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 e^(v) of its eccentricity eccG(v) such that
eccG(v)≤e^(v)≤eccG(v)+cδ 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,
eccG(v)≤eccT(v)≤eccG(v)+cδ. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of G, the smaller its
eccentricity is. 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∣2log2∣V∣)
time, where c′<c is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author