Schizophrenia is a chronic brain disorder associated with widespread alterations in functional brain connectivity. Although data-driven approaches such as independent component analysis are often used to study how schizophrenia impacts linearly connected networks, alterations within the underlying nonlinear functional connectivity structure remain largely unknown. Here we report the analysis of networks from explicitly nonlinear functional magnetic resonance imaging connectivity in a case-control dataset. We found systematic spatial variation, with higher nonlinear weight within core regions, suggesting that linear analyses underestimate functional connectivity within network centers. We also found that a unique nonlinear network incorporating default-mode, cingulo-opercular and central executive regions exhibits hypoconnectivity in schizophrenia, indicating that typically hidden connectivity patterns may reflect inefficient network integration in psychosis. Moreover, nonlinear networks including those previously implicated in auditory, linguistic and self-referential cognition exhibit heightened statistical sensitivity to schizophrenia diagnosis, collectively underscoring the potential of our methodology to resolve complex brain phenomena and transform clinical connectivity analysis.