A simple yet effective baseline for non-attributed graph classification

Graphs are complex objects that do not lend themselves easily to typical learning tasks. Recently, a range of approaches based on graph kernels or graph neural networks have been developed for graph classification and for representation learning on graphs in general. As the developed methodologies become more sophisticated, it is important to understand which components of the increasingly complex methods are necessary or most effective. As a first step, we develop a simple yet meaningful graph representation, and explore its effectiveness in graph classification. We test our baseline representation for the graph classification task on a range of graph datasets. Interestingly, this simple representation achieves similar performance as the state-of-the-art graph kernels and graph neural networks for non-attributed graph classification. Its performance on classifying attributed graphs is slightly weaker as it does not incorporate attributes. However, given its simplicity and efficiency, we believe that it still serves as an effective baseline for attributed graph classification. Our graph representation is efficient (linear-time) to compute. We also provide a simple connection with the graph neural networks. Note that these observations are only for the task of graph classification while existing methods are often designed for a broader scope including node embedding and link prediction. The results are also likely biased due to the limited amount of benchmark datasets available. Nevertheless, the good performance of our simple baseline calls for the development of new, more comprehensive benchmark datasets so as to better evaluate and analyze different graph learning methods. Furthermore, given the computational efficiency of our graph summary, we believe that it is a good candidate as a baseline method for future graph classification (or even other graph learning) studies.Comment: 13 pages. Shorter version appears at 2019 ICLR Workshop: Representation Learning on Graphs and Manifolds. arXiv admin note: text overlap with arXiv:1810.00826 by other author

FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees

The Gromov-Hausdorff distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Any finite tree with positive edge weight can be viewed as a metric tree where the weight is treated as edge length and the metric is the induced shortest path metric in the tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP-hard to approximate the Gromov-Hausdorff distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximate the Gromov-Hausdorff distance between two general metric trees within a multiplicative factor of 14. Interestingly, the development of our algorithm is made possible by a connection between the Gromov-Hausdorff distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary (it is a variant of the Reeb graphs and contour trees), and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we re-define the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a fixed-parameter tractable algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the Gromov-Hausdorff distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of O(sqrt{n}) even for trees with unit edge length

Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering

Hierarchical clustering is a popular method for analyzing data which associates a tree to a dataset. Hartigan consistency has been used extensively as a framework to analyze such clustering algorithms from a statistical point of view. Still, as we show in the paper, a tree which is Hartigan consistent with a given density can look very different than the correct limit tree. Specifically, Hartigan consistency permits two types of undesirable configurations which we term over-segmentation and improper nesting. Moreover, Hartigan consistency is a limit property and does not directly quantify difference between trees. In this paper we identify two limit properties, separation and minimality, which address both over-segmentation and improper nesting and together imply (but are not implied by) Hartigan consistency. We proceed to introduce a merge distortion metric between hierarchical clusterings and show that convergence in our distance implies both separation and minimality. We also prove that uniform separation and minimality imply convergence in the merge distortion metric. Furthermore, we show that our merge distortion metric is stable under perturbations of the density. Finally, we demonstrate applicability of these concepts by proving convergence results for two clustering algorithms. First, we show convergence (and hence separation and minimality) of the recent robust single linkage algorithm of Chaudhuri and Dasgupta (2010). Second, we provide convergence results on manifolds for topological split tree clustering