Before revision:  Finally, we  demonstrate via experiments the preservation of the up Laplacian during sparsification, as well as the utility of sparsification with respect to spectral clustering .
After revision:  Finally, we introduce higher-order generalizations of spectral clustering and label propagation for simplicial complexes and demonstrate via experiments the preservation of the up Laplacian during sparsification, as well as the utility of sparsification with respect to spectral clustering .