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1. DeepWalking Backwards: From Embeddings Back to Graphs.

   Chanpuriya, Sudhanshu; Musco, Cameron; Sotiropoulos, Konstantinos; Tsourakakis, Charalampos E. (January 2021, International Conference on Machine Learning (ICML))

   null (Ed.)

   Low-dimensional node embeddings play a key role in analyzing graph datasets. However, little work studies exactly what information is encoded by popular embedding methods, and how this information correlates with performance in downstream machine learning tasks. We tackle this question by studying whether embeddings can be inverted to (approximately) recover the graph used to generate them. Focusing on a variant of the popular DeepWalk method (Perozzi et al., 2014; Qiu et al., 2018), we present algorithms for accurate embedding inversion - i.e., from the low-dimensional embedding of a graph G, we can find a graph H with a very similar embedding. We perform numerous experiments on real-world networks, observing that significant information about G, such as specific edges and bulk properties like triangle density, is often lost in H. However, community structure is often preserved or even enhanced. Our findings are a step towards a more rigorous understanding of exactly what information embeddings encode about the input graph, and why this information is useful for learning tasks. 

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2. Anti-Section Transitive Closure

   Green, Oded; Du, Zhihui; Patel, Sanyamee; Xie, Zehui; Liu, Hang Liu; Bader, David A. (December 2021, The 28th IEEE International Conference on High Performance Computing, Data, and Analytics (HiPC))

   The transitive closure of a graph is a new graph where every vertex is directly connected to all vertices to which it had a path in the original graph. Transitive closures are useful for reachability and relationship querying. Finding the transitive closure can be computationally expensive and requires a large memory footprint as the output is typically larger than the input. Some of the original research on transitive closures assumed that graphs were dense and used dense adjacency matrices. We have since learned that many real-world networks are extremely sparse, and the existing methods do not scale. In this work, we introduce a new algorithm called Anti-section Transitive Closure (ATC) for finding the transitive closure of a graph. We present a new parallel edges operation – anti-sections – for finding new edges to reachable vertices. ATC scales to massively multithreaded systems such as NVIDIA’s GPU with tens of thousands of threads. We show that the anti-section operation shares some traits with the triangle counting intersection operation in graph analysis. Lastly, we view the transitive closure problem as a dynamic graph problem requiring edge insertions. By doing this, our memory footprint is smaller. We also show a method for creating the batches in parallel using two different techniques: dual-round and hash. Using these techniques and the Hornet dynamic graph data structure, we show our new algorithm on an NVIDIA Titan V GPU. We compare with other packages such as NetworkX, SEI-GBTL, SuiteSparse, and cuSparse. 

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3. Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones

   Bai, J; Ying, R; Ren, H; Leskovec, J (December 2021, Advances in neural information processing systems)

   Hierarchical relations are prevalent and indispensable for organizing human knowledge captured by a knowledge graph (KG). The key property of hierarchical relations is that they induce a partial ordering over the entities, which needs to be modeled in order to allow for hierarchical reasoning. However, current KG embeddings can model only a single global hierarchy (single global partial ordering) and fail to model multiple heterogeneous hierarchies that exist in a single KG. Here we present ConE (Cone Embedding), a KG embedding model that is able to simultaneously model multiple hierarchical as well as non-hierarchical relations in a knowledge graph. ConE embeds entities into hyperbolic cones and models relations as transformations between the cones. In particular, ConE uses cone containment constraints in different subspaces of the hyperbolic embedding space to capture multiple heterogeneous hierarchies. Experiments on standard knowledge graph benchmarks show that ConE obtains state-of-the-art performance on hierarchical reasoning tasks as well as knowledge graph completion task on hierarchical graphs. In particular, our approach yields new state-of-the-art Hits@1 of 45.3% on WN18RR and 16.1% on DDB14 (0.231 MRR). As for hierarchical reasoning task, our approach outperforms previous best results by an average of 20% across the three datasets. 

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4. Metaconcepts: Isolating Context in Word Embeddings

   https://doi.org/10.1109/MIPR.2019.00110  

   Sutor, Peter; Aloimonos, Yiannis; Fermuller, Cornelia; Summers-Stay, Douglas (March 2019, 2019 IEEE Conference on Multimedia Information Processing and Retrieval (MIPR))

   ord embeddings are commonly used to measure word-level semantic similarity in text, especially in direct word- to-word comparisons. However, the relationships between words in the embedding space are often viewed as approximately linear and concepts comprised of multiple words are a sort of linear combination. In this paper, we demonstrate that this is not generally true and show how the relationships can be better captured by leveraging the topology of the embedding space. We propose a technique for directly computing new vectors representing multiple words in a way that naturally combines them into a new, more consistent space where distance better correlates to similarity. We show that this technique works well for natural language, even when it comprises multiple words, on a simple task derived from WordNet synset descriptions and examples of words. Thus, the generated vectors better represent complex concepts in the word embedding space. 

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5. Capacity and Bias of Learned Geometric Embeddings for Directed Graphs

   Boratko, Michael; Zhang, Dongxu; Monath, Nicholas; Vilnis, Luke; Clarkson, Kenneth L; McCallum, Andrew (January 2021, NeurIPS 2021)

   A wide variety of machine learning tasks such as knowledge base completion, ontology alignment, and multi-label 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 real-world 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 region-based geometric representations on several families of synthetic and real-world 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. 

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