Hyperbolic Geometry for Optimal Placement of Patient Assistance Terminals in Wireless Mesh Networks: A Local Data-Driven Approach in Data-Scarce Remote Terrains

We present a novel framework for optimal placement of Patient Assistance Terminals (PATs) in wireless mesh networks using hyperbolic geometry modeling of remote terrains. The approach models terrain as a hyperbolic plane ${？}^2$ with Poincaré disk representation $\mathcal{D}=\left\{z\in ？:\left|z\right|<1\right\}$ to capture exponential distortions and tree-like hierarchies inherent in hub-centric settlement patterns. Let $\mathcal{G}=\left(\mathcal{V},？\right)$ represent the network graph where $\mathcal{V}$ denotes settlement hubs and internet spots, and $？$ encodes connectivity constraints. The optimization objective minimizes $\mathcal{J}\left(p\right)=\alpha {\displaystyle {\sum }_{i\in \mathcal{V}}\frac{1}{{\text{accessibility}}_i\left(p\right)}}+\beta {\displaystyle {\sum }_{\left(i,j\right)\in ？}{\text{latency}}_{ij}\left(p\right)}$ where