Visualizing Graphs with Structure Preserving Embedding

Structure Preserving Embedding (SPE) is a method for embedding graphs in lowdimensional Euclidean space such that the embedding preserves the graph’s global topological properties. Specifically, topology is preserved if a connectivity algorithm can recover the original graph from only the coordinates of its nodes after embedding. Given an input graph and an algorithm for linking embedded nodes, SPE learns a low-rank kernel matrix by means of a semidefinite program with linear constraints that captures the connectivity structure of the input graph. The SPE cost function ensures that the learned kernel is low-rank and thus the resulting embedding uses low-dimensional coordinates for each node that reproduce the original graph when processed by a connectivity algorithm (such as k-nearest neighbors, or b-matching). SPE provides significant improvements in terms of visualization and lossless compression of graphs, outperforming popular methods such as spectral embedding and spring embedding. Furthermore, we find that many classical graphs and networks can be properly embedded using only a few dimensions.