### Dimension Theory of Graphs and Networks

**Authors:**
Thomas Nowotny, Manfred Requardt

**Comments:** 20 pages, 7 figures, LaTex2e, uses amsmath, amsfonts, amssymb,
latexsym, epsfig

#### Abstract

Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face in this enterprise is to
find the discrete protoforms of the building blocks of continuum physics and
mathematics. A core concept is the notion of dimension. In the following we
develop such a notion for irregular structures like (large) graphs and networks
and derive a number of its properties. Among other things we show its stability
under a wide class of perturbations which is important if one has 'dimensional
phase transitions' in mind. Furthermore we systematically construct graphs with
almost arbitrary 'fractal dimension' which may be of some use in the context of
'dimensional renormalization' or statistical mechanics on irregular sets.