Fractals

One of the possible sources through which one comes across a knowledge about fractals is the study of Internet traffic behavior. Internet traffic can be said to behave in a self-similar manner. Self-similarity means that given a network traffic trace, the pattern of variability of various network metrics, such as end-to-end delay or variance, at various time-scales does not exhibit any change. In other words it means that given a time-series of network statistics, no-matter how much you zoom-in or zoom-out of  that time-series, the pattern you observe is the same.

Internet traffic behaves in such a way due to various reasons. Self-similarity is modeled using heavy-tailed or long-range statistical distributions. A heavy tailed distribution is one which has its hump on one side and it is skewed to another side. Its tail is rather thick and that is probably why it is called heavy. They are normally classified as belonging to some exponential family of statistical distributions. Pareto and Weibull distributions are examples of such statistical distributions. It has a finite mean and an infinite variance. It is the consequence of this infinite variance that anything modeled through it exhibits a self-similar behavior as described above. I presume that if something like a time-series of internet delays has an infinite variance, it will exhibit the same type of variability no-matter what time-scales you choose to look at it. This makes it self-similar.

The simplest way of understanding self-similarity is usually proposed to be the study of fractals or fractal art. A fractal in fractal art is a piece of art that apart from its aesthetic appeal has one additional quality. That quality is that if one zoom's in or out of the picture, one gets to see the same pattern repeating as was seen in the previous step.