  We present a novel spectral embedding of graphs that incorporates weights
assigned to the nodes, quantifying their relative importance. This spectral
embedding is based on the first eigenvectors of some properly normalized
version of the Laplacian. We prove that these eigenvectors correspond to the
configurations of lowest energy of an equivalent physical system, either
mechanical or electrical, in which the weight of each node can be interpreted
as its mass or its capacitance, respectively. Experiments on a real dataset
illustrate the impact of weighting on the embedding.
