When are correlations strong?

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.