#complexity quantification

The so-called Complexity Science has been around for a few decades now. Chaos, as well as complexity, were buzzwords in the 1980s, pretty much like Artificial Intelligence is today. There are Complexity Institutes in many countries. They speak of Complexity Theory. In a beautiful and recent video called “The End of Science”, Sabine Hossenfelder, speaks of Complexity Science and how it has not…

How Complex are Formations of Starlings?

From: https://gizmodo.com/this-starling-murmuration-video-is-the-most-stunning-iv-1679901420

There is a tendency to regard the phrase “the whole is greater than the sum of the parts” as a (sort of) definition of “complex systems” by those who claim to “study complexity”. Curiously enough, the said “complexity scientists” engage in an exercise which, inexplicably, does not contemplate a measure…

A Survey of “Complexity Measures”,,,,, Without Measures

We report verbatim a 1998 article A Survey of “Complexity Measures” by David P. Feldman and Jim Crutchfield.

A more adequate title for this evasive article would be “A Survey of How Science Avoids to Measure Complexity”. Once you’ve read it, ask yourself this: can I use any of these measures to actually measure the complexity of, say,

a piece of equipment, e.g. an electromechanical system

an…

Citing Wikipedia: “An advantage of terrain occurs when military personnel gain an advantage over an enemy utilizing, or simply in spite of, the terrain around them. The term does not exclusively apply to battles, and can be used more generally regarding entire campaigns or theaters of war. Mountains, for example, can block off certain areas, making it unnecessary to station troops within the inaccessible area. This deployment strategy can be applied with other formidable environmental features as well, such as forests and mountains. In the former instance, vegetation can provide concealment for tactical movements such as setting up an ambush. In the latter, the elevation can provide an advantage to soldiers using projectile weapons. Elevation itself is perhaps the most well-known example of terrain advantage, with gravity working to the advantage of the more elevated party.

This short blog has the intent of illustrating how with QCM technology it is possible to measure the complexity of terrain. Such information can be useful when planning campaigns, strategies, or deploying equipment. Clearly, a complex terrain will require a different approach to a less involved one.

Rough terrain

Smoother terrain

Until recently, QCM technology has been used to process data, such as real-time CAN bus data, stock market data, real-time patient data in ICUs, or air traffic radar output. In  other words, scalar multi-channel data sampled with frequencies ranging from a few Hertz to thousands of Hertz.

With the development of a Multi-thread version of the QCM algorithm, we can now use Supercomputers to process arrays representing terrain. In particular we are able to measure the complexity of 2D domains in which elevation, z, is provided in each point of the domain, i.e. z=f(x, y).

The images which follow are examples of functions of the type z=f(x, y) and of the corresponding Complexity Landscapes, i.e. c=g(x, y), where c is complexity. No analysis or comments shall be provided as the images are shown for illustrative purposes only. These are in fact the first ever Complexity Landscapes. Their synthesis, is expensive from a computational effort standpoint. Some landscapes illustrated below require more than one day of runtime.

The following are example of Complexity Landscapes of gravity along a plane passing through the source, i.e. F=-GMm/r^2, where r=sqrt(x^2 + y^2).

Finally, the examples below are Complexity Landscapes of regions of actual terrains described via elevation, z=f(x, y).

The bottom line: More complex terrain means potentially more effort, more energy expenditure, risk, more sophisticated equipment, and a more articulate strategy. The point is that today it is possible to measure terrain complexity (TERCOM) and to move from a qualitative to a quantitative assessment.

With the approach we have developed it is possible to process 3D data, representing any digital landscape or field, such as electromagnetic, in the form V=f(x, y, z). In terms of computational resources an order of magnitude more with respect to 2D domains will of course be necessary.

What are the Units of Complexity?

Complexity is a fundamental physical characteristic of every system that may be found in Nature. Its importance is similar to that of energy. There have been numerous attempts of measuring complexity. However, most are limited either to very simple systems, character strings or to computer code and have very little practical use. In many cases, measures of complexity have been equated to…