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 δhyperbolic 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 estimateê(v) of its eccentricity eccG(v) such that eccG(v) ≤ê(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 vertexThese 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 | 2 log 2 |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.

Easy computation of eccentricity approximating trees

“…We prove in what follows that such heuristics provide a quasi 2-approximation algorithm. Our analysis also extends to the recent construction given in [6] where the output is a shortest-path tree rooted at some almost central vertex.…”

“…Our main result will follow from this nice property. We note that all the constructions of eccentricity approximating spanning trees that are given in [6,10,18] are also based on shortest-path trees.…”

“…In particular, Prisner proved in [19] that every chordal graph admits an eccentricity 2-approximating spanning tree. His result has been recently generalized to larger classes of graphs [6,10]. -For instance, it was proved in [6] that we have ets(G) = O(δ(G)) for any graph G, with δ(G) being the graph hyperbolicity (see [15]).…”

“…His result has been recently generalized to larger classes of graphs [6,10]. -For instance, it was proved in [6] that we have ets(G) = O(δ(G)) for any graph G, with δ(G) being the graph hyperbolicity (see [15]). -However, the complexity of the following problem has been left open in [10]:…”

“…Our main result in this part is that any shortest-path tree that is rooted at some arbitrary almost central vertex in G is eccentricity O(ets(G))-approximating (Theorem 2). In particular, this is the case for all the spanning trees which are outputted by the quasi linear-time algorithms given in [6,10]. More generally, our approximation framework applies to any graph class where an almost central vertex can be computed efficiently (e.g., see [4,5]…”