The analytical description study results on probability of connectivity for the structures used to model the reliability of various complicated systems are presented. Expressions are formed to calculate the connectivity probability of systems that have structural redundancy. The characteristic components of the formulas are distinguished and they are systematized according to their increasing complexity and the number of elements. The features of the equations’ structure permitting to conveniently formulate the probability of the structures connectivity in the process of their construction and transformations are determined. The examples show the formation of formulas and their structural parts at various levels of complexity. The use of the ratio value of the network structure element’s unreliability and its reliability is justified, thus reducing the awkwardness of exact expressions for the connectivity probability of network structures and substantially improves the compactness and convenience of using the equations.

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The study results from the graphical representation on connectivity probability of systems that have structural redundancy for modeling. The structural reliability of different systems is presented. A oneparameter expression of a two-ring structure reliability with five sites is described. All the unique operating conditions of the structure are shown in a graphical form. The compact form of the exact expression of the redundant two-ring structure reliability with five sites is presented, which is convenient for computer modeling. The peculiarities of the dependences of many variables in the reliability model of the two-ring structure with five sites and four nodes are determined. It is shown that the graphic representation of the dependence of the five variables helps to study the properties of multiparameter dependencies. The components of a geometric model of various dimensions are considered in detail.

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Discrete modeling of continuous images by the static-geometric method in most cases is associated with certain errors. Therefore, it is relevant to study the formation of geometric images with a given accuracy, using a minimum amount of initial information. This will allow to create models with optimal discretization. Further development of this modeling method with rational decrease in initial information volume, is topical. The proposed line of research will open up new possibilities for its use in various industries, such as construction, engineering, economics etc. One of the objectives of this work is to continue research on the formation of discrete images of curved lines. The study is based on the classical finite difference method, static-geometric modeling method and the geometric apparatus of superpositions. The article proposes a method for determining the distribution functions of finite-difference values in the formation of discrete analogues of linear-fractional, exponential and hyperbolic functional dependencies using finite differences of different orders. These studies define a general method for obtaining similar patterns of distribution of the finite difference value in formation of discrete analogues of elementary functional dependencies using finite differences of different orders. Establishing a correspondence between the equation of the curve and the distribution function of a finite difference value can be the basis for assessing the accuracy of the formation of discrete frameworks of curves described by elementary functional dependencies. In the future, the results of this work will make it possible to determine coordinates of any point of a numerical sequence of the nth order and numerical sequences of elementary functional dependencies as a superposition of the coordinates of adjacent points. It can also be a superposition of arbitrarily given nodal points of these sequences, if the value of the final difference or its distribution function are already known.