Probabilistic-topological theory of systems with discrete interactions: I. System representation by a hypergraph

Abstract

The theory presented is applicable to any system with discrete interactions, i.e., one that lacks long-range crystal-like translational symmetry but is such that any of its structural elements interacts directly with only a finite (in most cases small) number of other elements, i.e., for materials such as cross-linked polymers, superpolymers (ferrofluids, wormlike micelles, colloidal necklaces), ceramics and glasses obtained by sol-gel processes, as well as for biophysical systems such as membrane receptors, cellular aggregates, neuronal branching patterns.The theory systematizes the information one needs to represent the system by a hypergraph, which then makes possible application of the so-called difference a posteriori (DAPOST) algorithm to calculate structural characteristics of the system and structure–property relationships. It is based on probabilistic and topological considerations; thus, it is applicable to systems far from thermodynamic equilibrium and to the analysis of spatiotemporal patterns.