## Abstract

The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness centrality of a node, is a popular importance measure in the study of social networks. We develop novel structural insights on the sparsifiability of the distance relation via weighted sampling. Based on that, we present highly practical algorithms with strong statistical guarantees for fundamental problems. We show that the average distance (and hence the centrality) for all nodes in a graph can be estimated using O(∈^{-2}) single-source distance computations. For a set V of n points in a metric space, we show that after preprocessing which uses O(n) distance computations we can compute a weighted sample S ⊂ V of size O(∈^{-2}) such that the average distance from any query point v to V can be estimated from the distances from v to S. Finally, we show that for a set of points V in a metric space, we can estimate the average pairwise distance using O(n + ∈^{-2}) distance computations. The estimate is based on a weighted sample of O(∈^{-2}) pairs of points, which is computed using O(n) distance computations. Our estimates are unbiased with normalized mean square error (NRMSE) of at most ∈. Increasing the sample size by a O(log n) factor ensures that the probability that the relative error exceeds ∈ is polynomially small.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015 |

Editors | Naveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 659-679 |

Number of pages | 21 |

ISBN (Electronic) | 9783939897897 |

DOIs | |

State | Published - 1 Aug 2015 |

Event | 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United States Duration: 24 Aug 2015 → 26 Aug 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 40 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 |
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Country/Territory | United States |

City | Princeton |

Period | 24/08/15 → 26/08/15 |

## Keywords

- Average distance
- Closeness centrality
- Metric space
- Weighted sampling