Accurate localization of seizure onset zones (SOZs) in patients with drug-resistant epilepsy is essential for improving prognostic outcomes. This process can be significantly enhanced through effective network representation and analysis of functional dependencies among brain regions. However, traditional network construction methods often lack generalizability due to individual variability. Furthermore, the independent design of the network construction and analysis modules restricts the overall optimization of localization frameworks. In this study, we propose a novel deep learning framework that integrates graph building and analysis modules for seizure localization. The graph building block adaptively generates customized network representations from Stereo-Electroencephalography (SEEG) data of individual patients by extracting feature vectors of each channel and calculating functional connectivity weights among channels with these vectors. While the GCN-and-LSTM-based graph analysis block identifies abnormal nodes corresponding to SOZs by aggregating spatial and temporal information in the network representations. The graph analysis block is trained alongside the graph building block via the seizure prediction task. Attention weights assigned to each channel are utilized to characterize epileptogenicity, facilitating precise localization of the SOZ. Our method demonstrates superior performance, surpassing baseline and state-of-the-art approaches in 9 of 13 patients from a public dataset and 11 of 14 patients from a clinical dataset. Visualization of the identified brain regions aligns well with labeled SOZs. Furthermore, the adaptive functional brain network reveals that the connectivity density among SOZ channels is greater than that of other brain regions, corroborating existing clinical findings and further confirming the model's reliability and interpretability.