  In this paper we study a random graph with $N$ nodes, where node $j$ has
degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with $\prob(D_j\leq x)=F(x)$. We
assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant
$c>0$. This graph model is a variant of the so-called configuration model, and
includes heavy tail degrees with finite variance.
  The minimal number of edges between two arbitrary connected nodes, also known
as the graph distance or the hopcount, is investigated when $N\to \infty$. We
prove that the graph distance grows like $\log_{\nu}N$, when the base of the
logarithm equals $\nu=\expec[D_j(D_j -1)]/\expec[D_j]>1$. This confirms the
heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the
random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are
characterized and shown to be uniformly bounded. In particular, we show
convergence in distribution of the centered graph distance along exponentially
growing subsequences.
