Ultrametric graphons model hierarchical community networks and yield closed-form Laplacian spectra that approximate those of sampled random graphs with high probability as hierarchy depth grows.
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The ultrametric phylogenetic Laplacian has closed-form eigenvalues that aggregate clade-weighted branch lengths and eigenvectors supported on individual clades, enabling linear-time spectral reconstruction and eigenmode analysis of traits.
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Ultrametric Graphons and Hierarchical Community Networks: Spectral Theory and Applications
Ultrametric graphons model hierarchical community networks and yield closed-form Laplacian spectra that approximate those of sampled random graphs with high probability as hierarchy depth grows.
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Spectral Geometry and Heat Kernels on Phylogenetic Trees
The ultrametric phylogenetic Laplacian has closed-form eigenvalues that aggregate clade-weighted branch lengths and eigenvectors supported on individual clades, enabling linear-time spectral reconstruction and eigenmode analysis of traits.