Conditional Network Embeddings
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Network embeddings map the nodes of a given network into d-dimensional Euclidean space R^d. Ideally, this mapping is such that `similar' nodes are mapped onto nearby points, such that the embedding can be used for purposes such as link prediction (if `similar' means being `more likely to be connected') or classification (if `similar' means `being more likely to have the same label'). In recent years various methods for network embedding have been introduced. These methods all follow a similar strategy, defining a notion of similarity between nodes (typically deeming nodes more similar if they are nearby in the network in some metric), a distance measure in the embedding space, and minimizing a loss function that penalizes large distances for similar nodes or small distances for dissimilar nodes. A difficulty faced by existing methods is that certain networks are fundamentally hard to embed due to their structural properties, such as (approximate) multipartiteness, certain degree distributions, or certain kinds of assortativity. Overcoming this difficulty, we introduce a conceptual innovation to the literature on network embedding, proposing to create embeddings that maximally add information with respect to such structural properties (e.g. node degrees, block densities, etc.). We use a simple Bayesian approach to achieve this, and propose a block stochastic gradient descent algorithm for fitting it efficiently. Finally, we demonstrate that the combination of information such structural properties and a Euclidean embedding provides superior performance across a range of link prediction tasks. Moreover, we demonstrate the potential of our approach for network visualization.
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