Correlations reveal the hierarchical organization of networks with latent binary variables

Deciphering the functional organization of large biological networks is a major challenge for current mathematical methods. A common approach is to decompose networks into largely independent functional modules, but inferring these modules and their organization from network activity is difficult, given the uncertainties and incompleteness of measurements. Typically, some parts of the overall functional organization, such as intermediate processing steps, are latent. We show that the hidden structure can be determined from the statistical moments of observable network components alone, as long as the functional relevance of the network components lies in their mean values and the mean of each latent variable maps onto a scaled expectation of a binary variable. Whether the function of biological networks permits a hierarchical modularization can be falsified by a correlation-based statistical test that we derive. We apply the test to three biological networks at different spatial scales, i.e., gene regulatory networks, dendrites of pyramidal neurons, and networks of spiking neurons.