Dimension theory of graphs and networks

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.

(Quantum) spacetime as a statistical geometry of fuzzy lumps and the connection with random metric spaces

“…This conjectured emergent geometric order can be viewed as kind of a discrete proto space-time or pregeometry carrying metrical, causal and dimensional structures (as to the cellular network approach cf. [4] to [8], the other point of view is expounded in e.g. the book of Roy; [19] to which we also refer for a more complete list of references).…”

Cellular networks as models for Planck-scale physics

“…We start to develop our personal concept in the last section of this paper. More information can be found in [8]. As the interrelation of these various approaches and frameworks is both very interesting and presently far from obvious we plan to compare them elsewhere.…”

“…6. For some remarks concerning more or less related ideas being scattered in the literature (known to us) see [8] That this definition is reasonable can be seen by applying it to ordinary cases like regular translation invariant lattices. It is however not evident that such a definition makes sense for arbitrary graphs, in other words, a (unique) limit point may not always exist.…”

“…many of the nodes which can be reached from, say, a node of ∂U m by one step are already contained in U m itself. On the other side we recently constructed ( [8]) tree graphs having an arbitrary fractal dimension (they are however not regular, i.e. having a varying node degree, in particular they have a lot of end points).…”

“…The mentioned ideas are sketched almost at the end of a long contribution and we stumbled over them only very recently (February 1998). One could perhaps venture to say that the framework we are going to develop in the following and in related papers (see [8])…”