DeepAI

# Fast approximation of centrality and distances in hyperbolic graphs

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 ecc_G(v)≤ê(v)≤ ecc_G(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, ecc_G(v)≤ ecc_T(v)≤ ecc_G(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|^2^2|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.

• 19 publications
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02/19/2020

### Eccentricity terrain of δ-hyperbolic graphs

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### Eccentricity function in distance-hereditary graphs

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### Fast deterministic algorithms for computing all eccentricities in (hyperbolic) Helly graphs

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### Distance problems within Helly graphs and k-Helly graphs

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### Perturb and Combine to Identify Influential Spreaders in Real-World Networks

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### Discrete Hyperbolic Random Graph Model

The hyperbolic random graph model (HRG) has proven useful in the analysi...
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### Hyperbolic grids and discrete random graphs

We present an efficient algorithm for computing distances in hyperbolic ...