ASAP : Towards Accurate, Stable and Accelerative Penetrating-Rank Estimation on Large Graphs

Abstract

Abstract. Pervasive web applications increasingly require a measure of similarity among objects. Penetrating-Rank (P-Rank) has been one of the promising link-based similarity metrics as it provides a comprehensive way of jointly encoding both incoming and outgoing links into computation for emerging applications. In this paper, we investigate P-Rank efficiency problem that encompasses its accuracy, stability and computational time. (1) We provide an accuracy estimate for iteratively computing P-Rank. A symmetric problem is to find the iteration number K needed for achieving a given accuracy ǫ. (2) We also analyze the stability of P-Rank, by showing that small choices of the damping factors would make P-Rank more stable and wellconditioned. (3) For undirected graphs, we also explicitly characterize the P-Rank solution in terms of matrices. This results in a novel non-iterative algorithm, termed ASAP , for efficiently computing P-Rank, which improves the CPU time from O(n 4 ) to O n 3 . Using real and synthetic data, we empirically verify the effectiveness and efficiency of our approaches.

On the Efficiency of Estimating Penetrating Rank on Large Graphs

“…Let Q be the backward transition matrix of G, whose entry q i,j = 1/|I(i)| if ∃ an edge (j, i) ∈ E, and 0 otherwise; and let P be the forward transition matrix of G, whose entry p i,j = 1/|O(i)| if ∃ an edge (i, j) ∈ E, and 0 otherwise. By virtue of our prior work [17], the P-Rank equation (1) then equivalently takes the simple form…”

“…P-Rank has become an appealing measure of similarity used in a variety of areas, such as publication network [26], top-k nearest neighbor search [13], and social graph [2,17]. The traditional method leverages a fixed-point iteration to compute P-Rank, yielding O(Kn 4 ) time in the worst case.…”

“…Closer to this work are [15,17,27]. Our prior work [17] focuses on P-Rank computations on undirected graphs by showing an O(n 3 )-time deterministic algorithm; however the optimization techniques in [17] rely mainly on the symmetry of the adjacency matrix.…”

“…To address this issue, Zhao et al [26] proposed to use P-Rank similarity to effectively incorporate both in-and out-links. Since then, P-Rank has attracted growing attention (e.g., [2,13,17,21]), and it can be widely used to any ubiquitous domain where SimRank is applicable, such as social graphs [2], and publication networks [13]. The intuition behind P-Rank is an improved version of SimRank: "two distinct vertices are similar if (a) they are referenced by similar vertices, and (b) they reference similar vertices".…”

“…Firstly, it is rather time-consuming to iteratively compute P-Rank on large graphs. Previous methods [17,26] deploy a fixed-point iterative paradigm for P-Rank computation. While these methods often attain good accuracy, they do not scale well for large graphs since they need to enumerate all n 2 vertex-pairs per iteration if there are n vertices in a graph.…”