Cellular automata models can be used to simulate social processes like crowd movement. The document describes a 2D cellular automata model with probabilistic rules to model pedestrian movement. Examples are provided of simulating movement with obstacles, evacuation scenarios, and migration patterns influenced by capital attractiveness. Optimization of these models is discussed, such as finding optimal configurations to improve traffic flow or evacuation times.

1. 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

2. 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.

3. 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

4. Game „Life‟by J.Conway
 Cells create a lattice.

5. Local aspects
 Neumann‟s and Moor‟s neighboorhood

6. 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:

7. 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.

8. Example
 Oscillator

9. Example
 Some solutions (Gosper glider gun):

10. 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

11. 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,

12. 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

13. 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],

14. 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,

15. 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

16. 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

17. 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

18. End of part A

19. 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.).

20. 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)

21. 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

22. 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

23. 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.

24. 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)

25. 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

26. 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

27. 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:
N
„north‟, „west‟, „south‟, „east‟ and „centre‟ places).
W C E
S

28. 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

29. 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

30. 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).

31. 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).

32. End of part B.

33. 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

34. 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.

35. 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.

36. Average achievment time
 Fig. 3. Dependence of average
achievement time for two pedestrian
streams from quantity of gaps in pass
147.00
146.00
145.00
time,t
144.00
143.00
142.00
141.00
0 1 2 3 4 5
quantity of gaps , nz
2 1

37. Example 3. Evacuation scenerio
example (1).
 A problem of evacuation of the working
personnel from office
 Geometry of event Iteration 10

 Iteration N100

38. Evacuation scenerio example
(2).

39. Evacuation scenerio example
(3)

40. Example 4. Migration simulation at country:
CASE OF CAPITAL ATTRACTIVITY
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
↑

41. End of part C

42. 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

43. 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.
 Preparingplans of evacuations in large-scales
emergencies: floodsfafts, forest fires, hurricains,
earthquecke, volcanos activities (example - region of

44. 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

45. End of part D.

46. 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

47. 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.

48. 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

49. 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,

50. 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

51. 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.

52. 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))

53. 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

54. End of part E.

55. 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

56. 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‟).

57. End of part F.

58. 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)

59. 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.

60. One of presumable sources of scenarious
origin in human systems by anticipation
accounting
 Possible branching of the solution of
X
models with anticipation in time
0 1 2 3 t

61. 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

62. End of part G.

63. 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)

64. CA Example A
 Came „Life”

65. Game “Life”: a brief description
Rule #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 Next step function:
f0(x) f 0 ( Sk ), Ck 0
0 0 0 1 0 0 0 0 0 Fk F (Sk ) , Fk {0,1}
f1 (S k ), Ck 1
f1(x)
0 0 1 1 0 0 0 0 0
Ck {0,1} - state of the k-th cell
Dynamics of a N-cell automaton:
t 1 t
C k F ( S ), k 1..N
k
t – discrete time

66. “LifeA” = “Life” with anticipation
Conway’s “Life” “Life” with anticipation
t t Fkt F ((1 ) S kt S kt 1 ), [0;1]
Fk F (S )
k weighted
Fkt F ( S kt S kt 1 ), IR
additive
Dynamics:
t 1 t
C k k F , k 1..N

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

68. LifeA: simulations
 Multivaluedness
Multivaluedness Choice Optimal management

69. LifeA: simulations maximum after several steps
The number of solutions reaches its
and then remains constant, while the solutions themselves may
change.

70. CA Example B
 Pedestriancrowd movement
and optimization by cellular
automata models

71. 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.
P2
P3 P1
Pk Pk (1 Pk ,occ )
Pk – probability of moving in direction k
P4
Pk,occ – probability of k-th cell of the
neighborhood being occupied (predicted)

72. Anticipating pedestrians
 Two basic variants of anticipation accounting were simulated:
and
v ) P )
Pk (1 Pk ,occ ) Pk (1 (1
vmax k ,occ
All pedestrians have Fast moving pedestrians have
equal rights a priority
 And two variants of calculation Pk,occ:
Observation- Model-based
based
P2
P3 P1
P2
P4
P3 P1
P4

73. Anticipating pedestrians:
simulations
E/P – equal rights/with priority;
O/M – observation-/model-based prediction

74. CA Modelling of Epydemy
(t=0) <Example C>

75. CA Modelling of Epydemy
(t=20)

76. CA Modelling of Epydemy
(t=60)

77. 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.

78. CA Applications. EXAMPLE
SOCCER
CELLULAR
AUROMATA MODELS

79. 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

80. Transition from continuous to
discret space
 Real movement and
movement in
cellular space

81. Some results of modeling
 Diminishing of player‟s health on time

82. Example of modeling results
 Screenshot
 of game