Isotree: Tree clustering via metric embedding

Abstract

One of the problems that hinders the spectral analysis of trees is that they have a strong tendency to be co-spectral. As a result, structurally distinct trees possess degenerate graph-spectra, and spectral methods can be reliably used to neither compute distances between trees nor to cluster trees. The aim of this paper is to describe a method that can be used to alleviate this problem. We use the ISOMAP algorithm to embed the trees in a Euclidean space using the pattern of shortest distances between nodes. From the arrangement of nodes in this space, we compute a weighted proximity matrix, and from the proximity matrix a Laplacian matrix is computed. By transforming the graphs in this way we lift the co-spectrality of the trees. The spectrum of the Laplacian matrix for the embedded graphs may be used for purposes of comparing trees and for clustering them. Experiments on sets of shock graphs reveal the utility of the method on real-world data.