  We perform an analytical analysis of the long-range degree correlation of the
giant component in an uncorrelated random network by employing generating
functions. By introducing a characteristic length, we find that a pair of nodes
in the giant component is negatively degree-correlated within the
characteristic length and uncorrelated otherwise. At the critical point, where
the giant component becomes fractal, the characteristic length diverges and the
negative long-range degree correlation emerges. We further propose a
correlation function for degrees of the $l$-distant node pairs, which behaves
as an exponentially decreasing function of distance in the off-critical region.
The correlation function obeys a power-law with an exponential cutoff near the
critical point. The Erd\H{o}s-R\'{e}nyi random graph is employed to confirm
this critical behavior.
