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(epsilon^{2}) singlesource 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 subset of V of size O(epsilon^{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+epsilon^{2}) distance computations. The estimate is based on a weighted sample of O(epsilon^{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 epsilon. Increasing the sample size by a O(log(n)) factor ensures that the probability that the relative error exceeds epsilon is polynomially small.
BibTeX  Entry
@InProceedings{chechik_et_al:LIPIcs:2015:5329,
author = {Shiri Chechik and Edith Cohen and Haim Kaplan},
title = {{Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
pages = {659679},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897897},
ISSN = {18688969},
year = {2015},
volume = {40},
editor = {Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5329},
URN = {urn:nbn:de:0030drops53291},
doi = {10.4230/LIPIcs.APPROXRANDOM.2015.659},
annote = {Keywords: Closeness Centrality; Average Distance; Metric Space; Weighted Sampling}
}
Keywords: 

Closeness Centrality; Average Distance; Metric Space; Weighted Sampling 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015) 
Issue Date: 

2015 
Date of publication: 

13.08.2015 