We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments.
The eigenvalue distribution for such a tree is a smooth bell-shaped curve starting from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4 are semi-global oscillations (like Fourier modes) over the entire tree or one of the branches; on the other hand, those corresponding to the eigenvalues above 4 are much more localized and concentrated (like wavelets) around junctions/branching vertices.
For a special class of trees called starlike trees, we obtained a complete understanding of such phase transition phenomenon. For a general graph, we proved the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly larger than 2.
Moreover, we also proved that if a graph contains a branching path, then the magnitudes of the components of any eigenvector corresponding to the eigenvalue greater than 4 decay exponentially from the center toward the leaf of that branch. We have also identified a unique class of trees that can have an eigenvalue exactly equal to 4.
If time permits, I will also discuss some applications, e.g., data analysis and missing data recovery.
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