  The inverse problem of statistical mechanics involves finding the minimal
Hamiltonian that is consistent with some observed set of correlation functions.
This problem has received renewed interest in the analysis of biological
networks; in particular, several such networks have been described successfully
by maximum entropy models consistent with pairwise correlations. These
correlations are usually weak in an absolute sense (e.g., correlation
coefficients ~ 0.1 or less), and this is sometimes taken as evidence against
the existence of interesting collective behavior in the network. If
correlations are weak, it should be possible to capture their effects in
perturbation theory, so we develop an expansion for the entropy of Ising
systems in powers of the correlations, carrying this out to fourth order. We
then consider recent work on networks of neurons [Schneidman et al., Nature
440, 1007 (2006); Tkacik et al., arXiv:0912.5409 [q-bio.NC] (2009)], and show
that even though all pairwise correlations are weak, the fact that these
correlations are widespread means that their impact on the network as a whole
is not captured in the leading orders of perturbation theory. More positively,
this means that recent successes of maximum entropy approaches are not simply
the result of correlations being weak.
