Construction of Growing Graphs with Given Power-Law Asymptotics of Vertex Degree Distributions

Two classes of graphs are considered in the article: preferential attachment graphs with linear weight function and hybrid model of Jackson-Rogers graphs. The research is carried out to find such graphs whose vertex degree distributions have power-law asymptotics with an exponent that belongs to the interval from two to three. It is established for the first time that each hybrid graph corresponds to a certain graph with a linear weight function that has exactly the same vertex degree distribution as a hybrid graph. As a result of the investigation, all graphs of the classes under consideration are revealed, which implement the desired power-law asymptotics of the studied distributions. A formula is derived that allows us to determine the value of the weight function parameter from the given value of the power asymptotics exponent. The reliability and practical significance of the obtained theoretical results are confirmed by an example of their application for graph calibration according to data on a simulated network of autonomous Internet systems.