We consider a configuration graph with N vertices whose degrees are independent and identically distributed according to the power law which depends on a slowly varying function. They are equal to the number of each vertex’s numbered semi-edges. The graph is constructed by joining all the semi-edges pairwise equiprobably to form edges. Such models can be used to adequately describe various communication networks and Internet topologies. The paper investigates the conditions under which the random configuration graph is asymptotically connected as N → ∞. In addition, the rate at which the probability that the graph is not connected convergences to zero was estimated.