Algebraic and Geometric Aspects of Flow Modeling and Prospects of Natural Science Applications
Every modern applied direction of natural-scientific research has access to a lot of data of network information—big data. These trends make it possible to assume higher standards for information processing and quality of developing models. In addition to using in traffic modeling applications, models of particle flows on complex networks are applied in different areas. The areas are, for example, material science for design of new materials, study of biological processes (metabolism), medicine for the study of drug action , computing (supercomputers), etc.
We consider regular networks being perturbation of standard rectangular networks or other regular graphs, where vertices are replaced with contours as support of the local movement of particles. The movement rules are postulated. Qualitative and quantitative flow characteristics are studied.
The BML model of cellular automata on Manhattan network on a torus was considered. There are two types of particles moving in one direction. The particles of first type move along meridians, and the particles of second type move along parallels. For the first time there was obtained an effect of self-organization of the system, and collapse conditions were formulated. The BML authors laid down the foundation of the spectral theory for the considered systems.
We are developing models of flows in more general periodic symmetric structures. There are proved results about particles velocities. Conditions of self-organization and collapse of the system have been formulated.
We consider a variant of general algebraic formulation for full graph with architecture of N-dial. The particle plans are given by recording of real number in N-ary number system. The results characterized by the quality behavior of the system have been obtained. We have developed approaches to evaluate the velocity of the plan implementation by the particle and value of intensity of the particle movement. Various generalizations of the considered statements are formulated. Interconnection of these problems with questions of classical branches of mathematics has been found, in particular, with the theory of Diophantine equations.
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