Closed form solution of similarity algorithms

Algorithms defining similarities between objects of an information network are important of 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 probably in a matrix form. Second, we show that P-Rank (SimRank is only the special case of P-Rank) has a unique solution of eeT 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.