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

Abstract

A systematic approach is proposed to the theme of safety, reliabil-ity and global quality of complex networks (material and immaterial) bymeans of special mathematical tools that allow an adequate geometriccharacterization and study of the operation, even in the presence of mul-tiple obstacles along the path. To that end, applying the theory of graphsto the problem under study and using a special mathematical model basedon stochastic geometry, in this article we consider some regular latticesin which it is possible to schematize the elements of the network, withthe fundamental cell with six, eight or 2(n + 2) obstacles, calculating theprobability of Laplace. In this way it is possible to measure the "degree ofimpedance" exerted by the anomalies along the network by the obstaclesexamined. The method can be extended to other regular or irregular ge-ometric gures, whose union together constitutes the examined network,allowing to optimize the functioning of the complex system considered.