NOTES/★/COURTNEY.BOLTON

Emergence across complex systems. Ripple patterns in a sand dune created by wind or water is an example of an emergent structure in nature. 

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. This characteristic of every system is called emergence and is true of any system, not just complex ones.

A system’s complexity may be of one of two forms: disorganized complexity and organized complexity. In essence, disorganized complexity is a matter of a very large number of parts, and organized complexity is a matter of the subject system (quite possibly with only a limited number of parts) exhibiting emergent properties.

Examples of complex systems include ant colonies, human economiesand social structures, climate, nervous systems, cells and living things, including human beings, as well as modern energy or telecommunication infrastructures. Indeed, many systems of interest to humans are complex systems.

Coney Island Parachute Jump, and Here is New York, by E.B. White (1949): “There are roughly three New Yorks.

There is, first, the New York of the man or woman who was born here, who takes the city for granted and accepts its size and turbulence as natural and inevitable.

Second, there is the New York of the commuter—the city that is devoured by locusts each day and spat out each night.

Third, there is the New York of the person who was born somewhere else and came to New York in quest of something. Of these three trembling cities the greatest is the last—the city of final destination, the city that is a goal. It is this third city that accounts for New York’s high-strung disposition, its poetical deportment, its dedication to the arts, and its incomparable achievements. Commuters give the city its tidal restlessness; natives give it solidity and continuity; but the settlers give it passion. And whether it is a farmer arriving from Italy to set up a small grocery store in a slum, or a young girl arriving from a small town in Mississippi to escape the indignity of being observed by her neighbors, or a boy arriving from the Corn Belt with a manuscript in his suitcase and a pain in his heart, it makes no difference: each embraces New York with the intense excitement of first love, each absorbs New York with the fresh eyes of an adventurer, each generates heat and light to dwarf the Consolidated Edison Company.”

Cantor dust: a multi-dimensional version of the Cantor set. It can be formed by taking a finite Cartesian product of the Cantor set with itself, making it a Cantor space. Like the Cantor set, Cantor dust has zero measure.

Strange attractors
While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the Rössler map, which experiences period-two doubling route to chaos, like the logistic map.

Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points - Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure.

The Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems.

The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic motion. (infinity 8 /limit /design dynamic systems /models)

Airplane vortex. Turbulence in the tip vortex from an airplane wing. Studies of the critical point beyond which a system creates turbulence was important for Chaos theory, analyzed for example by the Soviet physicist Lev Landau who developed the Landau-Hopf theory of turbulence. David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory. (chaos /limit /design dynamic systems /models)

Cessna, 182 model, wingtip vortex. The flow field around an airplane is a vector field in R³, here visualized by bubbles that follow the streamlines showing a wingtip vortex. 

Doppler effect. A source of waves moving to the left. The frequency is higher on the left than on the right.

Lorenz Attractor.

“Mulgo,” the Black swan. Black swan ~relates to Native American medicine; Grace; Leda+Swan/16th c. European symbolism in painting; the ugly duckling; Haley’s comet; Web2.0 Long Tail; mathematical theory; outliers.

The Black Swan Theory (in Nassim Nicholas Taleb’s version) refers to high-impact, hard-to-predict, and rare events beyond the realm of normal expectations. Unlike the philosophical “black swan problem,” the “Black Swan Theory” (capitalized) refers only to events of large magnitude and consequence and their dominant role in history. “Black Swan” events are considered extreme outliers.

Coping with Black Swan Events
The main idea in Taleb’s book is not to attempt to predict Black Swan events, but to build robustness to the negative ones, while being able to exploit positive ones. Taleb contends that banks and trading firms are very vulnerable to hazardous Black Swan events and are exposed to losses beyond that predicted by their defective models.

Taleb states that a Black Swan event depends on the observer—a Black Swan surprise for the turkey is not a Black Swan surprise for the butcher, hence his idea is to “avoid being the turkey” by finding out where one may be exposed to being a turkey and “turn the Black Swans white.”

Identifying a Black Swan Event
Based on the author’s criteria:

1. The event is a surprise.
2. The event has a major impact.
3. After the fact, the event is rationalized by hindsight, as if it had been expected.

Non-philosophical epistemological approach
Taleb’s black swan is different from the earlier (philosophical) versions of the problem as it concerns a phenomenon with specific empirical/statistical properties which he calls “the fourth quadrant.” Before Taleb, those who dealt with the notion of the improbable, like Hume, Mill and Popper, focused on the problem of induction in logic, specifically that of drawing general conclusions from specific observations. Taleb’s Black Swan has a central and unique attribute: the high impact. His claim is that almost all consequential events in history come from the unexpected—while humans convince themselves that these events are explainable in hindsight (bias).

One problem, labeled the ludic fallacy by Taleb, is the belief that the unstructured randomness found in life resembles the structured randomness found in games. This stems from the assumption that the unexpected can be predicted by extrapolating from variations in statistics based on past observations, especially when these statistics are assumed to represent samples from a Bell Curve. These concerns are often highly relevant in financial markets, where major players use value at risk models (which imply normal distributions) but market return distributions have fat tails.

More generally, decision theory based on a fixed universe or model of possible outcomes ignores and minimizes the impact of events which are “outside model”. For instance, a simple model of daily stock market returns may include extreme moves such as Black Monday (1987), but might not model the market breakdowns following the September 11 attacks. A fixed model considers the “known unknowns”, but ignores the “unknown unknowns.”

Taleb notes that other distributions are not usable with precision, but often more descriptive, such as the fractal, power law, or scalable distributions; awareness of these might help to temper expectations. Beyond this, he emphasizes that many events are simply without precedent, undercutting the basis of this type of reasoning altogether. Taleb also argues for the use of counterfactual reasoning when considering risk.