Starting from the hypothesis that both physics, in particular space-time andthe physical vacuum, and the corresponding mathematics are discrete on thePlanck scale we develop a certain framework in form of a class of ' cellularnetworks' consisting of cells (nodes) interacting with each other via bondsaccording to a certain 'local law' which governs their evolution. Both theinternal states of the cells and the strength/orientation of the bonds areassumed to be dynamical variables. We introduce a couple of candidates of suchlocal laws which, we think, are capable of catalyzing the unfolding of thenetwork towards increasing complexity and pattern formation. In section 3 thebasis 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 advancedgeometric/topological concepts and tools are introduced which allow to studyand classify such irregular structures as (random)graphs and networks. We showin particular that the systems under study carry in a natural way a 'groupoidstructure'. In section 5 a, as far as we can see, promising concept of'topological dimension' (or rather: ' fractal dimension') in form of a 'degreeof connectivity' for graphs, networks and the like is developed. Thepossibility of dimensional phase transitions is discussed.
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