Optimization processes of tangible and intangible networks through the Laplace problems for regular lattices with multiple obstacles along the way

A systematic approach is proposed to the theme of safety, reliability and global quality of complex networks (material and immaterial) by means of special mathematical tools that allow an adequate geometric characterization and study of the operation, even in the presence of multiple obstacles along the path. To that end, applying the theory of graphs to the problem under study and using a special mathematical model based on stochastic geometry, in this article we consider some regular lattices in which it is possible to schematize the elements of the network, with the fundamental cell with six, eight or 2(n+2) obstacles, calculating the probability of Laplace. In this way it is possible to measure the “degree of impedance” exerted by the anomalies along the network by the obstacles examined. The method can be extended to other regular and / or irregular geometric figures, whose union together constitutes the examined network, allowing to optimize the functioning of the complex system considered.