A Sharp PageRank Algorithm with Applications to Edge Ranking and Graph Sparsification

Abstract

Abstract. We give an improved algorithm for computing personalized PageRank vectors with tight error bounds which can be as small as Ω(n −p ) for any fixed positive integer p. The improved PageRank algorithm is crucial for computing a quantitative ranking of edges in a given graph. We will use the edge ranking to examine two interrelated problems -graph sparsification and graph partitioning. We can combine the graph sparsification and the partitioning algorithms using PageRank vectors to derive an improved partitioning algorithm.

Dirichlet PageRank and Trust-Based Ranking Algorithms

“…Presumably one's friends' actions carry a lot of weight in one's own decisions. Vertex v can efficiently compute a personalized PageRank vector as its ranking function using algorithms from [10], but PageRank alone will not take into account the implied trust between v and its friends. But using Dirichlet PageRank with boundary conditions, we can take v's trusted friends into account.…”

Finding and Visualizing Graph Clusters Using PageRank Optimization

“…Instead, we use an approximate PageRank algorithm as given in [4,10]. This approximation algorithm is much more tractable on large networks, because it can be computed using only the local graph structure around the starting seed vector s. Besides s and the jumping constant α, the algorithm requires an approximation parameter .…”

“…We remark that by using the sharp approximate PageRank algorithm in [10], the error bound δ for PageRank can be set to be quite small since the time Algorithm 1 PageRank-ClusteringA Input: G, k, Output: A set of centers C and partitions S, or nothing for all v ∈ G do compute pr(α, v) end for Find the roots of Φ (α) (There can be more than one root if G has a layered clustering structure. )…”

“…end if end for end for Algorithm 2 PageRank-ClusteringB Input: G, k, Output: A set of centers C and partitions S, or nothing for all v ∈ G do compute pr(α, v) end for Find the roots of Φ (α) within an error bound /2, by using sampling techniques from [32] involving O(log n) nodes, log(1/ ) values of α and δ-approximate PageRank vectors [4,10] where δ = /n 2 . (There can be more than one root if G has a layered clustering structure.…”

“…As long as G is connected, the PageRank vector will be nonzero on every vertex. Using the algorithms from [4,10], the approximation factor acts as a cutoff, and any node v with PageRank less than d v will be assigned zero. This is advantageous because the support of the approximate PageRank vector will be limited to the local community containing its seed.…”