We consider a configuration random graph with N vertices, whose degrees are independent and identically distributed according to power-law distribution with the parameter τ = τ(N). The properties of this graph depend on the value of the parameter τ. These values can be grouped into three zones: τ > 2, τ ∈ (1,2), τ < 1, in each of which the structure of the graph is similar for all values of τ, but differs greatly from the structure in the other two zones. This means that the values τ = 2 and τ = 1 are critical points. The most important characteristic of a graph is the number of edges. Limit distributions of this characteristic also differ between these three zones as N →∞ and τ is fixed. It is therefore important to study the behavior of the number of edges in transitional situations when τ changes in the neighborhood of the critical points. In this paper the local limit distributions of the number of edges of a graph as τ → 2,τ → 1, and also as τ →∞ were obtained.