We consider configuration graphs with N vertices whose degrees are independent
and identically distributed. The distribution of the random variable ξ, which is
defined as the degree of any vertex, is assumed to satisfy the condition
P{ξ = k} ∼ L / (k^τ ln^g k),
as k→∞, where L, g > 0, τ ∈ (2, 3). We study the local clustering coefficient c(s) which can be interpreted as the probability that two different vertices adjacent to
a vertex of degree s are also connected by an edge. The limit theorem is proved for
the c(s) as N →∞ and s = o(N^{(τ−2)/(τ−1)}).