Connectivity of configuration graphs in complex network models

The author considers configuration graphs whose degrees of vertices are independent and identically distributed according to the generalized power-law distribution. Connections between vertices are equiprobably formed in compliance with their degrees. Such random graphs are often used for modeling complex communication networks like the Internet and social networks. It is assumed that the distribution of vertex degrees is unknown because it depends on a slowly varying function with unknown properties. The conditions are found under which a graph is asymptotically almost surely connected as the number of vertices tends to infinity. Under these conditions, estimates of the convergence rate to zero of the probability that the graph is not connected are obtained. The results in the present paper are proved using the properties of stable distributions and slowly varying functions.