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Given a social network, which of its nodes are more central? This questionhas been asked many times in sociology, psychology and computer science, and awhole plethora of centrality measures (a.k.a. centrality indices, or rankings)were proposed to account for the importance of the nodes of a network. In thispaper, we approach the problem of computing geometric centralities, such ascloseness and harmonic centrality, on very large graphs; traditionally thistask requires an all-pairs shortest-path computation in the exact case, or anumber of breadth-first traversals for approximated computations, but thesetechniques yield very weak statistical guarantees on highly disconnectedgraphs. We rather assume that the graph is accessed in a semi-streamingfashion, that is, that adjacency lists are scanned almost sequentially, andthat a very small amount of memory (in the order of a dozen bytes) per node isavailable in core memory. We leverage the newly discovered algorithms based onHyperLogLog counters, making it possible to approximate a number of geometriccentralities at a very high speed and with high accuracy. While the applicationof similar algorithms for the approximation of closeness was attempted in theMapReduce framework, our exploitation of HyperLogLog counters reducesexponentially the memory footprint, paving the way for in-core processing ofnetworks with a hundred billion nodes using "just" 2TiB of RAM. Moreover, thecomputations we describe are inherently parallelizable, and scale linearly withthe number of available cores.

In-Core Computation of Geometric Centralities with HyperBall: A Hundred Billion Nodes and Beyond