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Spreading Phenomena Spreading Phenomena and and Excitable Media Excitable Media Christian Jacob AI AI Department of Computer Science University of Calgary CPSC 565 Winter 2003 Emergent Computing CPSC 565 Winter 2003 1

  1. Spreading Phenomena Spreading Phenomena and and Excitable Media Excitable Media Christian Jacob AI AI Department of Computer Science University of Calgary CPSC 565 — Winter 2003 Emergent Computing — CPSC 565 — Winter 2003 1 Christian Jacob, University of Calgary

  2. Two-Dimensional Two-Dimensional Cellular Automata Cellular Automata Emergent Computing — CPSC 565 — Winter 2003 2 Christian Jacob, University of Calgary

  3. Two-dimensional CA Neighbourhoods Two-dimensional CA Neighbourhoods von Neumann Moore Extended Moore r = 1 r = 1 r = 1, 2, 3 Formally, we can characterize the 2D Moore neighbourhood N ij ( r ) of radius r in a cell lattice L for a cell c ij by N ij ( r ) = {( k , l ) Œ L | | k - i | ≤ r and | l - j | ≤ r }. Emergent Computing — CPSC 565 — Winter 2003 3 Christian Jacob, University of Calgary

  4. Totalistic Cellular Automata Totalistic Cellular Automata • Consider the following 1D CA rule: 111 110 101 100 011 010 001 000 0 1 1 0 1 0 0 0 • This automaton changes c i into a 1 only if the sum of the cells c i-1 , c i , and c i+1 is exactly 2: 1, if c i -1 ( t ) + c i ( t ) + c i +1 ( t ) = 2 { c i ( t+ 1) = 0, otherwise This is an example of a totalistic rule. Emergent Computing — CPSC 565 — Winter 2003 4 Christian Jacob, University of Calgary

  5. Encoding of Totalistic Rules Encoding of Totalistic Rules • We first look at one-dimensional, binary CA rules ( k = 2) : c i ( t +1) = f ( ∑ j Œ N i(2) c i + j ( t ) ), where f : {0, 1, …, 2 r +1} Æ {0, 1}. • We can make a table of f in the form of a tuple ( f (0), f (1), f (2), …, f (2 r + 1) ). • We can encode this tuple by the following formula: C f = ∑ j=0, …, 2 r +1 f ( j ) · 2 j . Emergent Computing — CPSC 565 — Winter 2003 5 Christian Jacob, University of Calgary

  6. Encoding of Totalistic Rules (2) Encoding of Totalistic Rules (2) • With this encoding all legal rule codes are even numbers. – All 32 rules for k = 2 and r = 2 are encoded by all even numbers between 0 and 62. Note : In the previous lecture we used a different notation, with V instead of k , and K = 2 r +1. – Example: mod-2 rule Sum: 0 1 2 3 4 5 f (Sum): 0 1 0 1 0 1 C f : 0 · 2 0 + 1 · 2 1 + 0 · 2 2 + 1 · 2 3 + 0 · 2 4 + 1 · 2 5 = 42 Emergent Computing — CPSC 565 — Winter 2003 6 Christian Jacob, University of Calgary

  7. Encoding of Totalistic Rules (3) Encoding of Totalistic Rules (3) • This encoding for binary, totalistic CAs ( k = 2) C f = ∑ j=0, …, 2 r +1 f ( j ) · 2 j can be generalized to any number of k values per cell: C f = ∑ j=0, …, 2 r +1 f ( j ) · k j , where f maps to any of the k values for a neighbourhood of radius r : f : {0, 1, …, 2 r +1} Æ {0, 1, …, k }. Emergent Computing — CPSC 565 — Winter 2003 7 Christian Jacob, University of Calgary

  8. Some Example 2D Cellular Automata: 1022 1022 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 8 Christian Jacob, University of Calgary

  9. Some Example 2D Cellular Automata: 510 510 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 9 Christian Jacob, University of Calgary

  10. Some Example 2D Cellular Automata: 374 374 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 10 Christian Jacob, University of Calgary

  11. Some Example 2D Cellular Automata: 614 614 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 11 Christian Jacob, University of Calgary

  12. Some Example 2D Cellular Automata: 174 174 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 12 Christian Jacob, University of Calgary

  13. Some Example 2D Cellular Automata: 494 494 Some Example 2D Cellular Automata: 5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic 0-39 20 30 40 Emergent Computing — CPSC 565 — Winter 2003 13 Christian Jacob, University of Calgary

  14. Patterns from Seed Regions: 143954 143954 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 14 Christian Jacob, University of Calgary

  15. Patterns from Seed Regions: 50224 50224 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 15 Christian Jacob, University of Calgary

  16. Patterns from Seed Regions: 15822 15822 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 16 Christian Jacob, University of Calgary

  17. Patterns from Seed Regions: 85507 85507 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 17 Christian Jacob, University of Calgary

  18. Patterns from Seed Regions: 191044 191044 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 18 Christian Jacob, University of Calgary

  19. Patterns from Seed Regions: 93737 93737 Patterns from Seed Regions: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Single-Cell Seed Random Disordered Seed Emergent Computing — CPSC 565 — Winter 2003 19 Christian Jacob, University of Calgary

  20. Spreading Phenomena Spreading Phenomena Emergent Computing — CPSC 565 — Winter 2003 20 Christian Jacob, University of Calgary

  21. Spreading Phenomena Spreading Phenomena • Spreading is the process in which an object extends itself over an increasingly larger area by incorporating regions adjacent to itself. • The spreading or kinetic growth (KG) models describe a wide variety of natural processes: – Tumor growth – Epidemic spread – Gelation – Rumor-mongering – Fluid flow through porous media – … Emergent Computing — CPSC 565 — Winter 2003 21 Christian Jacob, University of Calgary

  22. Slow Diffusive Growth: 256746 256746 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 22 Christian Jacob, University of Calgary

  23. Slow Diffusive Growth: 736 736 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 23 Christian Jacob, University of Calgary

  24. Slow Diffusive Growth: 174826 174826 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 24 Christian Jacob, University of Calgary

  25. Slow Diffusive Growth: 175850 175850 Slow Diffusive Growth: 9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic Emergent Computing — CPSC 565 — Winter 2003 25 Christian Jacob, University of Calgary

  26. Eden Model Eden Model • The original kinetic growth model was the Eden model , introduced by the biologist M. Eden in the 1960s. • The Eden model simulates accretive growth of a cell cluster (originally for simulating the spread of tumor cells) in a square lattice by sequentially adjoining randomly selected cells from the perimeter. • In more detail: – We start with a cluster list consisting of a seed site. – A site is randomly chosen from the perimeter list, consisting of the nearest sites adjacent to the seed site (v. Neumann neighbourhood). – The cluster and perimeter lists are updated. – Another site is randomly selected from the perimeter list, and so on. Emergent Computing — CPSC 565 — Winter 2003 26 Christian Jacob, University of Calgary

  27. Variations of the Eden Model Variations of the Eden Model • We look at two variations of the Eden model: – The Invasion Percolation Model – The Single Percolation Cluster Model Emergent Computing — CPSC 565 — Winter 2003 27 Christian Jacob, University of Calgary

  28. The Invasion Percolation Model The Invasion Percolation Model • The invasion percolation model describes the flow of fluid through porous media. • Invasion Percolation Algorithm: – Each site in the perimeter list has a random number associated with it. – The cluster spreads by incorporating the perimeter site with the lowest associated random number. • This model describes a process of spreading that “follows the path of least resistance.” Emergent Computing — CPSC 565 — Winter 2003 28 Christian Jacob, University of Calgary

  29. Invasion Percolation Model: Spreading Invasion Percolation Model: Spreading Emergent Computing — CPSC 565 — Winter 2003 29 Christian Jacob, University of Calgary

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