Complexity_ A Guided Tour - Melanie Mitchell [51]
For example, a model of the message source of string 1 above could be very simple: “repeat A C”; thus its statistical complexity is low. However, in contrast to what could be done with entropy or algorithmic information content, a simple model could also be built of the message source that generates string 2: “choose at random from A, C, G, or T.” The latter is possible because models of statistical complexity are permitted to include random choices. The quantitative value of statistical complexity is the information content of the simplest such model that predicts the system’s behavior. Thus, like effective complexity, statistical complexity is low for both highly ordered and random systems, and is high for systems in between—those that we would intuitively consider to be complex.
Like the other measures described above, it is typically not easy to measure statistical complexity if the system in question does not have a ready interpretation as a message source. However, Crutchfield, Young, and their colleagues have actually measured the statistical complexity of a number of real-world phenomena, such as the atomic structure of complicated crystals and the firing patterns of neurons.
Complexity as Fractal Dimension
So far all the complexity measures I have discussed have been based on information or computation theoretic concepts. However, these are not the only possible sources of measures of complexity. Other people have proposed concepts from dynamical systems theory to measure the complexity of an object or process. One such measure is the fractal dimension of an object. To explain this measure, I must first explain what a fractal is.
The classic example of a fractal is a coastline. If you view a coastline from an airplane, it typically looks rugged rather than straight, with many inlets, bays, prominences, and peninsulas (Figure 7.2, top). If you then view the same coastline from your car on the coast highway, it still appears to have the exact same kind of ruggedness, but on a smaller scale (Figure 7.2, bottom). Ditto for the close-up view when you stand on the beach and even for the ultra close-up view of a snail as it crawls on individual rocks. The similarity of the shape of the coastline at different scales is called “self-similarity.”
The term fractal was coined by the French mathematician Benoit Mandelbrot, who was one of the first people to point out that the world is full of fractals—that is, many real-world objects have a rugged self-similar structure. Coastlines, mountain ranges, snowflakes, and trees are often-cited examples. Mandelbrot even proposed that the universe is fractal-like in terms of the distribution of galaxies, clusters of galaxies, clusters of clusters, et cetera. Figure 7.3 illustrates some examples of self-similarity in nature.
Although the term fractal is sometimes used to mean different things by different people, in general a fractal is a geometric shape that has “fine structure at every scale.” Many fractals of interest have the self-similarity property seen in the coastline example given above. The logistic-map bifurcation diagram from chapter 2 (figure 2.6) also has some degree of self-similarity; in fact the chaotic region of this (R greater than 3.57 or so) and many other systems are sometimes called fractal attractors.
Mandelbrot and other mathematicians have designed many different mathematical models of fractals in nature. One famous model is the so-called Koch curve (Koch, pronounced “Coke,” is the name of the