A wide variety of machine learning tasks such as knowledge base completion, ontology alignment, and multilabel classification can benefit from incorporating into learning differentiable representations of graphs or taxonomies. While vectors in Euclidean space can theoretically represent any graph, much recent work shows that alternatives such as complex, hyperbolic, order, or box embeddings have geometric properties better suited to modeling realworld graphs. Experimentally these gains are seen only in lower dimensions, however, with performance benefits diminishing in higher dimensions. In this work, we introduce a novel variant of box embeddings that uses a learned smoothing parameter to achieve better representational capacity than vector models in low dimensions, while also avoiding performance saturation common to other geometric models in high dimensions. Further, we present theoretical results that prove box embeddings can represent any DAG. We perform rigorous empirical evaluations of vector, hyperbolic, and regionbased geometric representations on several families of synthetic and realworld directed graphs. Analysis of these results exposes correlations between different families of graphs, graph characteristics, model size, and embedding geometry, providing useful insights into the inductive biases of various differentiable graph representations.
Same Stats, Different Graphs: Exploring the Space of Graphs in Terms of Graph Properties
Data analysts commonly utilize statistics to summarize large datasets. While it is often sufficient to explore only the summary statistics of a dataset (e.g., min/mean/max), Anscombe's Quartet demonstrates how such statistics can be misleading. We consider a similar problem in the context of graph mining. To study the relationships between different graph properties and summary statistics, we examine loworder nonisomorphic graphs and provide a simple visual analytics system to explore correlations across multiple graph properties. However, for larger graphs, studying the entire space quickly becomes intractable. We use different random graph generation methods to further look into the distribution of graph properties for higher order graphs and investigate the impact of various sampling methodologies. We also describe a method for generating many graphs that are identical over a number of graph properties and statistics yet are clearly different and identifiably distinct.
 Award ID(s):
 1839274
 Publication Date:
 NSFPAR ID:
 10179527
 Journal Name:
 IEEE Transactions on Visualization and Computer Graphics
 Page Range or eLocationID:
 1 to 1
 ISSN:
 10772626
 Sponsoring Org:
 National Science Foundation
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