Before revision:  HoPeS represents the 1-dimensional shape of a cloud at any scale by the Optimality Theorem. The Reconstruction Theorem gives conditions on a noisy sample of a graph when HoPeS provides a reconstructed graph with a correct homotopy type and within a small offset of the sample.
After revision:  HoPeS represents the 1-dimensional shape of a cloud at any scale . Other subgraphs of HoPeS reconstruct an unknown graph from its noisy point sample with a correct homotopy type and within a small offset of the sample.