Local computation of pagerank contributions

Abstract

Motivated by the problem of detecting link-spam, we consider the following graph-theoretic primitive: Given a webgraph G, a vertex v in G, and a parameter δ ϵ (0, 1), compute the set of all vertices that contribute to v at least a δ-fraction of v’s PageRank. We call this set the δ-contributing set of v. To this end, we define the contribution vector of v to be the vector whose entries measure the contributions of every vertex to the PageRank of v. A local algorithm is one that produces a solution by adaptively examining only a small portion of the input graph near a specified vertex. We give an efficient local algorithm that computes an ϵ-approximation of the contribution vector for a given vertex by adaptively examining O(1/ϵ) vertices. Using this algorithm, we give a local approximation algorithm for the primitive defined above. Specifically, we give an algorithm that returns a set containing the δ-contributing set of v and at most O(1/δ) vertices from the δ/2-contributing set of v, and that does so by examining at most O(1/δ) vertices. We also give a local algorithm for solving the following problem: If there exist k vertices that contribute a ρ-fraction to the PageRank of v, find a set of k vertices that contribute at least a (ρ−ϵ)-fraction to the PageRank of v. In this case, we prove that our algorithm examines at most O(k/ϵ) vertices.