2015/03/30 by Shiri Chechik, Edith Cohen, Chechik, Shiri +3 · 16 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Graph Theory Research #Algorithm #Centrality #Combinatorics #Complex Network Analysis Techniques #Complexity and Algorithms in Graphs #Computation #Computer science #Discrete mathematics #Distance matrix #Mathematics #Metric (unit) #Metric space #Pairwise comparison #Sample size determination #Set (abstract data type) #Statistics #Triangle inequality #cs.LG #cs.SI
paper · pdf · doi:10.48550/arxiv.1503.08528
published in arXiv (Cornell University) (Cornell University) · 21 pages, will appear in the Proceedings of RANDOM 2015
openalex publication_date 2015/03/30 · arxiv created 2015/06/26 · arxiv updated 2015/06/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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(ε-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(ε-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+ε-2) distance computations. The estimate is based on a weighted sample of O(ε-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 ε. Increasing the sample size by a O(log n) factor ensures that the probability that the relative error exceeds ε is polynomially small.