Model for the evolution of the degree distributions of the vertices of social network graphs

The paper presents a study of the structure of the network graph formed by friendships of the social network «VKontakte» between Russian cities, taking into account the gender and age structure of participants. This graph is interesting because its vertices are not points, but multidimensional objects corresponding to the amount of parameters the user is described by. In this case, we consider three-dimensional vertexes that correspond to gender, age, and region of residence. It turned out that the distribution of graph vertices by degrees strongly depends on which parameter the vertex corresponds to. Thus, the distribution of regional relationships without gender and age is close to uniform, and the distribution of age relationships without gender and region is triangular. As a result, the «urban» graph has a large fully connected core and sparse periphery, and the «age» graph has fully connected communities of 5–7 vertices that are weakly connected to each other. In this paper, model distributions of degrees of multidimensional vertices of a network graph are constructed and the dependence of the graph density on the rank of vertex parameters (large city, medium city, small city, popular age etc.) is studied. Various methods of vertex clustering and matrix evolution are also considered.