Complexity and
Self-Organized
Criticality
PJ Lamberson
University of Michigan
pjlamber@umich.edu
kkollman@umich.edu
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A good definition of complex systems
is hard to come by.
“I will define systems with large
variability as complex.”
- Per Bak, How Nature Works
“About thirty years ago, in the infancy of the
computer era, there was a rather extensive effort,
known as limits to growth, that had the goal of making
global predictions. The hope was to be able to
forecast, among other things, the growth of the
human population and its impact on the supply of
natural resources. The project failed miserably
because the outcome depended on unpredictable
factors not explicitly incorporated into the program.
Perhaps predictions on global warming fall into the
same category, since we are dealing with long-term
predictions in a complex system, even though we
have a good understanding of the physics of weather.”
- Per Bak, How Nature Works
“At most, the theory can explain why there is
variability, or what typical patterns may emerge, not
what the particular outcome of a particular system
will be.”
- Per Bak, How Nature Works
“Any given outcome of this model would differ from
others depending upon the random features inherent
in the rules. This does not mean that the model
makes no predictions. If we ran the model a hundred
or a thousand times, we would get a distribution of
outcomes - a cloud ... When we compare the
outcome of this model to a real-world situation, we
would be comparing the distribution of outcomes
produced by the model t another distribution of
outcomes produced by the real world - comparing
clouds to clouds.”
- Miller and Page, Complex Adaptive Systems p. 75
A good definition of complex systems
is hard to come by.
It’s clear that emergence is a crucial part of the puzzle.
“The hallmark of emergence is the sense
of much coming from little.”
- John Holland, Emergence
Emergence:
A macro level pattern that is not obvious from
the micro level interactions that lead to it.
Where is the problem with this definition?
obvious is subjective
Conway’s Game of Life
Live
Dead
Conway’s Game of Life
Each cell interacts with its 8 neighbors
Conway’s Game of Life
live cell with less than 2 live neighbors
• Any
dies (loneliness).
live cell with more than 3 live neighbors
• Any
dies (overcrowding).
live cell with 2 or 3 live neighbors
• Any
survives unchanged.
dead cell with exactly 3 live neighbors
• Any
comes to life.
Emergence
The game can produce:
•static patterns (still-lifes)
•repeating patterns (oscillators)
•translating patterns (spaceships)
•patterns that emit spaceships (guns)
•patterns that move and leave a trail (puffers)
•patterns that move and emit spaceships (rakes)
Power Laws
• “Big events” happen rarely.
events” happen much more
• “Small
frequently.
Power Laws
!
f(x)=cx
log f(x)=log c + ! log x
So, the graph of f(x) is
linear on a log-log scale.
Power Laws
a. Words in Moby Dick
b. Scientific citations
c. Website hits
d. Bestselling books
e. Telephone calls received
f. Earthquake magnitudes
g. Moon crater diameters
h. Solar flare intensity
i. War intensity
j. Individual wealth
k. Family name frequency
l. City population
“Darwin and Einstein correspondence patterns”
João Gama Oliveira and Albert-László Barabási
Nature 437 (2005)
Preferential Attachment
http://expertvoices.nsdl.org/cornell-info204/2007/04/15/networks-the-phenomenon-of-the-rich-get-richer/
The Web in Africa
http://www2002.org/CDROM/poster/164/
Firm Sizes
“Zipf Distribution of U.S. Firm Sizes”
Robert L. Axtell, et al.
Science 293, 1818 (2001)
Preferential Attachment
(The Yule Process)
Preferential Attachment
(The Yule Process)
Let us start with a single page, with a link to itself.
At each time step, a new page appears, with
outdegree 1. With probability ! < 1, the link for the
new page points to a page chosen uniformly at
random. With probability 1"!, the new page points
to page chosen proportionally to the indegree of the
page. Then the indegree (or outdegree) distribution
of the resulting web follows a power law.
Yule Process 2
Suppose that species appear, but never die. Species
are added to genera by splitting into to species.
Suppose this happens randomly at a constant rate, so
that a genus with k species gains new species at a
rate proportional to k. Every m speciation events, a
new genus is formed. Then the distribution of the
number of species per genus follows a power law.
Typing Monkeys
Typing Monkeys
A monkey types randomly on a keyboard with n
characters and a space bar. A space is hit with
probability q; all other characters are hit with equal
probability (1" q)/n. A space is used to separate
words.
Random Walks
(Gambler’s Ruin)
Imagine a gambler that starts a night of gambling with
no money. The friendly house lets him play his first
game for free, but kicks him out if he ever runs out
of money again. He plays a simple game: the
probability of winning is .5, and each time he wins he
gets a dollar, each time he loses he pays a dollar. The
probability distribution for the length of time the
gambler plays before running out of money is a
power law with exponent 3/2.
Percolation
(Forest Fires)
At the critical point (.59274...), the distribution of
cluster sizes follows a power law.
Self-Organized
Criticality
(Forest Fires 2)
Trees grow instantaneously at some constant rate.
Occasionally, (at some constant probability), lightning
strikes and starts a forest fire if it hits a tree. The
size of clusters and the size of fires converges to a
power law.
Trust me, anyone can come up with a power law
model. The challenge is figuring out how to show
your model is actually right.
-Michael Mitzenmacher
http://geomblog.blogspot.com/2005/10/darwins-and-einsteins-email.html