Weighted Spectral Embedding of Graphs

Thomas Bonald 1, 2 Alexandre Hollocou 3, 2 Marc Lelarge 3, 4, 2
3 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
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.
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Thomas Bonald, Alexandre Hollocou, Marc Lelarge. Weighted Spectral Embedding of Graphs. 56th Annual Allerton Conference on Communication, Control, and Computing, Oct 2018, Urbana-Champaign, United States. ⟨hal-01887680⟩

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