Butterfly Effect

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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: 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

Basic Principles
Natural Selection
Red Queen Effect
Swarm Intelligence
Self-Organized Criticality
Butterfly Effect
Control Loop
Lever Point
Frozen Accidents
Path Dependence
Complexity, Simplicity
Edge of Chaos
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The Butterfly effect is related to the principle of Path Dependence and Frozen Accidents. 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, 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 "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 Systems with adaptive 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 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 systems refers to a state near a phase transition or the edge of chaos: poised exactly between two kinds of organization or phases. In social systems and 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 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 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 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 (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 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 and Frozen Accidents 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


Main Wikipedia Sites

The Effect in Chaos Theory

The Movie


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

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