DIMENSION THEORY OF GRAPHS AND NETWORKS

Starting from the working hypothesis that both physics and the corresp onding 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 o f continuum physics and mathematics. A core concept is the notion of d imension. In the following we develop such a notion for irregular stru ctures such as (large) graphs and networks and derive a number of its properties. Among other things we show its stability under a wide clas s of perturbations which is important if one has 'dimensional phase tr ansitions' in mind. Furthermore we systematically construct graphs wit h almost arbitrary 'fractal dimension' which may be of some use in the context of 'dimensional renormalization' or statistical mechanics on irregular sets.