Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs
Abstract
Graphs (networks) are an important tool to model data in different domains. Realworld graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network $G$ and a vertex $r \in V(G)$, we propose an exact algorithm to compute betweenness score of $r$. Our algorithm precomputes a set $\mathcal{RV}(r)$, which is used to prune a huge amount of computations that do not contribute to the betweenness score of $r$. Time complexity of our algorithm depends on $\mathcal{RV}(r)$ and it is respectively $\Theta(\mathcal{RV}(r)\cdotE(G))$ and $\Theta(\mathcal{RV}(r)\cdotE(G)+\mathcal{RV}(r)\cdotV(G)\log V(G))$ for unweighted graphs and weighted graphs with positive weights. $\mathcal{RV}(r)$ is bounded from above by $V(G)1$ and in most cases, it is a small constant. Then, for the cases where $\mathcal{RV}(r)$ is large, we present a simple randomized algorithm that samples from $\mathcal{RV}(r)$ and performs computations for only the sampled elements. We show that this algorithm provides an $(\epsilon,\delta)$approximation to the betweenness score of $r$. Finally, we perform extensive experiments over several realworld datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.08739
 Bibcode:
 2017arXiv170808739H
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Social and Information Networks
 EPrint:
 Fundamenta Informaticae, Volume 182, Issue 3 (November 18, 2021) fi:8624