Scalable Hierarchical Embeddings of Complex Networks

Graph representation learning has become important in order to understand and predict intrinsic structures in complex networks. A variety of embedding methods has in recent years been developed including the Latent Distance Modeling (LDM) approach. A major challenge is scaling network embedding approaches to very large networks and a drawback of LDM is the computational cost invoked evaluating the full likelihood having O(N^2) complexity, making such analysis of large networks infeasible. We propose a novel multiscale hierarchical estimate of the full likelihood of LDMs providing high details where the likelihood approximation is most important while scaling in complexity at O(NlogN). The approach relies on a clustering procedure approximating the Euclidean norm of every node pair according to the multiscale hierarchical structure imposed. We demonstrate the accuracy of our approximation and for the first time embed very large networks in the order of a million nodes using LDM and contrast the predictive performance to prominent scalable graph embedding approaches. We find that our approach significantly outperforms these existing scalable approaches in the ability to perform link prediction, node clustering, and classification utilizing a surprisingly low embedding dimensionality of two to three dimensions whereas the extracted hierarchical structure facilitates network visualization and interpretation. The developed scalable hierarchical embedding approach enables accurate low dimensional representations of very large networks providing detailed visualizations that can further our understanding of their properties and structure.