Contextual analysis of the bipolar structures connectivity

The paper discusses an original approach to the analysis of network structures of various natures, which are represented in the form of graphs. It is assumed that the probability of connectivity of individual edges functionally depends on their physical characteristics (edge length) and tends to zero. The issue of increasing the connectivity probability of the entire graph is being addressed, which is defined as the probability of the existence of a sequential set of edges connecting the selected start and end vertices. A distinctive feature of the proposed method is the move away from the traditional reservation of obviously weak connection points towards a functional impact on individual connections of the structure, changing their physical characteristics. Based on the theory of systems functioning and the theory of dominants, contextual approach is being developed aimed at identifying the dominant connections of the graph, reflecting the functional and meaningful meaning of the connections between the vertices of the original modeled object. As a result, a set of dominants $\mathcal{S}$ is identified, consisting of edges whose parameters meet the given criteria. The selection criterion is the length of the edge included in the asymptotic relation, which characterizes the connectivity probability of the entire graph. It has been proven that changing the length of edges from a given set by $\varepsilon>0$ increases the probability of connectedness of the original graph in the maximum case in $h^{- \varepsilon}$ or $ h^{-\varepsilon \beta} $ times, where $h\to 0$ and $ \beta $ is a parameter depending on the number of graph paths passing through this edge. Various methods have been proposed to increase the connectivity probability of the graph under consideration: from the point of view of the point effect, by reducing the length of a single edge from the set $\mathcal{S}$; from the position of influencing the maximum set of edges from $\mathcal{S}$, allowing to obtain the overall maximum effect. A comparative analysis of the proposed ways to increase the probability of graph connectivity has been carried out, and the appropriate conditions have been identified to achieve the maximum effect from the selected methods of influence.