Psychotic experiences have been proposed to lie on a spectrum, ranging from subclinical experiences to treatment-resistant schizophrenia. We aimed to characterize functional connectivity and brain network characteristics in relation to the schizophrenia spectrum and bipolar disorder with psychosis to disentangle neural correlates to psychosis. Additionally, we studied antipsychotic medication and lithium effects on network characteristics. We analyzed functional connectivity strength and network topology in 487 resting-state functional MRI scans of individuals with schizophrenia spectrum disorder (SCZ), bipolar disorder with a history of psychotic experiences (BD), treatment-naïve subclinical psychosis (SCP), and healthy controls (HC). Since differences in connectivity strength may confound group comparisons of brain network topology, we analyzed characteristics of the minimum spanning tree (MST), a relatively unbiased backbone of the network. SCZ and SCP subjects had a lower connectivity strength than BD and HC individuals but showed no differences in network topology. In contrast, BD patients showed a less integrated network topology but no disturbances in connectivity strength. No differences in outcome measures were found between SCP and SCZ, or between BD patients that used antipsychotic medication or lithium and those that did not. We conclude that functional networks in patients prone to psychosis have different signatures for chronic SCZ patients and SCP compared to euthymic BD patients, with a limited role for medication. Connectivity strength effects may have confounded previous studies, as no functional network alterations were found in SCZ after strict correction for connectivity strength.