  We study pair correlations in cooperative systems placed on complex networks.
We show that usually in these systems, the correlations between two interacting
objects (e.g., spins), separated by a distance $\ell$, decay, on average,
faster than $1/(\ell z_\ell)$. Here $z_\ell$ is the mean number of the
$\ell$-th nearest neighbors of a vertex in a network. This behavior, in
particular, leads to a dramatic weakening of correlations between second and
more distant neighbors on networks with fat-tailed degree distributions, which
have a divergent number $z_2$ in the infinite network limit. In this case, only
the pair correlations between the nearest neighbors are observable. We obtain
the pair correlation function of the Ising model on a complex network and also
derive our results in the framework of a phenomenological approach.
