An unsupervised framework for comparing graph embeddings

Graph embedding is the transformation of vertices of a graph into set of vectors. A good embedding should capture the graph topology, vertex-to-vertex relationship and other relevant information about the graph, its subgraphs and vertices. If these objectives are achieved, an embedding is a meaningful, understandable and compressed representations of a network. Finally, vector operations are simpler and faster than comparable operations on graphs. The main challenge is that one needs to make sure that embeddings well describe the properties of the graphs. In particular, a decision has to be made on the embedding dimensionality which highly impacts the quality of an embedding. As a result, selecting the best embedding is a challenging task and very often requires domain experts. In this article, we propose a 'divergence score' that can be assigned to embeddings to help distinguish good ones from bad ones. This general framework provides a tool for an unsupervised graph embedding comparison. In order to achieve it, we needed to generalize the well-known Chung-Lu model to incorporate geometry which is an interesting result in its own right. In order to test our framework, we did a number of experiments with synthetic networks as well as real-world networks, and various embedding algorithms.