Mass-Aware Spectral Clustering on Adaptive Supernodes

Spectral clustering can recover clusters of arbitrary shape, but only from the graph it is given, and graphs built from local affinities or compact representatives poorly describe extended clusters with unequal sample support. Density peaks provide complementary cues about which regions are well supported and well separated, but local prominence alone does not determine which regions share a label. We treat these cues as evidence for graph construction rather than as clustering decisions. The result is mass-aware spectral clustering (MASC) on adaptive supernodes: hierarchical grouping forms supernodes that follow the manifold structure and remain linked where they touch. Each nonroot supernode connects to its nearest predecessor in the mass order, forming a sparse peak-affinity tree whose weights are the reciprocal of mass times separation, so that heavy, prominent groups can be cut at low cost. A mass-normalized spectral objective then determines memberships jointly, and mass-weighted rounding propagates the same sample counts to the final labels. We prove that the reduced problem is exactly RatioCut on a lifted sample-level graph over supernode-respecting partitions and derive a boundary perturbation bound whose boundary-to-gap ratio predicts spectral stability. MASC tunes a single resolution parameter besides the cluster count and requires no kernel bandwidth or neighborhood size, and the eigenproblem scales with the number of supernodes rather than the number of samples. On real datasets, MASC attains the highest ACC and NMI against density-peak and spectral baselines, with average gains of 9.11/4.17 points over the best density-peak baseline and 9.62/5.79 over the best spectral baseline on each dataset.