## Abstract

Betweenness centrality is a fundamental measure in social network analysis, expressing the importance or influence of individual vertices (or edges) in a network in terms of the fraction of shortest paths that pass through them. Since exact computation in large networks is prohibitively expensive, we present two efficient randomized algorithms for betweenness estimation. The algorithms are based on random sampling of shortest paths and offer probabilistic guarantees on the quality of the approximation. The first algorithm estimates the betweenness of all vertices (or edges): all approximate values are within an additive factor \(\varepsilon \in (0,1)\) from the real values, with probability at least \(1-\delta \). The second algorithm focuses on the top-K vertices (or edges) with highest betweenness and estimate their betweenness value to within a multiplicative factor \(\varepsilon \), with probability at least \(1-\delta \). This is the first algorithm that can compute such approximation for the top-K vertices (or edges). By proving upper and lower bounds to the VC-dimension of a range set associated with the problem at hand, we can bound the sample size needed to achieve the desired approximations. We obtain sample sizes that are independent from the number of vertices in the network and only depend on a characteristic quantity that we call the vertex-diameter, that is the maximum number of vertices in a shortest path. In some cases, the sample size is completely independent from any quantitative property of the graph. An extensive experimental evaluation on real and artificial networks shows that our algorithms are significantly faster and much more scalable as the number of vertices grows than other algorithms with similar approximation guarantees.

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Note that even if \(p_\emptyset =\emptyset \), the set \(\{p_\emptyset \}\) is not empty. It contains one element.

We use the normalized version of betweenness as we believe it to be more suitable for presenting approximation results.

After this work was accepted for publication, Bergamini and Meyerhenke (2015) presented an improved upper bound for VD(G) for undirected weighted graphs.

It is a well-known open problem whether there is an algorithm to perform a single

*s*,*t*-shortest path computation between a pair of vertices with smaller worst-case time complexity than the Single Source Shortest Path computation.Bounded-distance betweenness is also known as

*k*-betweenness. We prefer the former denomination to avoid confusion with*k*-path betweenness as defined by Kourtellis et al. (2012).We take the average, rather than the sum, as defined in the original work by Kourtellis et al. (2012), to normalize the value so that it belongs to the interval [0, 1]. This has no consequences on the results.

The implementations are available at http://cs.brown.edu/~matteo/centrsampl.tar.bz2. An independent implementation of our algorithms is available in NetworKit Staudt et al. (2014).

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## Acknowledgments

This project was supported, in part, by the National Science Foundation under award IIS-1247581. E. M. Kornaropoulos was supported, in part by the Kanellakis Fellowship at Brown University. We are thankful to Eli Upfal for his guidance and advice and to the anonymous reviewers of WSDM’14 and DMKD whose comments helped us improving this work.

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Responsible editor: Tina Eliassi-Rad.

A preliminary version of this work appeared in the proceedings of ACM WSDM’14 (Riondato and Kornaropoulos 2014).

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Riondato, M., Kornaropoulos, E.M. Fast approximation of betweenness centrality through sampling.
*Data Min Knowl Disc* **30**, 438–475 (2016). https://doi.org/10.1007/s10618-015-0423-0

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DOI: https://doi.org/10.1007/s10618-015-0423-0