Defining and identifying the optimal embedding dimension of networks

Network embedding is a general-purpose machine learning technique that encodes network structure in vector spaces with tunable dimension. Choosing an appropriate embedding dimension -- small enough to be efficient and large enough to be effective -- is challenging but necessary to generate embeddings applicable to a multitude of tasks. Unlike most existing strategies that rely on performance maximization in downstream tasks, here we propose a principled method for the identification of an optimal dimension such that all structural information of a network is parsimoniously encoded. The method is validated on various embedding algorithms and a large corpus of real-world networks. Estimated values of the optimal dimension in real-world networks suggest that efficient encoding in low-dimensional spaces is usually possible.