Weighted Spectral Embedding of Graphs
This work addresses the need for weighted graph embeddings in machine learning, but it appears incremental as it builds on existing spectral methods by adding node weights.
The authors tackled the problem of incorporating node importance into graph spectral embeddings by introducing a weighted spectral embedding method based on normalized Laplacian eigenvectors, proving its equivalence to low-energy configurations in physical systems, and demonstrating its impact through experiments on a real dataset.
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.