Graph Augmentation via Metric Embedding

Abstract

Kleinberg [17] proposed in 2000 the first random graph model achieving to reproduce small world navigability, i.e. the ability to greedily discover polylogarithmic routes between any pair of nodes in a graph, with only a partial knowledge of distances. Following this seminal work, a major challenge was to extend this model to larger classes of graphs than regular meshes, introducing the concept of augmented graphs navigability. In this paper, we propose an original method of augmentation, based on metrics embeddings. Precisely, we prove that, for any ε> 0, any graph G such that its shortest paths metric admits an embedding of distorsion γ into ℝ^d can be augmented by one link per node such that greedy routing computes paths of expected length $O(\frac1\varepsilon\gamma^d\log^{2+\varepsilon}n)$ between any pair of nodes with the only knowledge of G. Our method isolates all the structural constraints in the existence of a good quality embedding and therefore enables to enlarge the characterization of augmentable graphs.