Cellular Automata Models of Social Processes


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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.
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Cellular Automata Models of Social Processes

  1. 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. 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. 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. 4. Game „Life‟by J.Conway  Cells create a lattice.
  5. 5. Local aspects  Neumann‟s and Moor‟s neighboorhood
  6. 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. 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. 8. Example  Oscillator
  9. 9. Example  Some solutions (Gosper glider gun):
  10. 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. 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. 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. 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. 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. 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. 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. 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. 18. End of part A
  19. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 32. End of part B.
  33. 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. 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. 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. 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. 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. 38. Evacuation scenerio example (2).
  39. 39. Evacuation scenerio example (3)
  40. 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. 41. End of part C
  42. 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. 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. 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. 45. End of part D.
  46. 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. 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. 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. 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. 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. 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. 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. 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. 54. End of part E.
  55. 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. 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. 57. End of part F.
  58. 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. 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. 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. 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. 62. End of part G.
  63. 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. 64. CA Example A  Came „Life”
  65. 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. 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. 67. LifeA: simulations “Life”: linear dynamics “LifeA”: multiple solutions
  68. 68. LifeA: simulations  Multivaluedness Multivaluedness Choice Optimal management
  69. 69. LifeA: simulations maximum after several steps The number of solutions reaches its and then remains constant, while the solutions themselves may change.
  70. 70. CA Example B  Pedestriancrowd movement and optimization by cellular automata models
  71. 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. 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. 73. Anticipating pedestrians: simulations E/P – equal rights/with priority; O/M – observation-/model-based prediction
  74. 74. CA Modelling of Epydemy (t=0) <Example C>
  75. 75. CA Modelling of Epydemy (t=20)
  76. 76. CA Modelling of Epydemy (t=60)
  77. 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.
  79. 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. 80. Transition from continuous to discret space  Real movement and movement in cellular space
  81. 81. Some results of modeling  Diminishing of player‟s health on time
  82. 82. Example of modeling results  Screenshot  of game