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GEM2505M

Taming Chaos. GEM2505M. Frederick H. Willeboordse frederik@chaos.nus.edu.sg. The Bigger Picture. Lecture 6. Today’s Lecture. Cellular Automata revisited Self-organized Criticality Scale-free Networks. Cellular Automata – the Stephen Wolfram way. ANKOS - in short

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GEM2505M

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  1. Taming Chaos GEM2505M Frederick H. Willeboordse frederik@chaos.nus.edu.sg

  2. The Bigger Picture Lecture 6

  3. Today’s Lecture • Cellular Automata revisited • Self-organized Criticality • Scale-free Networks

  4. Cellular Automata – the Stephen Wolfram way • ANKOS - in short • Doing calculations with CA • Rule 110 • Computational Equivalence • ANKOS - the principle claims

  5. A New Kind of Science (ANKOS) The book that has caused quite some stir. “For what I have found is that with the new kind of science I have developed it suddenly becomes possible to make progress on a remarkable range of fundamental issues that have never successfully been addressed by any of the existing sciences before.” (p1)

  6. Steven Wolfram • Born in 1959 in London • First paper at age 15 • Ph.D. at 20 • Youngest recipient of MacArthur ‘young genius’ award • Worked at Caltech and Princeton • Owner of Mathematica (Wolfram Research) • Fantastic publication record … until … • 1988 when he stopped publishing in scientific journals From his web site …

  7. Computing with Cellular Automata Thus far, Cellular Automata were discussed as rule-based systems and hence as simple computer programs. One could turn that around and look at a Cellular Automaton as a computing device where the initial conditions are the input and the state after some time steps the output. Let us have a look at some examples:

  8. Computing with Cellular Automata Rule 132 can be used to decide whether a number is even or odd Rule 132 Odd Odd

  9. Computing with Cellular Automata Rule 132 can be used to decide whether a number is even or odd Rule 132 Even Even

  10. Computing with Cellular Automata Rule 129 can be used to obtain the powers of two. 129 2 4 8 16

  11. CA as a Computer In the previous slides we saw that a cellular automaton can be used for computation if one has a specific rule for a specific task. That’s not very convenient. Just imagine if we would need a different computer for spell-checking, writing, printing etc. Ergo, the question arises: Are there cellular automata that can act similarly to our desktop computers?

  12. Universal Cellular Automaton A universal cellular automaton is a cellular automaton which is capable of universal computation, i.e. it can compute anything another computational device (like our PC) can compute too. Interestingly enough, one of the elementary cellular automata, rule 110, can be proven to be universal. In other words, it is possible to configure the initial conditions of rule 110 such that it can do any computation that is theoretically possible.

  13. Rule 110

  14. Rule 110 After starting from random initial conditions There are many localized structures that interact in various ways. The idea is to use these structures to build up blocks that can be used for computations.

  15. Rule 110 Implications Wolfram believes (and I think he is right to do so) that the discovery of such a simple system displaying universality is very significant. Why? It makes it quite conceivable that many systems, including many natural systems are universal.

  16. Self-organized Criticality Background Basically, traditional Physics is reductionist. That is to say, it assumes that the whole can be understood by its parts. Why did the big bang not lead to a nice gas (just think of it, if you put some oxygen molecules into an empty bottle they will not form ‘oxygen’ galaxies)

  17. Self-organized Criticality Background Sometimes, the parts can explain the whole extremely well. E.g. crystals, gasses This is due to their uniformity. Nature around us, however, is complex. Why is that so?

  18. Self-organized Criticality What is it? In the theory of self-organized criticality, it is argued that the complexity in nature is an effect of the tendency of systems with many parts to evolve into what is called a “critical” state. That is to say, the dynamical interactions among the elements of the system automatically and without outside intervention drive it towards that critical state.

  19. Self-organized Criticality What is critical? But then, what is a critical state? Let us look at an example. Dominos: Take a diamond grid and randomly place domino pieces on a given fraction of the total number of grid point.

  20. Self-organized Criticality After placing the dominos randomly on the grid. E.g. the blue squares below. Knock the dominos on the bottom row over and see what happens. Setup End result

  21. Self-organized Criticality Super-critical Sub-critical If we place a great many dominos: If we place only a few dominos:

  22. Self-organized Criticality Critical If we place not too many and not too few dominos:

  23. Self-organized Criticality Sandpile A great example from nature are piles of granular materials like sand piles of rice piles. Mostly, small perturbations have no or little effect. But sometimes, big avalanches can occur.

  24. Self-organized Criticality What is complex? According to Per Bak: “I will define systems with large variability as complex” (How Nature Works, p. 5) Complexity Theory A theory of complexity can explain why there is a certain variability but not what the outcome of a system will be. Hence a theory of complexity is abstract and probabilistic.

  25. Power Laws Big event are rare but small events are common. A power law is obtained when one observes a straight line in a plot of ‘the number of events’ versus ‘how often they occur’. Speaking of Internet…. From: http://ginger.hpl.hp.com/shl/papers/ranking/ranking.html

  26. Scale-free Networks A nice example of a big network is the global communication network. Even though we know that it functions quite well, it is made of large numbers of different types (both physically as well as software-technically) of nodes and connections. Nodes Routers Satellites Computers Connections Cables EM-Waves

  27. Scale-free Networks Internet map

  28. Scale-free Networks Metabolic Networks Archaea Eukaryotes Bacteria H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000)

  29. Robustness node failure Scale-free Networks 1 Robustness S Complex systems maintain their basic functions even under errors and failures fc 0 1 Fraction of removed nodes, f

  30. Scale-free Networks failure Achilles Heel attack Internet Protein network An attack against a well chosen node leads to rapid collapse of the network! R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

  31. Key Points of the Day • Simple Rules. • Amazing Dynamics!

  32. Think about it! • Am I self-organized?

  33. References http://mathworld.wolfram.com/ http://www.nd.edu/~networks/

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