Sig-SR: SimRank Search over Singular Graphs

SimRank is an attractive structural-context measure of similarity between two objects in a graph. It recursively follows the intuition that "two objects are similar if they are referenced by similar objects". The best known matrix-based method [1] for calculating SimRank, however, implies an assumption that the graph is non-singular, i.e., its adjacency matrix is invertible. In reality, non-singular graphs are very rare; such an assumption in [1] is too restrictive in practice. In this paper, we provide a treatment of [1], by supporting similarity assessment on non-invertible adjacency matrices. Assume that a singular graph G has n nodes, with r (< n) being the rank of its adjacency matrix. (1) We show that SimRank matrix S on G has an elegant structure: S can be represented as a rank r matrix plus a scaled identity matrix. (2) By virtue of this, an efficient algorithm over singular graphs, Sig-SR, is proposed for calculating all-pairs SimRank in O(r(n(2) + Kr-2)) time for K iterations. In contrast, the only known matrix-based algorithm that supports singular graphs [2] needs O(r(4)n(2)) time. The experimental results on real and synthetic datasets demonstrate the superiority of Sig-SR on singular graphs against its baselines.