Mismatch negativity (MMN) has been consistently found deficit in schizophrenia, which was considered as a promising biomarker for assessing the impairments in preattentive auditory processing. However, the functional connectivity between brain regions based on MMN is not clear. This study provides an in-depth investigation in brain functional connectivity during MMN process among patients with first-episode schizophrenia (FESZ), chronic schizophrenia (CSZ) and healthy control (HC). Electroencephalography (EEG) data of 128 channels is recorded during frequency and duration MMN in 40 FESZ, 40 CSZ patients and 40 matched HC subjects. We reconstruct the cortical endogenous electrical activity from EEG recordings using exact low-resolution electromagnetic tomography and build functional brain networks based on source-level EEG data. Then, graph-theoretic features are extracted from the brain networks with the support vector machine (SVM) to classify FESZ, CSZ and HC groups, since the SVM has good generalization ability and robustness as a universally applicable nonlinear classifier. Furthermore, we introduce the graph neural network (GNN) model to directly learn for the network topology of brain network. Compared to HC, the damaged brain areas of CSZ are more extensive than FESZ, and the damaged area involved the auditory cortex. These results demonstrate the heterogeneity of the impacts of schizophrenia for different disease courses and the association between MMN and the auditory cortex. More importantly, the GNN classification results are significantly better than those of SVM, and hence the EEG-based GNN model of brain networks provides an effective method for discriminating among FESZ, CSZ and HC groups.