Fast approximation of centrality and distances in hyperbolic graphs

Abstract

We show that the eccentricities (and thus the centrality indices) of all vertices of a δ-hyperbolic graph G=(V, E) can be computed in linear time with an additive one-sided error of at most cδ, i.e., after a linear time preprocessing, for every vertex v of G one can compute in O(1) time an estimate ê(v) of its eccentricity ecc G (v) such that (Formula presented) for a small constant c. We prove that every δ-hyperbolic graph G has a shortest path tree, constructible in linear time, such that for every vertex v of G, (Formula presented). We also show that the distance matrix of G with an additive one-sided error of at most c′δ can be computed in O(|V| 2 log 2 |V|) time, where c′