Numerical Methods of Multifractal Analysis in Information Communication Systems and Networks

Theory Of Fractals And Multifractals

The term "fractal” was used for the first time in Benoît Mandelbrot’s work (Mandelbrot, 1982). The word fractal is derived from the Latin fractus meaning "fractured" or "broken." Mandelbrot used the term "fractals" for geometric objects that have strongly fragmented shape and can possess the property of self-similarity. It is possible to generalize the concept of fractal to any object (image, speech, telecommunication traffic, etc.) some parameters of which are remain invariant with change in scale or time. Thus, the principal property of such objects (i.e. self-similarity) implies that at augmentation, its parts are similar (in some specified sense) to its total shape.

The property of exact self-similarity is a characteristic of the regular fractals only. If an element of randomness is to be included in the algorithm of their creation instead of the determined method of construction (as it happens, for example, in many processes of diffusion growth of clusters, voltage failure, etc.), then the so-called incidental fractals appear. Their basic difference from regular ones is that the property of self-similarity holds true only after a corresponding averaging on the base of all statistically independent realizations of the object. For quantitative description of fractals, a single value is enough - a fractal dimension (Hausdorff dimension) or the index of scaling which is determined as follows