Closed form solution of similarity algorithms

Abstract

Algorithms defining similarities between objects of an information network are important to many IR tasks. SimRank algorithm and its variations are popularly used in many applications. Many fast algorithms are also developed. In this note, we first reformulate them as random walks on the network and express them using forward and backward transition probability in a matrix form. Second, we show that P-Rank (SimRank is just the special case of PRank) has a unique solution of ee T when decay factor c is equal to 1. We also show that SimFusion algorithm is a special case of P-Rank algorithm and prove that the similarity matrix of SimFusion is the product of PageRank vector. Our experiments on the web datasets show that for P-Rank the decay factor c doesn't seriously affect the similarity accuracy and accuracy of P-Rank is also higher than SimFusion and SimRank. SIMRANK AND P-RANK ALGORITHMSSimRank[1] is a method of measuring linkage-based similarity between objects in a graph that models the object-to-object relationships for a particular domain. The intuition behind SimRank similarity is that two objects are similar if they are linked by the similar objects. However, SimRank only consider in-link information on the information network but in fact out-link information is also useful for the similarity calculation on the real network. Thus, P-Rank[2] extends SimRank intuition and consider both out-link and in-link information. The intuition of P-Rank is that "two objects are similar if (1) they are linked by the similar objects; and (2) they link the similar objects."We proceed to present the formula to compute P-Rank. Given a graph G(V, E) consisting of a set of nodes V and a set of links E, the P-Rank similarity between objects a and b, denoted as S(a,b), is computed recursively as follows: | ( )|| ( )| 1 1 | ( 1 1 ( ) ( ( ), ( )) + | ( ) || ( ) | ( , ) (1-) ( ( ), ( )) | ( )|| ( )|

A comprehensive structural-based similarity measure in directed graphs

“…In fact, SimFusion can be easily transformed into SimRank with minor modification [4]. NetSim [39] in the worst case to O(n 2 d), which is further reduced to min(O(n 2 d), O(n 3 / log 2 n)) by using cross summation [26], where n is the node number of a given graph and d is the average degree.…”

On the Efficiency of Estimating Penetrating Rank on Large Graphs

“…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.…”

“…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".…”

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

“…There has been a surge of studies (e.g., [2,15,3,9,4,5,16,6]) on link-based analysis over information networks in recent years. PageRank became popular since the famous result of Page et al [6] was used by the Google search engine for ranking web sites based on link structures.…”

“…Since then, a host of new ranking algorithms for web pages have been developed. The famous results include the HITS algorithm [2] proposed by Jon et al (now used www.ask.com), SimRank [17,10,5], SimFusion [15,4] and P-Rank algorithm [7].…”