Fractal dimension of infinitely growing discrete sets

The definition of fractal dimension of infinite growing sets is presented. It is based on a box-counting approach and coincides with the elasticity coefficient for number of boxes as function to the window length size. The finite difference forms of this dimension are proposed. They are applicable to infinite growing discrete sets, such as scale-free networks. The scale-free network model having strict constant fractal dimension is constructed. Those networks are discrete exactly self-similar sets. Applying the concept of non-unit fractal dimension of scale-free networks, i.e. the concept of different relative growth rates for edges and vertices, makes it possible to generate scale-free models for dense networks.