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

Abstract

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 ℍ 2 with Poincaré disk representation D = { z ∈ ℂ : | z | < 1 } to capture exponential distortions and tree-like hierarchies inherent in hub-centric settlement patterns. Let G = ( V , ℰ ) represent the network graph where V denotes settlement hubs and internet spots, and ℰ encodes connectivity constraints. The optimization objective minimizes J ( p ) = α ∑ i ∈ V 1 accessibility i ( p ) + β ∑ ( i , j ) ∈ ℰ latency i j ( p ) where p ∈ ℍ 2 represents PAT positions and ( α , β ) is multi-objective weights. Distortion-minimizing embeddings ϕ : G → ℍ 2 preserve local community knowledge while enabling global optimization. Experimental validation of the framework demonstrates a high reliability in brutal, unforgiving environments through local human knowledge integration, achieving low-latency mesh networks that scale without noise or constraint issues.