Summary: Let \((M,g_{1})\) be a complete \(d\)-dimensional Riemannian manifold for \(d>1\). Let \(\mathcal{X}_{n}\) be a set of \(n\) sample points in \(M\) drawn randomly from a smooth Lebesgue density \(f\) supported in \(M\). Let \(x,y\) be two points in \(M\). We prove that the normalized length of the power-weighted shortest path between \(x,y\) through \(\mathcal{X}_{n}\) converges almost surely to a constant multiple of the Riemannian distance between \(x,y\) under the metric tensor \(g_{p}=f^{2(1-p)/d}g_{1}\), where \(p>1\) is the power parameter.