Euclidean Distances, soft and spectral Clustering on Weighted Graphs

We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the “raw coordinates” encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow data, properly conditioned, illustrate the procedure as well as visualization aspects.

💡 Research Summary

The paper introduces a novel class of Euclidean distances specifically designed for weighted graphs and demonstrates how these distances enable a thermodynamic soft‑clustering framework. Traditional spectral clustering embeds graph nodes into a low‑dimensional Euclidean space using the eigenvectors of the graph Laplacian and then applies hard clustering such as k‑means. This approach suffers from two major drawbacks: the distance metric derived from the raw eigenvectors does not fully capture the weighted connectivity of the graph, and hard assignments ignore the possibility that a node may belong partially to several clusters, which is especially problematic in networks with overlapping community structure.

To overcome these limitations, the authors first define “raw coordinates” as the set of selected Laplacian eigenvectors ({u_k}). They then introduce a scaling function (f(\lambda_k)) applied to each eigenvalue (\lambda_k) and construct a distance between nodes (i) and (j) as
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