Spectral distances on graphs

Jiao Gu, Bobo Hua, and Shiping Liu

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Submission date: 25. Feb. 2014
Pages: 27
published in: Discrete applied mathematics, 190-191 (2015), p. 56-74 
DOI number (of the published article): 10.1016/j.dam.2015.04.011
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Abstract:
By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using L^p Wasserstein distances between probability measures, we define the corresponding spectral distances d_p on the set of all graphs. This approach can even be extended to measuring the distances between infinite graphs. We prove that the diameter of the set of graphs, as a pseudo-metric space equipped with d₁, is one. We further study the behavior of d₁ when the size of graphs tends to infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation between d₁ and the evolutionary distance of biological networks is observed in simulations.