LASAGNE: Locality and Structure Aware Graph Node Embedding

Abstract

Recent work has attempted to identify structure in social and information graphs by using the following approach: first, use random walk methods to explore the neighborhood of a node; second, use ideas from natural language processing to use this neighborhood information to learn vector representations of these nodes reflecting properties of the graph. Informally, the idea is that if a node is a member of a meaningful cluster or community, then the vector representation should be higher-quality, thereby leading to improved learning.In this paper, we identify and remedy an important shortcoming of this approach. In particular, we show that the performance of existing methodologies depends strongly on the structural properties of the graph, e.g., the size of the graph, whether the graph has a flat or upward-sloping Network Community Profile (NCP), whether the graph is expander-like, whether the classes of interest are more k-core-like or more peripheral, etc. For larger graphs with flat NCPs that are strongly expander-like, existing methods lead to random walks that expand rapidly, touching many dissimilar nodes, thereby leading to lower-quality vector representations that are less useful for downstream tasks.Based on our findings, we propose Lasagne, a methodology to learn locality and structure aware graph node embeddings in an unsupervised way. Rather than relying on global random walks or neighbors within fixed hop distances, Lasagne exploits strongly local Approximate Personalized PageRank stationary distributions to more precisely engineer local information into node embeddings. This leads, in particular, to more meaningful and more useful vector representations of nodes in poorly-structured graphs. We show that Lasagne leads to significant improvement in downstream multi-label classification for larger graphs with flat NCPs, that it is comparable for smaller graphs with upward-sloping NCPs, and that is comparable to existing methods for link prediction tasks. * nodes E = {f (v i )|v i ∈ V } in the d-dimensional vector space R d still reflects structural properties of G. For instance, such structural properties could be the similarity of the neighborhoods of two nodes v i and v j . The neighborhood N (v) of a node v is defined as the set of nodes having the highest probabilities to be visited by a random walk starting from node v, a geodesic walk starting from v, or some other related process. This means if N (The intuition behind these representation learning methods is that nodes having similar neighborhoods are similar to each other, and thus one can use information in the neighbors of a node to make predictions for a given node. Defining the right neighborhood for each node, however, is a challenging task. For example, in unsupervised multi-label classification, the labels of the nodes define the underlying local structure for a particular class, but often this does not necessarily overlap significantly with the local structure defined by the edge connectivity of the graph. Alternatively, realistic graphs...

On the Use of Unrealistic Predictions in Hundreds of Papers Evaluating Graph Representations

“…• Faerman et al (2018): "Precisely, this method uses the actual number of labels k each test instance has. ...…”

“…... However, in realworld situations it is fairly uncommon to have such prior knowledge of m." To be realistic, Faerman et al (2018); Liu and Kim (2018) predict labels by checking the sign of decision values. 5 We name this method and give its details as follows.…”

On the Use of Unrealistic Predictions in Hundreds of Papers Evaluating Graph Representations

“…... However, in real-world situations it is fairly uncommon to have such prior knowledge of m." To be realistic, Faerman et al (2018); Liu and Kim (2018) predict labels by checking the sign of decision values. 5 We name this method and give its details as follows.…”

“…For example, out of the works that criticized the unrealistic setting (see Section 2), Faerman et al (2018) used a fixed regularization parameter for comparing with past works, but Liu and Kim (2018) conducted cross-validation in their one-vs-rest implementation. Therefore, a more appropriate baseline should be the following extension of one-vs-restbasic:…”

5* Knowledge Graph Embeddings with Projective Transformations

“…The similarity of nodes in a KG is local, i.e. nodes of a neighborhood are more likely to be semantically more similar [5,6] than nodes at higher distance. A projective transformation is a bijective conformal mapping, i.e.…”