Complex Systems Winter 1390 - 2011

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2. 2 Complexity Theory is a new area of science which some people have said could change our lives as much as Michael Faraday's discovery of electricity and its properties. Professor Stephen Hawking has said, "Complexity will be the science of the 21st Century."

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6. 6 Although the bulk of the development has been in the last 30 years or so, there are people whose work foreshadowed the understandings we are now developing.

7. Back around 1870 the King of Sweden announced a mathematical competition offering a prize for the person who could calculate the three body problem. When two celestial bodies are in motion with one in orbit around the other, we simply need to use Newton's Laws of motion to understand and predict their motion. 7

8. When a third body is added, so one body orbits around a central body and the third body orbits the second body as in the case of the moon, earth and sun, then calculated where the bodies will be becomes far more complicated. Newton's Laws have been sufficient to enable us to get humans to the moon, but a fully accurate solution to the three body problem is not as straight forward. 8

9. In fact, Henri Poincar� (1854 -1912) was able to prove that the three body problem could in fact not be solved. As soon as the earth moves, it changes the distances between the other bodies, which alters the gravitational forces. All three bodies interact with each other in such complicated ways as to defy calculation. 9

10. If we cannot even calculate �the motions of three bodies, how can we possibly predict the outcome of systems we see about us everyday with millions, trillions or more of intensely interacting parts? 10

11. In the 1940's the field of cybernetics developed.� Louis Kauffmann, President of the American Society for Cybernetics, defined Cybernetics as the study of systems and processes that interact with themselves and produce themselves from themselves. Complexity Theory has been one of those disciplines that has taken inspiration from it,� but continued to develop in it's own way. 11

12. Ludwig von Bertalanffy was one of the prime movers of General Systems Theory, which developed around the same time.� He emphasized the fact that the traditional closed system could not explain the types of systems that are found about us in our world. 12

13. his work influenced cybernetics and obviously points to the work done on dissipative systems. General Systems theory emphasizes holism over reductionisms and organism over mechanism. 13

14. Von Bertalanffy saw his work as particularly relevant to social systems and has been used in the fields of anthropology, economics, political science and psychology. Margaret Mead and Gregory Bateson helped develop General Systems Theory in the social sciences. 14

15. Ludwig Von Bertalanffy Homeschooled in biology, Von Bertalanffy attended meetings of the Vienna Circle � a group of scientists and philosophers devoted to logical positivism organized by Morris Schlick, Professor of Philosophy at the University of Vienna. Although opposed to the positivist philosophy of science, Von Bertalanffy earned a Ph.D. with Schlick, and became professor of biology at the University of Vienna. His first books were devoted to theoretical biology (1928, 1932). Around the time of Von Bertalanffy's immigration to North America in 1950, his ideas coalesced into GST, in which all systems were considered similar, whether physical, biological, or social. The main ideas of GST were holism, organicism, and open systems, as found in the biological sphere, but Von Bertalanffy generalized grandly to social systems, and world cultural history. On arriving in North America after 1950, Von Bertalanffy moved about widely, visiting Stanford in 1954. At Stanford, with Ralph Gerard, Kenneth Boulding, and Anatol Rapoport, Von Bertalanffy created the Society for General Systems Research (SGSR) in 1956. (Davidson, 1983) 15

16. In the 1960s Meteorologist Ed Lorenz was using an early computer to run a simulation of the weather. One day, when he was rushed for time, he set the computer to round off the numbers to be calculated so a result would be found sooner. He was expecting that the rounding off would have little or no effect on the final results. 16

17. However, surprisingly, what he found was that the final results were dramatically different. He found small changes in the state of a system can cause major changes in the final output (sensitivity to initial conditions). We had been used to thinking large changes need large forces. He found that small forces could have large effects. This has become known as the butterfly effect. 17

18. It has been said (although it is an exaggeration) that a butterfly flapping its wings in Hong Kong could cause a tornado in Texas. The picture above is the mathematical depiction of the attractor he found investigating the weather and is known as the butterfly attractor. 18

19. In the 1980s, Benoit Mandelbrot used a home computer to mathematically create what he was to call fractals. He found the Mandelbrot Set in 1980.� A fractal is a shape that is self similar, that is that repeats the same basic shape at smaller levels within the same structure. 19

20. for example look at a fern and you will find that the sub branches have the same basic shape as the whole fern and the sub branches off the sub branches also have the same basic shape. 20

21. Complexity has turned out to be very difficult to define. The dozens of definitions that have been offered all fall short in one respect or another, classifying something as complex which we intuitively would see as simple, or denying an obviously complex phenomenon the label of complexity. Moreover, these definitions are either only applicable to a very restricted domain, such as computer algorithms or genomes, or so vague as to be almost meaningless. 21

22. But it seems that there is a common, "objective" core in the different concepts of complexity. Let us go back to the original Latin word complexus, which signifies "entwined", "twisted together". This may be interpreted in the following way: in order to have a complex you need two or more components, which are joined in such a way that it is difficult to separate them. 22

23. Similarly, the Oxford Dictionary defines something as "complex" if it is "made of (usually several) closely connected parts". Here we find the basic duality between parts which are at the same time distinct and connected. Intuitively then, a system would be more complex if more parts could be distinguished, and if more connections between them existed. 23

24. The aspects of distinction and connection determine two dimensions characterizing complexity. Complexity can only exist if both aspects are present: neither perfect disorder (which can be described statistically through the law of large numbers), nor perfect order (which can be described by traditional deterministic methods) are complex. It thus can be said to be situated in between order and disorder. 24

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26. Definition of complexity as midpoint between order and disorder depends on the level of representation: what seems complex in one representation, may seem ordered or disordered in a representation at a different scale. For example, a pattern of cracks in dried mud may seem very complex. When we zoom out, and look at the mud plain as a whole, though, we may see just a flat, homogeneous surface. When we zoom in and look at the different clay particles forming the mud, we see a completely disordered array. 26

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32. Complexity can then be characterized by lack of symmetry or "symmetry breaking", by the fact that no part or aspect of a complex entity can provide sufficient information to actually or statistically predict the properties of the others parts. This again connects to the difficulty of modeling associated with complex systems. 32

33. Havel (1995) calls a system "scale-thin" if its distinguishable structure extends only over one or a few scales. For example, a perfect geometrical form, like a triangle or circle, is scale-thin: if we zoom out, the circle becomes a dot and disappears from view in the surrounding empty space; if we zoom in, the circle similarly disappears from view and only homogeneous space remains. A typical building seen from the outside has distinguishable structure on 2 or 3 scales: the building as a whole, the windows and doors, and perhaps the individual bricks. A fractal or self-similar shape, on the other hand, has infinite scale extension: however deeply we zoom in, we will always find the same recurrent structure. 33

34. A complex system is a system composed of interconnected parts that as a whole exhibit one or more properties (behavior among the possible properties) not obvious from the properties of the individual parts. 34

35. "A system comprised of a (usually large) number of (usually strongly) interacting entities, processes, or agents, the understanding of which requires the development, or the use of, new scientific tools, nonlinear models, out-of equilibrium descriptions and computer simulations." 35

36. A system that can be analyzed into many components having relatively many relations among them, so that the behavior of each component depends on the behavior of others. 36

37. "A system that involves numerous interacting agents whose aggregate behaviors are to be understood. Such aggregate activity is nonlinear, hence it cannot simply be derived from summation of individual components behavior." 37

38. complex system is a set of elements connected in order to perform a unique function that cannot be achieved by any of the parts alone. In their view, a complex system may be approached at different levels of abstraction, each with its own techniques for problem-solving. 38

39. A system is complex when it is composed of a group of related units for which the degree and nature of the relationships is imperfectly known. In this case, emergent behavior is difficult to predict, even when the behavior of every subsystem is readily predictable. 39

40. A more rigorous definition for complexity is proposed by Bob Rosen and Don Mikulecky, professors of Physiology at the Medical College of Virginia Commonwealth University. 40

41. Rosen and Mikulecky define complexity as �the property of a real world system that is manifest in the inability of any one formalism being adequate to capture all its properties�. That is, researchers single out a small part of the natural system and convert it to a formal system. However, the world is complex and any formal system they choose to try to capture what happens in the real world can only be partially successful. They argue that complexity theory was born out of the need to come up with explanations for aspects of real world systems that the Newtonian paradigm is thus unable to encompass, given its reliance on Cartesian Reductionism. 41

42. Emergence Emergence is the formation of complex but regular patterns from the interaction of the many simple parts of a system. The emergent collective behaviour of a system cannot be predicted merely by understanding its individual elements, or from understanding the interactions between these elements, but it can in principle be predicted by seeing how all the elements work together. It is this element of regularity in the emergent behaviour that distinguishes complex systems from complicated and chaotic systems. 42

43. 43 Self organization Self organization is closely related to emergence and refers to the ability of the system to organize itself. The emergent features of the system appear spontaneously. There is no one in control of the system.

44. Many strongly interdependent variables, with multiple inputs contributing to observed outputs 44

45. Chaotic behavior: extreme sensitivity to initial conditions, fractal geometry, and self-organized criticality 45

46. Feedback loops: Where change in a variable results in either an amplification (positive feedback) or a dampening (negative feedback) of that change 46

47. A non-Gaussian distribution of outputs: often where outcomes that are far away from the average are more likely than you might think. Estimates and predictions of system future behavior, particularly Gaussian estimates, formed by observations collected over short time periods provide an incorrect picture of large-scale fluctuations. 47

48. Complexity theory studies and analyzes complex systems and aims at understanding their structure and behavior. A complex system is characterized by emergent behavior resulting from the interaction among its parts and, for that reason it cannot be fragmented without losing its identity and purposefulness. 48

49. Nature is visually complex. Capturing and reproducing that complexity in synthetic imagery is one of the principal research problems in complexity Theory. Fractal geometry is a potent language of complex visual form. It is wonderful in that it reduces much of the staggering complexity we see in Nature to some very simple mathematics. 49

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51. Definition: A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. 51

52. Fractal: a complex object, the complexity of which arises from the repetition of a given shape at a variety of scales. Fractal: A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature. 52

53. Characteristics: A fractal often has the following features: It has a fine structure at arbitrarily small scales. It is too irregular to be easily described in traditional Euclidean geometric language. It is self-similar It has a simple and recursive definition. 53

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56. Fractal-generating programs: There are many fractal generating programs available, both free and commercial. Some of the fractal generating programs include: Apophysis - open source software for Microsoft Windows based systems Electric Sheep - open source distributed computing software Fractint - freeware with available source code Sterling - Freeware software for Microsoft Windows based systems SpangFract - For Mac OS Ultra Fractal - A proprietary fractal generator for Microsoft Windows based systems XaoS - A cross platform open source realtime fractal zooming program 56

57. Apophysis is an open source fractal flame editor and renderer for Microsoft Windows: 57

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62. Why has the problematic of complexity appeared so late? Classical science rejected complexity in virtue of three fundamental explanatory principles: 62

63. The principle of universal determinism, illustrated by Laplace�s Daemon, capable, thanks to his intelligence and extremely developed senses, of not only knowing all past events, but also of predicting all events in the future. The principle of reduction, that consists in knowing any composite from only the knowledge of its basic constituting elements. The principle of disjunction, that consists in isolating and separating cognitive difficulties from one another, leading to the separation between disciplines, which have become hermetic from each other. 63

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