Weighted Spectral Embedding of Graphs

TL;DR

This paper introduces a weighted spectral embedding method for graphs that incorporates node importance through weights, based on eigenvectors of a normalized Laplacian, with physical system analogies and real data experiments.

Contribution

It proposes a novel weighted spectral embedding approach that accounts for node importance and provides physical interpretations, enhancing graph analysis.

Findings

Weighted embedding affects node positioning significantly.

Physical analogies help interpret the embedding configurations.

Experiments demonstrate the impact of node weights on the embedding.

Abstract

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.