  Starting from the hypothesis that both physics, in particular space-time and
the physical vacuum, and the corresponding mathematics are discrete on the
Planck scale we develop a certain framework in form of a class of ' cellular
networks' consisting of cells (nodes) interacting with each other via bonds
according to a certain 'local law' which governs their evolution. Both the
internal states of the cells and the strength/orientation of the bonds are
assumed to be dynamical variables. We introduce a couple of candidates of such
local laws which, we think, are capable of catalyzing the unfolding of the
network towards increasing complexity and pattern formation. In section 3 the
basis is laid for a version of 'discrete analysis' on 'graphs' and 'networks'
which, starting from different, perhaps more physically oriented principles,
manages to make contact with the much more abstract machinery of Connes et al.
and may complement the latter approach. In section 4 several more advanced
geometric/topological concepts and tools are introduced which allow to study
and classify such irregular structures as (random)graphs and networks. We show
in particular that the systems under study carry in a natural way a 'groupoid
structure'. In section 5 a, as far as we can see, promising concept of
'topological dimension' (or rather: ' fractal dimension') in form of a 'degree
of connectivity' for graphs, networks and the like is developed. The
possibility of dimensional phase transitions is discussed.
