Sublinear algorithms for local graph centrality estimation

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 - δ) computes a multiplicative (1 ± ϵ)-approximation of its score by examining only Õ(min(m 2/3 Δ 1 / 3 d -2 / 3 , m 4 / 5 d -3 / 5 )) nodes/arcs, where Δ and d are respectively the maximum and average outdegree of the graph (omitting for readability poly(ϵ -1 ) and polylog(δ -1 ) factors). A similar bound holds for computational cost. We also prove a lower bound of Ω(min(m 1/2 Δ 1/2 d -1/2 , m 2/3 d -1/3 )) for both query complexity and computational complexity. Moreover, our technique yields a Õ(n 2 / 3 )-queries algorithm for an n-node graph in the access model of Brautbar et al. [1], 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.