GEM2505M

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Taming Chaos

• GEM2505M

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
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The Bigger Picture

• Lecture 6

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Todays Lecture

• Cellular Automata revisited
• Self-organized Criticality
• Scale-free Networks

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Cellular Automata the Stephen Wolfram way

• ANKOS - in short
• Doing calculations with CA
• Rule 110
• Computational Equivalence
• ANKOS - the principle claims

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




































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




































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Computing with Cellular Automata
Rule 129 can be used to obtain the powers of two.
129
2
4
8
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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.
Thats 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?
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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.
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Rule 110
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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.
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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.
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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)
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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?
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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.
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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.
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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
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Self-organized Criticality
Super-critical
Sub-critical
If we place a great many dominos
If we place only a few dominos
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Self-organized Criticality
Critical
If we place not too many and not too few dominos
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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.
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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.
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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
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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
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Scale-free Networks
Internet map
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Scale-free Networks
Metabolic Networks
Archaea
Bacteria
Eukaryotes
H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and
A.L. Barabasi, Nature, 407 651 (2000)
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Robustness
Scale-free Networks
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Robustness
S
Complex systems maintain their basic functions
even under errors and failures

fc
0
1
Fraction of removed nodes, f
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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)
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Key Points of the Day

• Simple Rules.
• Amazing Dynamics!

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Think about it!

• Am I self-organized?

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References
http//mathworld.wolfram.com/
http//www.nd.edu/networks/