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

2. 2 The Bigger Picture Lecture 6

3. GEM2505M 3 Today’s Lecture Cellular Automata revisited Self-organized Criticality Scale-free Networks

4. GEM2505M 4 Cellular Automata – the Stephen Wolfram way ANKOS - in short Doing calculations with CA Rule 110 Computational Equivalence ANKOS - the principle claims

5. GEM2505M 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. GEM2505M 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. GEM2505M 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. GEM2505M 8 Computing with Cellular Automata Rule 132 Rule 132 can be used to decide whether a number is even or odd Odd

9. GEM2505M 9 Computing with Cellular Automata Rule 132 Rule 132 can be used to decide whether a number is even or odd Even

10. GEM2505M 10 Computing with Cellular Automata 129 Rule 129 can be used to obtain the powers of two. 2 4 8 16

11. GEM2505M 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. Ergo, the question arises: Are there cellular automata that can act similarly to our desktop computers? That’s not very convenient. Just imagine if we would need a different computer for spell- checking, writing, printing etc.

12. GEM2505M 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. GEM2505M 13 Rule 110

14. GEM2505M 14 Rule 110 There are many localized structures that interact in various ways. After starting from random initial conditions The idea is to use these structures to build up blocks that can be used for computations.

15. GEM2505M 15 Rule 110 Wolfram believes (and I think he is right to do so) that the discovery of such a simple system displaying universality is very significant. Implications Why? It makes it quite conceivable that many systems, including many natural systems are universal.

16. GEM2505M 16 Self-organized Criticality Basically, traditional Physics is reductionist. That is to say, it assumes that the whole can be understood by its parts. Background 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. GEM2505M 17 Self-organized Criticality Sometimes, the parts can explain the whole extremely well. Background Nature around us, however, is complex. Why is that so? E.g. crystals, gasses This is due to their uniformity.

18. GEM2505M 18 Self-organized Criticality 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. What is it? 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. GEM2505M 19 Self-organized Criticality But then, what is a critical state? Let us look at an example. What is critical? Take a diamond grid and randomly place domino pieces on a given fraction of the total number of grid point. Dominos:

20. GEM2505M 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. GEM2505M 21 Self-organized Criticality If we place a great many dominos: Super-critical If we place only a few dominos: Sub-critical

22. GEM2505M 22 Self-organized Criticality If we place not too many and not too few dominos: Critical

23. GEM2505M 23 Self-organized Criticality A great example from nature are piles of granular materials like sand piles of rice piles. Sandpile Mostly, small perturbations have no or little effect. But sometimes, big avalanches can occur.

24. GEM2505M 24 Self-organized Criticality According to Per Bak: “I will define systems with large variability as complex” (How Nature Works, p. 5) What is complex? 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. Complexity Theory

25. GEM2505M 25 Power Laws From: http://ginger.hpl.hp.com/shl/papers/ranking/ranking.htmlhttp://ginger.hpl.hp.com/shl/papers/ranking/ranking.html 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….

26. GEM2505M 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. GEM2505M 27 Scale-free Networks Internet map

28. GEM2505M 28 Scale-free Networks H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) Archaea Bacteria Eukaryotes Metabolic Networks

29. GEM2505M 29 node failure fcfc 01 Fraction of removed nodes, f 1 S Robustness Complex systems maintain their basic functions even under errors and failures Scale-free Networks Robustness

30. GEM2505M 30 Scale-free Networks Internet Protein network failure attack R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000) Achilles Heel An attack against a well chosen node leads to rapid collapse of the network!

31. GEM2505M 31 Simple Rules. Amazing Dynamics! Key Points of the Day

32. GEM2505M 32 Am I self-organized? Think about it!

33. GEM2505M 33 References http://mathworld.wolfram.com/ http://www.nd.edu/~networks/