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AACIMP 2010 Summer School lecture by Alexander Makarenko. "Applied Mathematics" stream. "General Tasks and Problems of Modelling of Social Systems. Problems and Models in Sustainable Development" course. Part 7.More info at http://summerschool.ssa.org.ua
Citation preview
Cellular Automata Models
of Social Processes
Alexander Makarenko
Institute for applied systems analysis
NTUU „KPI”, Prospect Pobedy 37, 03056,
Kiev-56, Ukraine
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