Sublinear algorithms for local graph centrality estimation

Marco Bressan, Enoch Peserico, Luca Pretto

We study the complexity of local graph centrality estimation, with the goal of approximating the centrality score of a given target node while exploring only a sublinear number of nodes/arcs of the graph and performing a sublinear number of elementary operations. We develop a technique, that we apply to the PageRank and Heat Kernel centralities, for building a low-variance score estimator through a local exploration of the graph. We obtain an algorithm that, given any node in any graph of $m$ arcs, with probability $(1-\delta)$ computes a multiplicative $(1\pm\epsilon)$-approximation of its score by examining only $\tilde{O}(\min(m^{2/3} \Delta^{1/3} d^{-2/3},\, m^{4/5} d^{-3/5}))$ nodes/arcs, where $\Delta$ and $d$ are respectively the maximum and average outdegree of the graph (omitting for readability $\operatorname{poly}(\epsilon^{-1})$ and $\operatorname{polylog}(\delta^{-1})$ factors). A similar bound holds for computational complexity. We also prove a lower bound of $\Omega(\min(m^{1/2} \Delta^{1/2} d^{-1/2}, \, m^{2/3} d^{-1/3}))$ for both query complexity and computational complexity. Moreover, our technique yields a $\tilde{O}(n^{2/3})$ query complexity algorithm for the graph access model of [Brautbar et al., 2010], widely used in social network mining; we show this algorithm is optimal up to a sublogarithmic factor. These are the first algorithms yielding worst-case sublinear bounds for general directed graphs and any choice of the target node.

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