Hyperbolic Embeddings for Near-Optimal Greedy Routing

Abstract

Greedy routing computes paths between nodes in a network by successively moving to the neighbor closest to the target with respect to coordinates given by an embedding into some metric space. Its advantage is that only local information is used for routing decisions. We present different algorithms for generating graph embeddings into the hyperbolic plane that are well suited for greedy routing. In particular, our embeddings guarantee that greedy routing always succeeds in reaching the target, and we try to minimize the lengths of the resulting greedy paths. We evaluate our algorithm on multiple generated and real-world networks. For networks that are generally assumed to have a hidden underlying hyperbolic geometry, such as the Internet graph [3], we achieve near-optimal results (i.e., the resulting greedy paths are only slightly longer than the corresponding shortest paths). In the case of the Internet graph, they are only 6% longer when using our best algorithm, which greatly improves upon the previous best known embedding, whose creation required substantial manual intervention. In addition to measuring the stretch, we empirically evaluate our algorithms regarding the size of the coordinates of the resulting embeddings and observe how it impacts the success rate when coordinates are not represented with very high precision. Since numerical difficulties are a major issue when performing computations in the hyperbolic plane, we consider variations of our algorithm that improve the success rate when using coordinates with lower precision.