Electronic health records (EHRs) capture the complex web of interactions among healthcare providers, patients, and clinical events, offering a rich foundation for predicting critical outcomes in intensive care units (ICUs). However, existing approaches often fail to fully leverage higher-order relational structures and nonlinear dependencies within these networks. To address this gap, we propose LAI-Pool, a persistence-aware pooling method that integrates our self-attention–guided persistence images (SAPI) with graph neural networks (GNNs) via cross attention. First, we model patient–provider interactions as ego networks constructed from EHRs, where nodes represent patients, admissions, diagnoses, procedures, providers, ICU stays, and ICU units, and edges encode the temporal order of care interactions. Then, we apply persistent homology to extract multi-scale topological features from these networks and fuse them with GNN representations through LAI-Pool. We benchmark our approach against persistent homology–only, GNN-only, and hybrid PH+GNN baselines on the MIMIC-IV dataset comprising 65,366 ICU patients. Our results demonstrate that LAI-Pool achieves superior performance in mortality prediction, and highlight the value of combining topological data analysis with deep graph models for advancing patient outcome prediction in critical care.