Alzheimer's disease (AD) is associated with local (e.g. brain tissue atrophy) and global brain changes (loss of cerebral connectivity), which can be detected by high-resolution structural magnetic resonance imaging. Conventionally, these changes and their relation to AD are investigated independently. Here, we introduce a novel, highly-scalable approach that simultaneously captures $\textit{local}$ and $\textit{global}$ changes in the diseased brain. It is based on a neural network architecture that combines patch-based, high-resolution 3D-CNNs with global topological features, evaluating multi-scale brain tissue connectivity. Our local-global approach reached competitive results with an average precision score of $0.95\pm0.03$ for the classification of cognitively normal subjects and AD patients (prevalence $\approx 55\%$).