Dynamics equations of end arc degrees in growing graphs

The study was made of random graphs grown by nonlinear preferential attachment rule with stochastic graph differentials. Such graphs are used in network theory as mathematical models of real large networks with millions of elements (for example, social, telecommunicational, financial and other networks). End degrees of randomly selected arc, i.e. the degrees of its two incident vertices, are considered. Two-dimensional probability distribution of end arc degrees is determined. The equations for changes of the named two-dimensional distribution during the graph growth are derived. Final probability distribution of end arc degrees is found. Arc degrees probability distributions for two real large network models are investigated. The obtained results expand the opportunities for adequate description and investigation of real large networks; in particular, they allow calibrating large network models due to their structural characteristics. With the obtained results, one can study and compare different scenarios affecting real large networks.