Weighted K-Harmonic Means Clustering: Convergence Analysis and Applications to Wireless Communications

We propose the \emph{weighted K-harmonic means} (WKHM) clustering algorithm, a regularized variant of K-harmonic means designed to ensure numerical stability while enabling soft assignments through inverse-distance weighting. Unlike classical K-means and constrained K-means, WKHM admits a direct interpretation in wireless networks: its weights are exactly equivalent to fractional user association based on received signal strength. We establish rigorous convergence guarantees under both deterministic and stochastic settings, addressing key technical challenges arising from non-convexity and random initialization. Specifically, we prove monotone descent to a local minimum under fixed initialization, convergence in probability under Binomial Point Process (BPP) initialization, and almost sure convergence under mild decay conditions. These results provide the first stochastic convergence guarantees for harmonic-mean-based clustering. Finally, through extensive simulations with diverse user distributions, we show that WKHM achieves a superior tradeoff between minimum signal strength and load fairness compared to classical and modern clustering baselines, making it a principled tool for joint radio node placement and user association in wireless networks.