All-Distances Sketches, Revisited: HIP Estimators for Massive Graphs Analysis

Graph datasets with billions of edges, such as social and Web graphs, are prevalent. To be feasible, computation on such large graphs should scale linearly with graph size. All-distances sketches (ADSs) are emerging as a powerful tool for scalable computation of some basic properties of individual nodes or the whole graph. ADSs were first proposed two decades ago (Cohen 1994) and more recent algorithms include ANF (Palmer, Gibbons, and Faloutsos 2002) and hyperANF (Boldi, Rosa, and Vigna 2011). A sketch of logarithmic size is computed for each node in the graph and the computation in total requires only a near linear number of edge relaxations. From the ADS of a node, we can estimate neighborhood cardinalities (the number of nodes within some query distance) and closeness centrality. More generally we can estimate the distance distribution, effective diameter, similarities, and other parameters of the full graph. We make several contributions which facilitate a more effective use of ADSs for scalable analysis of massive graphs. We provide, for the first time, a unified exposition of ADS algorithms and applications. We present the Historic Inverse Probability (HIP) estimators which are applied to the ADS of a node to estimate a large natural class of queries including neighborhood cardinalities and closeness centralities. We show that our HIP estimators have at most half the variance of previous neighborhood cardinality estimators and that this is essentially optimal. Moreover, HIP obtains a polynomial improvement over state of the art for more general domain queries and the estimators are simple, flexible, unbiased, and elegant. The ADS generalizes Min-Hash sketches, used for approximating cardinality (distinct count) on data streams. We obtain lower bounds on Min-Hash cardinality estimation using classic estimation theory. We illustrate the power of HIP, both in terms of ease of application and estimation quality, by comparing it to the HyperLogLog algorithm (Flajolet et al. 2007), demonstrating a significant improvement over this state-of-the-art practical algorithm. We also study the quality of ADS estimation of distance ranges, generalizing the near-linear time factor-2 approximation of the diameter.