Drawing Large Graphs by Low-Rank Stress Majorization

Optimizing a stress model is a natural technique for drawing graphs: one seeks an embedding into Rd which best preserves the induced graph metric. Current approaches to solving the stress model for a graph with |V| nodes and |E| edges require the full all-pairs shortest paths (APSP) matrix, which takes O(|V|2 log |E|+ |V||E|) time and O(|V|2) space. We propose a novel algorithm based on a low-rank approximation to the required matrices. The crux of our technique is an observation that it is possible to approximate the full APSP matrix, even when only a small subset of its entries are known. Our algorithm takes time O(k|V|+ |V| log |V|+ |E|) per iteration with a preprocessing time of O(k3 + k(|E|+ |V| log |V|)+ k2|V|) and memory usage of O(k|V|), where a user-defined parameter k trades off quality of approximation with running time and space. We give experimental results which show, to the best of our knowledge, the largest (albeit approximate) full stress model based layouts to date.