ARCI′: A Cost-Penalized Connectivity Index for Sparse Network Optimization

This paper introduces ARCI′ (Adjusted Rural Connectivity Index), a novel graph-theoretic metric designed to evaluate and optimize sparse networks by balancing construction cost and population coverage. Traditional minimum spanning tree (MST) approaches focus solely on minimizing cost, often neglecting social factors such as population distribution. ARCI′ incorporates a penalty factor to weigh network cost against population served, allowing flexible prioritization. We analyze ARCI′ mathematically, study its sensitivity to the penalty parameter, and demonstrate its utility through computational experiments on synthetic network data. Our results highlight ARCI′ as a promising metric for multi-objective network design in rural infrastructure, sensor placement, and related fields.