions II.

All models are discrete in space and

time.

Route-choice is pre-determined.

The irrational behaviour is rare.

Persons are not strongly competitive,

that is, they don‟t hurt each other.

Individual distinctions can be

represented by parameters determining

the movement behaviour.

Ilustration to geometry of searching

The black disks in the squares represents

the pedestrians

Case of Neumann-type neighboorhood

(the black cells – obstacles, gray cells

correspond to searching of neighboorhood

of given pedestrian)

Some rules of CA approach to

crowd movement I.

Each particle in group wishes to move

in the certain direction. If it is

impossible to move in this direction

(presence of obstacles or other person)

the particle will try to change a

direction of movement keeping the

basic direction.

Each particle can move with the certain

speed which can be no more than the

greatest possible - vmax.

Some rules of CA approach II.

The lattice of the cellular automata represents set of two rectangular matrixes (F; V),

where F is a matrix of values f(i,j), where f(i,j) from {0; 1} is a value which accords to the presence (1) or absence (0) of pedestrians in the given cell.

V is a matrix of values v(i,j), where v(i,j) from {0,1} is a value which accords to the presence (1) or absence (0) of obstacles in the given cell.

Some rules of CA approach III.

The model description is done for Neumann's

neighbourhood relation (the change of the cell

condition is influenced by four its neighbours;

the cell‟s position in Neumann‟s neighbours is given

by the first letters of the parties of the world: N, W, C,

E, S. (The letters correspond to next directions:

„north‟, „west‟, „south‟, „east‟ and „centre‟ places). N

W C E

S

Some rules of CA approach IV.

The entered variable α can have values N, W, C, E, S and it is accepted corresponding designations for conditions of neighbours of the chosen cell:

f(i+1,j) = f(i,j) (N), f(i,j+1) = f(i,j) (E), … ,

f(i,j)=f(i,j) (C)

Similar designations are entered for the values of elements of matrix V which are neighbours of the chosen cell.

Some rules of CA approach V.

The rules of moving from the given cell to the next one are given below (they are applied only to cells for which f(i,j) = 1).

On each step for every сell of cellular automata which contains the particle the probabilities of motion from the given position to one of the around cells are calculated.

These probabilities are equal to zero in case of the corresponding cell is occupied.

For “free” directions it is made "viewing" on distance r, it is took into account a quantity of occupied/available cells.

Some rules of CA approach VI.

First of all, it is prohibited to move to the occupied cells and cells which contain obstacles :

P′(i,j) = (1/4)(1- f(i,j)(α))(1-v(i,j)(α)) (1)

For remained directions it is made "viewing" on distance r (parameter of model): it is calculated a number of cells which lay in the given direction and have a zero-condition 0 (free).

Some rules of CA approach VII.

For realization of this it is calculated probabilities of moving to the next cells P′′(i,j), they are reduced in those directions where a lot of cells occupied by particles or obstacles:

P′′(i,j)(N)=(1 –( 1/r)(∑ f(i,j+k)+r-r* ))P′(i,j) (N)

P′′(i,j)(S)=(1 –( 1/r)(∑ f(i,j-k)+r-r* ))P′(i,j) (S)

P′′(i,j)(E)=(1 –( 1/r)(∑ f(i+k,j)+r-r* ))P′(i,j) (E)

P′′(i,j)(W)=(1 –( 1/r)(∑ f(i-k,j)+r-r* ))P′(i,j) (W)(2)

where r – a distance of particle viewing, r* - distance from the given cell to the nearest cells in the given direction which contains an obstacle, P′(i,j) (α ) - the probabilities calculated by formulas (1).

End of part B.

C. Examples of simulation

results

For evaluation of simulation results following characteristics are chosen:

(1) -density of a pedestrian stream: ρ = n / S pedestrians / cells (n- quantity of pedestrians S - square);

(2) - flow of pedestrians - j: j = ρ *v W pedestrians of cells Lengths / sec. (W- width of pass, v – velocity of movement);

(3) - average time of achievement of the goal by pedestrians: tavg = ti / n (tavg. - average time of achievement of the goal by pedestrians, ti - time of achievement of the goal by i-th pedestrian, n - quantity of pedestrians in the stream).

Example 1. Movement with

obstacles in corridor

The geometry can be presented by a

simple variant or more complex one, it

may move one or two streams of

people

Fig. 1. Movement with obstacles. The

Jam.

Example 2. Corridor with obstacles

and with corner Fig. 2. Application of the model –

investigation of the influence of

obstacles configuration in the pass.

Simulation results.

Average achievment time

Fig. 3. Dependence of average

achievement time for two pedestrian

streams from quantity of gaps in pass

141.00

142.00

143.00

144.00

145.00

146.00

147.00

0 1 2 3 4 5

quantity of gaps, nz

time,

t

2 1

Example 3. Evacuation scenerio

example (1).

A problem of evacuation of the working

personnel from office

Geometry of event Iteration 10

Iteration N100

Evacuation scenerio example

(2).

Evacuation scenerio example

(3)

1

↓

1

↓

1

↓

3

→

1

→

2

→

5 1

←

3

←

1

↓

1

↑

2

↑

1

↑

1

←

1

↓

1

↓

2

←

1

↑

1

→

1

→

4

→

5 1

←

1

↑

1

←

1

↑

2

↑

2

↑

1

←

1

↑

Example 4. Migration simulation at country:

CASE OF CAPITAL ATTRACTIVITY

End of part C

D. Optimisational aspects I.

Goals of optimisation investigations:

A. Theoretical

B. Practical

B. Optimisation problems in traffic processes

1.Searching optimal solution in normal conditios

2.Searching the evacuation ways in emergency

3.Optimal design of large objects

4. Risks evaluation

D. Optimisational aspects II.

Considered models of CA type may serve as

bacground for practical problems of many

scales:

Local design of obstacles placing in crowds

movement in evacuation from ships, trains,

buildings;

Design of safe large objects: buildings,

stadiums, new reilway and metro stations etc.

Preparing plans of evacuations in large-scales

emergencies: floodsfafts, forest fires, hurricains,

earthquecke, volcanos activities (example - region of

Vesuvium with about one million peoples in

D. Optimisational aspects III.

In theory:

The social objects, included crowds are difficult to formalise

The data is non-accurate or absent

Mentality is important in considerations

Considered CA models may help in such case:

1. Scenarious are prepared by CA models; using of genetical optimisation

2. Tolerance is the tool for reducing the calculations volume

End of part D.

E. The Problems of Mentality

Accounting in Trafficking

In Sections B and C we have presented

results restricted by the approach of

CA without special accounting of the

mentality properties for pedestrian

movements.

The accounting of mentality of

participants of social processes

(including trafficking) is one of the

main tendencies in developing more

adequate models.

Mentality accounting

There are many presumable ways of

doing such accounting

– from the attempts to model the

human consciousness and

decision – making in artificial

intelligence

to the simplest statistical rules.

Toward mentality accounting

The general questions are:

A. What? (The properties that we would like to account for in methodology)

B. How? (The approaches for formalisation and basic ideas of methodologies)

C. Where? (In what models and how to introduce mentalty into models)

PRESUMABLE RESULTS: qualitative understanding of systems and processes, quantitative modeling, forecasting, scenarios, optimisation and management

Some Possibilities

A. Behavior, choice, psichology, education

experience and memory, intelligence,…

B. Data formalisation, statistics,

questionnaire, sensor data plus modeling

concepts (econometrics, mathematical

modeling, gaming and simulation, artificial

intelligence, game theory,…)

Differential equations, statistical analysis,

multi-agent approach, cellular automata,

…

Models of neural network type

Earlier in the frame of the models with

associative memory we have found a

particular way and new prospects in

accounting and interpretation of mentality in

the models of large socio–economical

systems [15].

As the first step of mentality accounting we

suggest to incorporate the Hopfield neural

network model as the internal structure of

cells (elements).

A part of approach could be incorporated

into the CA traffic models.

Mentality aspects in movement

Of course many aspects related to the mentality accounting should be represented in the of the traffic:

monitoring and recognition of traffic situation;

decision – making process on movement direction,

velocity and goals;

possibilities of movement implementation etc.

ANTICIPATION PROPERTY

One of the most interesting properties

in social systems is the anticipation

property.

The anticipation property is the

property that the individual makes a

decision accounting the prediction on

future state of the system [15, 16].(see

R.Rosen (1985); D.Dubois (2000))

Anticipating in trafficing

Concerning the specific case of the traffic problems we stress that the anticipatory property is intrinsic for traffic.

At the local level each participant of the traffic process tries to anticipate the future state of traffic in local neighbourhood when he makes the decision on movement.

Also the macro neighbourhood of traffic participants might be accounted

End of part E.

F. CA and anticipation

The adequate accounting of

anticipatory property

in the CA methodologies is a difficult

problem because

it requires also complication of CA

models

by introducing

the internal states of CA cells and

special internal dynamical laws

for mental parameters.

New Self-organization

phenomena Self-organisation – emerging of structures in the

distributed systems (I.Prigogine; H.Haken)

Many structures are known experimentally for traffic problems: jams, spiral waves, vortices. Also some models exist (see D.Helbing, I.Prigogine etc.). But many problems are far from solutions.

Here we would like to remark some general new possibilities.

A new class of research problems is the investigation of self–organization processes in the anticipating media, in particular in discrete chains, lattices, networks constructed from anticipating elements (including the so-called ágents‟).

In such a case the main problems are self–organization, emergent of structures including

End of part F.

G. Multivaluedness and decision-

making

The outline of decision-making theory:

A. Many possibilities of system behaviour

(sometimes named scenarios)

B. Decision – making for choise of

variant(s)

C. Risks evaluations

Possibilities

A. Considering all possible variants by

testing all possible initial conditions or

calculation at least three scenarious:

optimistic, pessimistic or neutral in risk

evaluation

Normative and descriptive theories, utility

functions, artificial intelligence, behavioral

finance, stochastic concepts, etc.

Calculations of probabilities and risks.

One of presumable sources of scenarious

origin in human systems by anticipation

accounting

Possible branching of the solution of

models with anticipation in time

1 2 3 t

X

0

Decision-making and scenarios

Picture at previous slide show the set of trajectories for discrete time systems with anticipation.

Time is represented in abscissa axes. The ordinates correspond to the possible state of a single element

(but it may schematically represent multi – state of the whole system).

The thin lines correspond to all possible trajectories and

fat line corresponds for single chosen trajectory of the system.

End of part G.

References

1. Toffoli T., Margolis N.: Cellular automata computation. Mir, Moscow (1991)

2. Gilbert N., Troitzsch K.: Simulation for the social scientist. Open University press, Surrey, UK (1999)

3. Wolfram S.: New kind of science. Wolfram Media Inc., USA (2002)

4. Benjamin S. C., Johnson N. F. Hui P. M.: Cellular automata models of traffic flow along a highway containing a junction. J. Phys. A: Math Gen 29 (1996) 3119-3127

5. Nagel K., Schreckenberg M.: A cellular automation model for freeway traffic. Journal of Physics I France 2(1992) 2221- 2229

6. Schreckenberg M., Sharma S.D. (eds.): Pedestrian and evacuation dynamics. Springer–Verlag, Berlin (2001) 173-181

7. Helbing D., Molnar P., Schweitzer F.: Computer simulations of pedestrian dynamics and trail formation. Evolution of Natural Structures, Sonderforschungsbereich 230, Stuttgart (1998) 229-234

8. Thompson P.A., Marchant E.W.: A computer model for the evacuation of large building populations. Fire Safety Journal 24 (1995) 131 -148

9. Stepantsov M.E.: Dynamic model of a group of people based on lattice gas with non-local interactions. Applied nonlinear dynamics (Izvestiya VUZOV, Saratov) 5 (1999) 44-47

10. Wang F.Y. et al.: A Complex Systems Approach for Studying Integrated Development of Transportation Logistics, and Ecosystems. J. Complex Systems and Complexity Science 2. 1 (2004) 60–69

11. Ahuja R.K., Magnanti T.L., Orlin J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall (1993)

12. Kreighbaum E., Barthels K.M.: A Qualitative Approach for Studying Human Movement, Third Edition, Biomechanics. Macmillan, New York (1990)

13. Klupfel H.: A Cellular Automaton Model for Crowd Movement and Egress Simulation. PhD Thesis, Gerhard-Mercator-Universitat, Duisburg-Essen (2003)

14. Kirchner A., Schadschneider A.: Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics. Physica A 312 (2002) 260-276

15. Makarenko A.: Anticipating in modelling of large social systems - neuronets with internal structure and multivaluedness. International .Journal of Computing Anticipatory Systems 13 (2002) 77 - 92

16. Rosen R.: Anticipatory Systems. Pergamon Press, London (1985)

CA Example A

Came „Life”

Game “Life”: a brief descriptionRule #1: if a dead cell has 3 living neighbors, it turns to “living”.

Rule #2: if a living cell has 2 or 3 living neighbors, it stays alive, otherwise it “dies”.

Formalization:

x 0 1 2 3 4 5 6 7 8

f0(x)0 0 0 1 0 0 0 0 0

f1(x)0 0 1 1 0 0 0 0 0

}1,0{,1

0

),(

),()(

1

0

k

k

k

k

k

kk FC

C

Sf

SfSFF

NkSFC t

k

t

k ..1),(1

Next step function:

- state of the k-th cell}1,0{kC

Dynamics of a N-cell automaton:

t – discrete time

“LifeA” = “Life” with anticipationConway’s “Life”

NkFC t

k

t

k ..1,1

“Life” with anticipation

]1;0[),)1(( 1tk

tk

tk SSFF

)( t

k

t

k SFF

IRSSFF t

k

t

k

t

k ),( 1

weighted

additive

Dynamics:

LifeA: simulations“Life”: linear dynamics “LifeA”: multiple solutions

LifeA: simulations Multivaluedness

Multivaluedness Choice Optimal management

LifeA: simulations The number of solutions reaches its maximum after several steps

and then remains constant, while the solutions themselves may change.

CA Example B

Pedestrian crowd movement

and optimization by cellular

automata models

How anticipation can be introduced

into pedestrian traffic models?

One of the possible ways:

Supposition: the pedestrians avoid blocking each other. I.e. a person tries not to move into a particular cell if, as he predicts, it will be occupied by other person at the next step.

P1P3

P2

P4

kP )1( ,occkk PPPk – probability of moving in direction kPk,occ – probability of k-th cell of the neighborhood being occupied (predicted)

Anticipating pedestrians

Two basic variants of anticipation accounting were simulated:

)1( ,occkk PP ))1(1( ,max

occkk PvvP

and

All pedestrians have equal rights

Fast moving pedestrians have a priority

And two variants of calculation Pk,occ:

P1P3

P2

P4

P1P3

P2

P4

Observation-based

Model-based

Anticipating pedestrians:

simulations

E/P – equal rights/with priority;

O/M – observation-/model-based prediction

CA Modelling of Epydemy

(t=0) <Example C>

CA Modelling of Epydemy

(t=20)

CA Modelling of Epydemy

(t=60)

References

Makarenko A., Goldengorin B., Krushinskiy D., Smelianec N. Modeling of Large-Scale crowd‟s traffic for e_Government and decision-making. Proceed. 5th Eastern European eGov Days, Prague, Czech Republic 2007. p. 5

Makarenko A., Samorodov E., Klestova Z. Sustainable Development and eGovernment. Sustainability of What, Why and How. Proceed. 8th Eastern European eGov Days, Prague, Czech Republic 2010. p. 5 (accepted)

Makarenko A., New Neuronet Models of Global Socio- Economical Processes. In 'Gaming /Simulation for Policy Development and Organisational Change' (J.Geurts, C.Joldersma, E.Roelofs eds.), Tillburg Univ. Press. 1998. p.133- 138,

Makarenko A., Sustainable Development and Risk Evaluation: Challenges and Possible new Methodologies, In. Risk Science and Sustainability: Science for Reduction of Risk and Sustainable Development of Society, eds. T.Beer, A.Izmail- Zade, Kluwer AP, Dordrecht, 2003. pp. 87- 100.

CA Applications. EXAMPLE

SOCCER CELLULAR

AUROMATA MODELS

Some rules of players behavior in soccer

Free movements of

players

Movement toward the cell

with ball

Movement of player with

ball

Movement of near players

Transition from continuous to

discret space

Real movement and

movement in

cellular space

Some results of modeling

Diminishing of player‟s health on time

Example of modeling results

Screenshot

of game