The paper deals with the development of the topology of ultra-large communication networks, i.e. networks containing several thousand vertices. In this case, the coordinates of the vertices of the undirected graph are somehow predetermined and a set of edges must be constructed. The main point of the options we are considering for developing the network topology is the minimum of the sum of weights of the edges; however, we note in advance that this criterion of minimality is often not the only objective function in the practical problems we are considering. In our previous papers, two realistically considered tasks were formulated. However, everything is not so simple, and we cannot use the direct version of Kruskal’s algorithm. The complexity of this algorithm depends on the representation of the data, i.e. the data structures used. In our situation (when the number of considered vertices is approximately 5000 to 10000), the operation of a simple version of the algorithm takes about a half an hour, which, of course, is acceptable for a one-time solution to the problem under consideration, but it is unacceptable in the case when such solutions are constructed repeatedly (in particular, iteratively). Some temporary improvements to the practical operation of the algorithm provide different options for using complex data structures. The subject of this article can be formulated as follows. We are moving from exact algorithms (in particular, Kruskal’s algorithm) to some heuristics. Moreover, for the starting problem that we are considering, we cannot work without heuristic algorithm at all. However, we describe two specific variants of a simple implementation of Kruskal’s algorithm for problems of large dimensions; in our titles, those are “the 1st and the 2nd algorithms of the usual implementation”. We have formulated two heuristics (“the 3rd and 4th algorithms”). In our opinion, one of these algorithms turned out to be quite acceptable; we present some practical results of computational experiments. And it is very important that these two heuristics will be useful not only for such a “0-th problem”, but also for much more complex problems.

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The physical data layer transmits bits over physical communication channels, such as coaxial cable or twisted pair. That is, it is this level that directly transmits data. At this level, the characteristics of electrical signals that transmit discrete information are determined. After that it is necessary to consider the control of the communication network, its various algorithms. When designing a communication network, a prerequisite is to calculate its stability, and in the case of large scale communication networks, this is a big problem. The most common deterministic, as well as fairly fast approximate method, which is often implemented at the present time, is the method by which the stability of the communication direction which is estimated by analyzing independent paths only. The main disadvantage of this method is obtaining an understated estimate of stability due to unaccountable dependent routes of communication directions. And this leads to inefficient use of resources. Our proposed methodology allows to take into not only independent paths, but also dependent ones, which is the basis for obtaining a significantly more correct estimate. It is based on an algorithm for checking the presence of a certain path and, based on it, an algorithm for obtaining an exact assessment of stability. The paper also provides analytical and statistical analysis of the considered algorithms. In particular, a special parameter was introduced that characterizes the probability of a failure event of communication lines, in which the number of failed communication lines lies in a certain specified range; after which a study of the function describing this parameter was carried out.

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We consider two frequently encountered problems in the modeling of the communication network, related to the construction of a graph of the communication network that meets certain conditions. Heuristic algorithms for solving these problems are proposed. The question of uniqueness of the solution of the set tasks is investigated. A positive result of solving the problem was obtained and sufficient conditions of uniqueness were found. The research and development of appropriate software are of practical importance in the design of real communication networks.