Jfromm: New page: The Butterfly Effect is a (poetic) phrase that encapsulates the observation that local influcences can have far-reaching effects: a small event on one side of the world can have a direct...

2008-08-30T10:57:28Z

New page: The Butterfly Effect is a (poetic) phrase that encapsulates the observation that local influcences can have far-reaching effects: a small event on one side of the world can have a direct...

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The Butterfly Effect is a (poetic) phrase that encapsulates the observation
that local influcences can have far-reaching effects: a small event on one side
of the world can have a direct impact on a large event happening on the other side.
The size of the effect is much bigger than the size of the cause.
This is possible due to amplifying, strengthening or expansive forces, positive feedback
or epidemic spreading, which can blow up a tiny microscopic event to a large-scale
macroscopic effect.
A more technical term or notion is ''sensitive dependence on initial conditions''
in chaos theory. It is based on the idea that small variations in the initial conditions of a dynamical
system produce large variations in the long term behavior of the system.
The slightest change in one minor place causes
huge major differences elsewhere, local disturbances
can have an effect on global scales.

== History of the Term ==

Ray Bradbury seems to be the first who described the
Butterfly Effect in his short story "A sound of thunder"
(which can be found in his book "The Golden Apples
of the Sun", Simon & Schuster, 1999).
The story revolves around a group of trime travellers,
who travel back through time to kill a T-Rex. The
man from the company ("Time Safary, Inc.") which
carefully arranges the trip explains why nobody is
allowed to change anything during the visit, except
the T-Rex, which has been selected and marked before
in another time travel because he will die anyway:

"We don't want to change the Future. We don't belong here
in the Past. [..] Not knowing it, we might kill an
important animal, a small bird, a roach, a flower even,
thus destroying an important link in a growing species.
[..] Say we accidentally kill one mouse here. That means
all the future families of this one particular mouse
are destroyed. [..] And all the families of the families
of that one mouse! With a stmap of your foot, you
annihilate first one, then a dozen, then a thousand,
a million, a billion possible mice! [...] What about
the foxes that'll need those mice to survive? For want
of ten mice, a fox dies. For want of ten foxes, a lion
starves. For want of a lion, all manner of insects,
vultures, infinite billions of life forms are thrown
into chaos and destruction.
Eventually it all boils down to this: fifty-nine million
years later, a cave man, one of a dozen on the entire world,
goes hunting wild boar or saber-tooth tiger for food. But
you have stepped on all the tigers in that region. By
stepping on one single mouse. So the cave man starves.
And the cave man, please note, is not just any expendable
man, no! He is an entire future nation. From his loins would
have sprung ten sons. From their loins one hundred sons,
and thus onward to a civilization. Destroy this one man,
and you destroy a race, a people, an entire history of
life. It is comparable to slaying some of Adam's
grandchildren. The stomp of your foot on one mouse, could
start an earthquake, the effects of which could shake our
earth and destinies down through time, to their very
foundations. With the death of that cave man, a billion
others yet unborn are throttled in the womb. Perhaps
Rome never rises on its seven hills. Perhaps Europe
is forever a dark forest, and only Asia waxes healthy
and teeming. Step on a mouse and you crush the Pyramids.
Step on a mouse and you leave your print, like a Grand
Canyon, across Eternity. Queen Elizabeth might never be
born, Washington might not cross the Delaware, there might
never be a United States at all. So be careful."

Despite all the warnings, the main character Eckels
gets frightened when he sees the T-Rex and stomps
accidentally on a butterfly during his hasty retreat.
As the group arrives in the present time again, they
notice subtle changes everywhere, and this is the end
of the story:

"Eckels felt himself fall into a chair. He fumbled crazily
at the thick slime on his boots. He held up a clod of dirst,
trembling. "No, it can't be. Not a little thing like that.
No!" Embedded in the mud, glistening green and gold and black,
was a butterfly, very beautiful, and very dead. "Not a little
thing like that! Not a butterfly!" cried Eckels. It fell to
the floor, an exquisite thing, a small thing that could
upset balances and knock down a line of small dominoes and
then big dominoes and then gigantic dominoes all down the
years across Time. Eckels' mind whirled. It couldn't change
things. Killing one butterfly couldn't be that important!
Could it?"

== Chaos Theory ==

Does the "Butterfly Effect" from Chaos Theory has a
real effect in history ? Serious mathematical books about
Chaos Theory do not mention the term Butterfly Effect at all.
They speak of ''sensitive dependence on initial conditions'',
which means if two identical systems are started with
similar initial conditions, which vary only by a small amount,
their dynamical states will diverge from each other very quickly in
phase space. Small uncertainties or differences are amplified
enormously (exponentially) fast. This divergence is measured by the Lyapunov
exponents. Does this mean that history in general is based on
contingencies and random events ? If persons
like Gandhi and Darwin would not have been born,
would the world be different ? Not completely. Even in Chaos Theory
itself the behavior is not completely random. Exponential divergence
alone would result in a kind of "explosion", and there would be
no stable structures at all. Only if there is also an
attractive "force", visible for example through an
attractor, can stable structures emerge.

The fascinating thing in Chaos Theory is the complexity
of the strange attractors, which arises from order
in disorder or determinism in randomness or chaos.
Small variations in initial conditions do lead to
large variations in the long run, but the trajectories
stay always near the attractor. Only the motion ''on the attractor'' exhibits
sensitive dependence on initial conditions.
We can predict the motion or the path of the trajectories
for very small time scales (through corresponding differential
equations), and the average position for large time scales
(somewhere on the attractor), but we can not predict the dynamical
state for intermediate times scales (where exactly on the
attractor is not known).
In other words the future may be different in details - the
position on the attractor - but certain in general
aspects.

It depends on
your point of view: if you watch only at a Poincare cut
through the phase space, the behavior seems chaotic and
unrelated. If you look at the complete picture in the
right dimensions, the periodic behavior can
be explained well by a complex strange attractor.
This is what we call [[Complexity|complexity]]: order in chaos,
simplicity in intricacy, regularity in irregularity
or unity in diversity. Although we can not predict it,
the weather is not completely chaotic. It is complex.
There are certain regularities and irregularities.

== Butterfly Effect and the Weather ==

The butterfly effect has been most commonly associated with the Weather system.
Weather prediction is an extremely difficult problem:
it is not easy to construct a mathematically model of 5 million billion
tonnes of air and water vapour which make up the turbulent atmosphere
that wraps around our planet in a thin layer.
Meteorologists can predict the weather for short periods of time,
two-three days at most, but beyond that predictions are generally poor.
In weather forecasts, the error becomes very large very rapidly,
like the errors in Chaos Theory.
It is doubtful if a small local air movement can affect
the weather thousands of miles away.
It is probably not a single butterfly which causes randomness
in weather patterns, it is more the accumulated effects
of myriads of small derivations, disturbances and
influences.

Can a butterfly flapping its wings in Africa really cause a storm in Florida?
We know that there are hurricanes over Florida, which are in fact created near the
coast of West Africa, if all conditions are right. If there would be a butterly
''at the right place and the right time'', it could in fact trigger a hurricane.
Yet in most cases, the butterfly would simply cause and change nothing.
The weather is very complex, there are many amplifying and magnifying forces,
but also many constraining and damping influences.

On some scale the weather is quite predictable:
daily weather patterns on the small scale of a few hours or the next day,
seasonal weather patterns on the large scale of the seasons,
and even historic patterns on the huge time scales of ice ages.
You do not need even need a computer
for one-day forecasts. The statement "tomorrow weather
will be similar to today" is often true.
But a weather forecast two or three weeks ahead is
almost impossible to make. Yet there are certain
things and phenomena we can describe.
We can describe the short-term movement of low- and
high-pressure systems, emergent phenomena which are well-known.
We can classify and characterize zones with certain weather
by different climates.

== History and Frozen Accidents ==

{{SelfOrg}}

The Butterfly effect is related to the principle of [[Path Dependence]]
and [[Frozen Accident]]s. Path-dependence exists
when the outcome of a process depends on its past history,
and is certainly a property of complex adaptive systems.
Frozen accidents are according to Murray Gell-Mann
accidents with widespread ramifications and many
diverse consequences all traceable to one chance
event that could have turned out differently.
They seem to shape history, but in fact it is
more the other way round: history forms frozen
accidents. Branching processes and bifurcations,
e.g. through positive feedback, can turn small
fluctuations and random events into frozen accidents.

You have to look at the whole picture and observe
the complete system. In history certain things were due
to come about when they did, because their time had
come, regardless of the particular actors involved.
For example, certain individuals like Gandhi and
Darwin (although they were certainly great persons)
just happened to occupy a niche that needed to be filled.
There were successful exactly because they fit perfectly
into the role they had to play. If another individual would
have played the role, the details might have been
different, but certain events had deeper reasons
and came out of larger circumstances.
Thus the effect of the Butterfly Effect in history
in limited. There are unambiguous constraints and damping
effects in nature which prevent the complete domination of path
dependence or "Butterfly Effect". The laws of nature (at the
scale we can observe now) for instance are unique, they
existed before us and will exist after us, and they can not
be changed.

Many complex adaptive systems are often at the
[[Edge of Chaos|edge of chaos]], at the border between order and randomness.
The "Butterfly Effect" in these systems is not as
strong as in dynamical non-linear mathematical systems.
Complex systems are usually based on self-similar or
scale-free networks and structures on many scales.
A small change in initial conditions will not effect
all scales at the same time.

== Accidents and Destiny ==

When the butterfly effect is mentioned in films (for
example in Jurassic Park), it is often amusing and simplifying,
but not brain-stimulating at all. Yet there is a film named
[http://www.imdb.com/title/tt0289879/ "Butterfly Effect"]
with Amy Smart and Ashton Kutcher which
is quite different. I think it is kind of weird, intense,
interesting and a bit thought-provoking. It is also gruesome
and there is some violence, murder and sexual content:
sexual abuse, a child stabbed in the back, a dog getting burned,
a man beaten to death, a man loosing his arms and a baby which
is blown up. But this is not what the film is about. It is
about the historical version of the Butterfly Effect:
even if you could change the past, every time you would change
it even in the slightest way, the future would also be massively
effected. For complex systems in general, there is no linear
relationship between the size of the cause and the size of
the effect.

How much can you really change reality and the course of
history through a single moment, a few actions or a couple
of words? Can you really make everything better just by
knowing better? What if we could go back and change bad
decisions -- would that result in a better world for you
or for your friends ? Is a better world for all possible
or is life a zero sum game (if someone gains something
someone else will lose something), i.e. is there always
a winner and a looser due to natural selection and fierce
competition ? Which things are inevitable, predetermined,
unavoidable and inescapable destiny (things you just can
change because they were meant to be), and which things
are arbitrary, random and accidental incidents ?

These are some of the questions which the film provokes.
The main character in the film finds out, that he can
actually tap into his past by reading his journals/notebooks
and change what happened in these events. But no matter what
he tries, it is never going to end happily. Everytime he
tries to get it right for one person, the history turns out
bad for another: if he tries to help his girlfriend, her
brother goes astray, if he tries to save his mother, his
school day friend goes insane,..

The problem is you do not only change your life, your friends
and your environment, your life, your friends and your
environment are also changing you constantly. The things
in complex systems are deeply interconnected, intertwined
and interrelated. Would you like to change your life before
it has changed you? The film says it's much harder than
you think, because the effect of these changes are
unpredictable.

The concept of the "Butterfly Effect" as it is known
from Chaos Theory in general is a good metaphor for
the behavior of complex path dependent [[Multi-Agent System|Multi-Agent Systems]]
with adaptive [[Agent|agents]]. While most of
the events that happen are certainly insignificant,
there are certain key events which have profound effect
on the personal or global history. The key events which
produce a radically different different outcome are
often catastrophic, disastrous or traumatic events,
where other agents (or people) get killed or murdered,
get excluded or barred, or get finally insane and
mentally ill.. And these catastrophic key events
are mostly random, unpredictable and accidental
(otherwise you could prevent them).

Thus you can expect a scale-free or power-law distribution
for the importance and significance of events, which
is also related to arbitrariness and randomness. Most
of the events are insignificant and predetermined,
but a few events are very important and often accidental.
Can we quantify the concept of "destiny" in
[[Multi-Agent System|Multi-Agent Systems]] ?

In reality, the fine balance between inescapable destiny and
accidental incidents certainly shifts during the course
of life. The genetic setup, the kind of upbringing and
education you get, the tastes and preferences of your
parents, the particular context (the time and place)
you live in, your personal history, all this is part of
your destiny. These are constraining forces which shape
you as a person (and give you a chance to succeed) but
which also restrict your possibilities in later life.
The older you get, the more your life is determined
by restrictions, and the less is your chance to make
influential changes.

The younger you are, the more you have the freedom
to shape your life. Ironically, when your chance to
change your life is biggest, your ability to do it
is lowest. If you are young, the influence of
arbitrary, random and accidental incidents is very
high. The younger you are, the higher the influence
of random events. If your parents got accidentally
murdered, you might want to become a lawyer. If you
are treated badly as a child, you might become a criminal.
If you are traumatized, you might become more or
less mentally ill.

In this sense, life is similar to simulated annealing
(like evolutionary algorithms a generic probabilistic
heuristic approach for difficult optimization problems).
Initially there is a state with high disorder where
large changes are possible, then more and more
accidents freeze in, and finally the system becomes
more and more ordered, crystalline and fixed.
Randomness becomes destiny, accidents become fate,
and as the psychologists say, fluid intelligence
becomes crystalline intelligence.

== Tipping Point ==

The term '''critical point''' in [[System|systems]]
refers to a state near a [[Phase Transition|phase transition]]
or the [[Edge of Chaos|edge of chaos]]:
poised exactly between two kinds of organization or phases.
In social systems and [[Complex Network|complex networks]],
esp. small world networks, this point is often called '''tipping point'''
(Gladwell, 2002), the point where little things can make a big difference.
The idea of the tipping point is that tiny and apparently insignificant
changes can have huge consequences, if they happen near a threshold,
a critical value or tipping point. Above this tipping point,
a disease/infection/idea spreads explosively, below it that
point, it disappears and fades away.
The tipping point itself is where the process is
self-sustaining.
When it comes to [[Epidemic Computing|epidemics]], very tiny
influences can have startling effects.

== Chaos in Computer Performance ==

If you think the complex microchips that drive modern computers are
models of deterministic precision, think again. Their general behavior is
of course deterministic, but their concrete performance is inherently
unpredictable and chaotic, a property one normally associates
with the weather.
Intel's widely used Pentium 4 microprocessor has 42 million transistors
and the newer Itanium 2 has no fewer than 410 million. "Their
performance can be highly variable and difficult to predict," says
Hugues Berry of the National Research Institute for Information and
Automation in Orsay, France.

Berry, Perez and Temam say that chaos theory can explain
the unpredictable behaviour, see
[http://arxiv.org/abs/nlin.AO/0506030 Chaos in computer performance].
The team ran a standard program repeatedly
on a simulator which engineers routinely use to design and test
microprocessors, and found that the time taken to complete the task
varied greatly from one run to the next.
But within the irregularity, the team detected a pattern, the
mathematical signature of "deterministic chaos", a property that governs
other chaotic systems such as weather. Such systems are extremely
sensitive - a small change at one point can lead to wide fluctuations at
a later time. For complex microprocessors, this means that the precise
course of a computation, including how long it takes, is sensitive to
the processor's state when the computation began

== Butterfly Effect in Multi-Agent Systems ==

As we know from the movie [http://www.butterflyeffectmovie.com/ The Butterfly Effect] and personal experience, there are in fact decisions that create a wave or an avalanche of consequences over a lifetime. In a [[Multi-Agent System|Multi-Agent System (MAS)]] (especially in MAS with adaptive agents that learn from experience, and those with very complex environments) a few decisions of the agents will have large importance or significance,
but most of the actions will have no important effect at all. For members of social systems and humans in general some decisions are of tremendous importance (which job you will take, which subject you study, which person you marry, etc.) and can shape the entire life of the corresponding person, while most other decisions are less important. You can expect a scale-free or power-law distribution for the importance of the various actions.

The '''Butterfly Effect''' relies on positive [[Feedback|feedback ]]
and the amplification of microscopic differences.
If positive feedback exists, then local disturbances can be
amplified and small local fluctuations can have cascading
effects across the entire system. There are a few classic
examples of positive feedback in MAS:

-Schellings model of segregation.
Schelling's original model of segregation describes amplification of small fluctuations through positive feedback which results in total segregation: unsatisfactory environments lead to migration of agents, and migrating agents lead to unsatisfactory environments, etc.

-Bubbles and Buzz through imitation.
If agents imitate the behavior of other agents (for instance at electronic stock markets without considering if the behavior is appropriate or not), bubbles and buzz can arise, especially if frequent behavior is considered as good, and good behavior is imitated. Frequent behavior is imitated, and imitated behavior increase the frequency, etc.

The last example is related to W. Brian Arthur's model of increasing returns in the economy, an example
for [[Path Dependence|path-dependence]] and [[Frozen Accident]]s or lock-in. In this model,
a product is considered as good and valuable if it is frequent, and if it is valuable, it is frequently bought.
This was the case for VHS-Tapes, IBM PCs, QWERTY-Keyboards, DOS and Microsoft Windows, and in many other
systems and markets where

* imitation is useful,
* compatibility and data exchange is very important,
* the costs for obtaining, installing and learning to use a new technology are high

== Links ==

Main Wikipedia Sites

* [http://en.wikipedia.org/wiki/Butterfly_Effect Main Wikipedia Entry for Butterfly Effect]
* [http://en.wikipedia.org/wiki/Tipping_Point Main Wikipedia Entry for Tipping Point]

The Effect in Chaos Theory

* [http://www.fortunecity.com/emachines/e11/86/beffect.html The Butterfly Effect]
* [http://www.pha.jhu.edu/~ldb/seminar/butterfly.html John Hopkins University - Larry Bradley: The Butterfly Effect]
* [http://mathworld.wolfram.com/ButterflyEffect.html Butterfly Effect -- From MathWorld]
* [http://www.cmp.caltech.edu/%7Emcc/chaos_new/Lorenz.html Butterfly Effect and Lorenz Attractor]

The Movie

* [http://www.imdb.com/title/tt0289879/ IMDB Site]
* [http://www.butterflyeffectmovie.com/ Film Homepage]

== Books ==

Malcolm Gladwell, ''The Tipping Point: How Little Things Can Make a Big Difference'', Back Bay Books, 2002, ISBN 0316346624

[[Category:Basic Principles]]

Butterfly Effect - Revision history

Jfromm: New page: The Butterfly Effect is a (poetic) phrase that encapsulates the observation that local influcences can have far-reaching effects: a small event on one side of the world can have a direct...