martes, 11 de noviembre de 2008


Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system.

Emergent phenomena occur due to the pattern of interactions between the elements of the system over time.

Emergent phenomena are observable at a macro-level, even though they are generated by micro-level elements.

First and Second Order Emergence.

Emergence is one of the characteristics of a complex system and has become such a important idea in complex systems that perhaps it deserves a special mention.
Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system. Emergent phenomena occur due to the pattern of interactions (non-linear and distributed) between the elements of the system over time. One of the main points about emergent phenomena is that they are observable at a macro-level, even though they are generated by micro-level elements.
In terms of social organizations, emergent behavior is an important concept. For example, we could consider the occurrence or social norms within group emergent phenomena. Note that from a modeling point of view the identification of some behavior as being emergent depends of what has been modeled. To continue the example of social norms above, social norms *could* have been one of the elements modeled in a system. Only in the case where they were not modeled explicitly, could they be considered emergent phenomena.

Nigel Gilbert (editor of JASSS) makes an interesting comment about emergent behavior in human social organizations compared to non-human social organizations (such as a collection of ants). The differentiation between the two organizations lies in the ability to reason. Specifically, ‘people have the ability to recognize, reason about and react to human institutions, that is, to emergent features. Behavior which takes into account such emergent features might be called second-order emergence’ (as opposed to first-order emergent behavior). This has implications for modeling human social situations since we might need to model what *effect* the macro-level properties have on agents actions.

History of Complex Systems

Henri Poincaré showed the new conceptual difficulty: how a completely causal system could have indeterminate behavior.

Non-linear systems & neural networks


Distributed & self-organised systems

Artificial Life and Agent-based Societies

The theories of complex systems have been developed along three complementary, but nevertheless distinct, axes: the theory of non linear systems, the neural network approach and the theory of distributed or self organized systems.

Historically, the notion of complex systems was born at the beginning of the century when H. Poincaré worked on the equations used to predict the trajectory of planets. H. Poincaré showed that it was mathematically impossible to find an exact solution to these equations even for a system as simple as that containing three planets interacting in a non-linear fashion.

Poincaré revealed to the scientific community a new conceptual difficulty: even a completely causal system (a system where the behavioral rules are perfectly known) could have indeterminate behavior. Put another way, he showed how a simple system can explode into complex and unpredictable behavior.

The research soon faced many obstacles: one of them in the form of cultural difficulties and H. Poincaré himself reassessed the epistemological consequence of his work on non-linear systems. The point that would have intrigued H. Poincaré the most is how a perfectly determinable system, from a functional point of view, could have non-predictable behavior. Another difficulty came from the lack of computing power which would have been needed to find approximate solutions to problems that do not have exact solutions and thus explore the new field of complex systems.

Nevertheless, the school of non-linear systems brought many new conceptual and methodological insights. These contributions were not directly applicable to the study of socio-technical systems which are the systems of interest to ergonomists, designers and sociologists.

Non linear systems properties were also investigated through the development of neural networks. Neural networks researches emerge during the 50s with the perceptron. They were used in order to mimic the behavior of real neurons and to explore their classification capabilities. Due to their non linear properties, these systems have very interesting properties of classification and extrapolation which has been used as a metaphor for cognitive processes.

Later, the study of distributed and self organized systems overcame this difficulty and provided new perspectives in modeling social and cognitive systems.

Basically, the theory of distributed and self organized systems is based on the fact that a population of independent and autonomous agents interacting only locally may produce “intelligent” global behavior. The system is then said to have properties of self organization.

This approach has a long lineage beginning with the study of connectionist systems to artificial life and agent based societies. The methodological and philosophical roots of the distributed and self organized systems are drastically different from the classical analytical approaches and mainly due to the fact this paradigm does not use the concepts of representation.

The distributed and self organized approach found many applications in fields ranging from the study of micro societies (ethology) to the study of human organizations and anthropology.

Henri Poincaré, a mathematical physicist, attempted to answer the question of whether the solar system was stable forever, or if some planets would just simply drift off. This required an attempt to solve the celestial 3-body problem.
The 3 Body problem: Given 3 bodies (e.g. Sun, moon, Earth) and their initial positions and velocities, the problem is to determine the motion of the 3 bodies attracting one another according to Newtons law of gravity. Whilst the it sounds quite straightforward, the problem is surprisingly difficult to solve.
Issac Newton had solved the 2-Body problem and a solution was sought for the 3-Body problem and more generally the N-Body problem).
Given the deterministic way of thought, people believed that they could predict into the future provided they have sufficient information. Thus, given sufficient information they could easily solve the 3-Body problem.
In 1887 the King of Sweden and Norway, Oscar II, initiated a mathematical competition to celebrate his 60th Birthday in 1889. Henri Poincaré selected the 3-Body problem (actually, he considered a 9-Body problem: the then known about 8 planets plus the Sun. However, he realized that the minor components of the solar system would produce perturbations on the planets and thus the problem was closer to a 50-Body problem. He immediately saw the difficulty with this and restricted himself to the 3-Body problem.)
Poincaré was familiar with the then current algebraic techniques and their limitations. However, he started to look at the problem from a different point of view and decided to try a geometric approach. This approach was ground-breaking and although he had failed in solving the problem, he was awarded the prize.
His revolutionary work was to be published in Acta Mathematica. However, during the publication process, Edvard Phragmen (a Swedish Mathematician) noticed a serious error in Poincarés work. The editor of Acta Mathematica immediately stopped publication and asked Poincaré to review his work. With further effort, Poiincaré looked again at his data (primarily patterns on the slices in phase space) and realised that the orbit of a planet in a case such as his could not be calculated far into the future. He was shocked by the results and rewrote his paper.

After the final publication, Poincaré abandoned the 3-Body problem, although the strange results he had obtained bothered him and he made constant referrals to them throughout his lifetime. Given the lack of interest from the scientific community and the advent of computers to analyze efficiency the work, the 3-Body problem and it’s bizarre implications went out of vogue. It wasn’t until much later that the scientific community realized that Poincaré had predicted chaotic motion and broke the ground in the new field of chaos.

In order to explain how non linear systems are related to complexity, we will consider a very simple example of a non linear world: two concurrent populations of mice and cats living in a closed environment. Cats eat mice, but if they are not enough mice, cats start to starve and their population decreases allowing the mouse population to increase again. How can we simply model such an equilibrium?

In the early 1970s a mathematician, R. May, studied an equation which could approximately represent such interaction. This well known equation is called logistic equation :

Xn+1 = kXn - kX2n = kXn (1-Xn)

It says that the population of mice at year n+1 (Xn+1) is subject to two opposing trends : a growing factor (k) due to their breeding rate and a decreasing factor (-kX2n) which says that the mice population cannot grow too much because cats eat them. Note that, at this time we are not interested if this equation mimics exactly the interaction but only to its dynamic properties. We could first notice that this equation is non linear due to the corrective factor (-kXn2). Non linear means that if you double the input (Xn), you will not double the output.

Now we have our model, we can see it is perfectly deterministic (that means if we know the initial population and its growing factor k, we can always compute how the population will evolve with time. To do so, we can choose to use a time diagram, such as that shown in the leftmost figure at the top of the slide, and we will see that the mice and cats population oscillate approximately in synchrony which can easily be understood: when mice are seldom, the number of cats will decrease due to a lack of their preferred food and mice will take this opportunity to breed again, and so on. Alternatively, we can represent the same phenomena on a phase diagram which is an equivalent type of representation (the rightmost figure at the top of the slide).

Now we can analyze what happens when there is a change in the value of the mice breeding factor k.

If k is small, e.g. k=1.2 (mice do not reproduce very fast), the mice population will stabilize over the following years (Fig 2). If the growing factor increases a little bit, (k=1.5), the system behaves gently: the mice population increases in consequence. However, when k = 2.3 something new happens, Fig 2 shows that the mice population starts to oscillate between two values (oscillation of period 2); for k = 2.5, the oscillation is of period 4 which means that it takes four years for the population to come back to the same value. Finally, at k = 3, the process is no longer periodic. The mice population jumps incessantly among an infinite number of values in a way which is deterministic but cannot be predicted over a long period of time.

The bottom four diagrams show the transition from order to chaos for two populations interacting in a non linear way. Here, the mice population has been reported in relationship to its growing factor k. It is possible to see that when k=3, the mice population changes in an erratic (chaotic) way over the years.

From this simple example, we can already see some interesting results:
1) Chaotic behaviour can arise even in a very simple system. In our case, the two populations where related to a simple non linear equation which is fully deterministic.

2) Complexity can arise only from two facts: iteration (feedback from one year to the other) and non linearity in the feedback mechanism. Then, it is not necessary to have many interacting systems in order to get complexity.

3) Even a fully deterministic system (the mouse population at year N is fully specified if we know it at year N-1) can show chaotic behavior which means unpredictability over a certain period of time.

4) Deterministic behavior can be seen as a special case of chaotic behavior. If we take k=2.85 we can observe a small window of stability. If the mice-cat population has this growing factor value in this window, its behavior will be perfectly deterministic. It will be possible a find out rules or equations that allow a perfect computation of mice population over the time. This phenomenon is called intermittency (a period of order in a universe of randomness).
This characteristic behavior raises interesting questions such as: to what extent are an ordered system and its chaotic version both faces of one indivisible process? Is our familiar rule based world just an island of intermittency in the midst of chaotic universe?


The N-Body Problem:

Chaos and Henri Poincaré:
The 3 body problem

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