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., 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.