Fast incremental SimRank on link-evolving graphs

SimRank is an arresting measure of node-pair similarity based on hyperlinks. It iteratively follows the concept that 2 nodes are similar if they are referenced by similar nodes. Real graphs are often large, and links constantly evolve with small changes over time. This paper considers fast incremental computations of SimRank on link-evolving graphs. The prior approach [12] to this issue factorizes the graph via a singular value decomposition (SVD) first, and then incrementally maintains this factorization for link updates at the expense of exactness. Consequently, all node-pair similarities are estimated in O(r4n2) time on a graph of n nodes, where r is the target rank of the low-rank approximation, which is not negligibly small in practice. In this paper, we propose a novel fast incremental paradigm. (1) We characterize the SimRank update matrix ΔS, in response to every link update, via a rank-one Sylvester matrix equation. By virtue of this, we devise a fast incremental algorithm computing similarities of n2 node-pairs in O(Kn2) time for K iterations. (2) We also propose an effective pruning technique capturing the “affected areas” of ΔS to skip unnecessary computations, without loss of exactness. This can further accelerate the incremental SimRank computation to O(K(nd+|AFF|)) time, where d is the average in-degree of the old graph, and |AFF| (≤ n2) is the size of “affected areas” in ΔS, and in practice, |AFF| ≪ n2. Our empirical evaluations verify that our algorithm (a) outperforms the best known link-update algorithm [12], and (b) runs much faster than its batch counterpart when link updates are small

Gauging Correct Relative Rankings For Similarity Search

© 2015 ACM.One of the important tasks in link analysis is to quantify the similarity between two objects based on hyperlink structure. SimRank is an attractive similarity measure of this type. Existing work mainly focuses on absolute SimRank scores, and often harnesses an iterative paradigm to compute them. While these iterative scores converge to exact ones with the increasing number of iterations, it is still notoriously difficult to determine how well the relative orders of these iterative scores can be preserved for a given iteration. In this paper, we propose efficient ranking criteria that can secure correct relative orders of node-pairs with respect to SimRank scores when they are computed in an iterative fashion. Moreover, we show the superiority of our criteria in harvesting top-K SimRank scores and bucket orders from a full ranking list. Finally, viable empirical studies verify the usefulness of our techniques for SimRank top-K ranking and bucket ordering

Dynamical SimRank search on time-varying networks

SimRank is an appealing pair-wise similarity measure based on graph structure. It iteratively follows the intuition that two nodes are assessed as similar if they are pointed to by similar nodes. Many real graphs are large, and links are constantly subject to minor changes. In this article, we study the efficient dynamical computation of all-pairs SimRanks on time-varying graphs. Existing methods for the dynamical SimRank computation [e.g., LTSF (Shao et al. in PVLDB 8(8):838–849, 2015) and READS (Zhang et al. in PVLDB 10(5):601–612, 2017)] mainly focus on top-k search with respect to a given query. For all-pairs dynamical SimRank search, Li et al.’s approach (Li et al. in EDBT, 2010) was proposed for this problem. It first factorizes the graph via a singular value decomposition (SVD) and then incrementally maintains such a factorization in response to link updates at the expense of exactness. As a result, all pairs of SimRanks are updated approximately, yielding (Formula presented.) time and (Formula presented.) memory in a graph with n nodes, where r is the target rank of the low-rank SVD. Our solution to the dynamical computation of SimRank comprises of five ingredients: (1) We first consider edge update that does not accompany new node insertions. We show that the SimRank update (Formula presented.) in response to every link update is expressible as a rank-one Sylvester matrix equation. This provides an incremental method requiring (Formula presented.) time and (Formula presented.) memory in the worst case to update (Formula presented.) pairs of similarities for K iterations. (2) To speed up the computation further, we propose a lossless pruning strategy that captures the “affected areas” of (Formula presented.) to eliminate unnecessary retrieval. This reduces the time of the incremental SimRank to (Formula presented.), where m is the number of edges in the old graph, and (Formula presented.) is the size of “affected areas” in (Formula presented.), and in practice, (Formula presented.). (3) We also consider edge updates that accompany node insertions, and categorize them into three cases, according to which end of the inserted edge is a new node. For each case, we devise an efficient incremental algorithm that can support new node insertions and accurately update the affected SimRanks. (4) We next study batch updates for dynamical SimRank computation, and design an efficient batch incremental method that handles “similar sink edges” simultaneously and eliminates redundant edge updates. (5) To achieve linear memory, we devise a memory-efficient strategy that dynamically updates all pairs of SimRanks column by column in just (Formula presented.) memory, without the need to store all (Formula presented.) pairs of old SimRank scores. Experimental studies on various datasets demonstrate that our solution substantially outperforms the existing incremental SimRank methods and is faster and more memory-efficient than its competitors on million-scale graphs

Exact Single-Source SimRank Computation on Large Graphs

SimRank is a popular measurement for evaluating the node-to-node similarities based on the graph topology. In recent years, single-source and top-$k$ SimRank queries have received increasing attention due to their applications in web mining, social network analysis, and spam detection. However, a fundamental obstacle in studying SimRank has been the lack of ground truths. The only exact algorithm, Power Method, is computationally infeasible on graphs with more than $10^6$ nodes. Consequently, no existing work has evaluated the actual trade-offs between query time and accuracy on large real-world graphs. In this paper, we present ExactSim, the first algorithm that computes the exact single-source and top-$k$ SimRank results on large graphs. With high probability, this algorithm produces ground truths with a rigorous theoretical guarantee. We conduct extensive experiments on real-world datasets to demonstrate the efficiency of ExactSim. The results show that ExactSim provides the ground truth for any single-source SimRank query with a precision up to 7 decimal places within a reasonable query time.Comment: ACM SIGMOD 202

High quality graph-based similarity search

SimRank is an influential link-based similarity measure that has been used in many fields of Web search and sociometry. The best-of-breed method by Kusumoto et. al., however, does not always deliver high-quality results, since it fails to accurately obtain its diagonal correction matrix D. Besides, SimRank is also limited by an unwanted "connectivity trait": increasing the number of paths between nodes a and b often incurs a decrease in score s(a,b). The best-known solution, SimRank++, cannot resolve this problem, since a revised score will be zero if a and b have no common in-neighbors. In this paper, we consider high-quality similarity search. Our scheme, SR#, is efficient and semantically meaningful: (1) We first formulate the exact D, and devise a "varied-D" method to accurately compute SimRank in linear memory. Moreover, by grouping computation, we also reduce the time of from quadratic to linear in the number of iterations. (2) We design a "kernel-based" model to improve the quality of SimRank, and circumvent the "connectivity trait" issue. (3) We give mathematical insights to the semantic difference between SimRank and its variant, and correct an argument: "if D is replaced by a scaled identity matrix, top-K rankings will not be affected much". The experiments confirm that SR# can accurately extract high-quality scores, and is much faster than the state-of-the-art competitors

Fast Exact CoSimRank Search on Evolving and Static Graphs

In real Web applications, CoSimRank has been proposed as a powerful measure of node-pair similarity based on graph topologies. However, existing work on CoSimRank is restricted to static graphs. When the graph is updated with new edges arriving over time, it is cost-inhibitive to recompute all CoSimRank scores from scratch, which is impractical. In this study, we propose a fast dynamic scheme, \DCoSim for accurate CoSimRank search over evolving graphs. Based on \DCoSim, we also propose a fast scheme, \FCoSim, that greatly accelerates CoSimRank search over static graphs. Our theoretical analysis shows that \DCoSim and \FCoSim guarantee the exactness of CoSimRank scores. On the static graph G, to efficiently retrieve CoSimRank scores \mathbfS , \FCoSim is based on three ideas: (i) It first finds a "spanning polytree»» T over G. (ii) On T, a fast algorithm is designed to compute the CoSimRank scores \mathbfS (T) over the "spanning polytree»» T. (iii) On G, \DCoSim is employed to compute the changes of \mathbfS (T) in response to the delta graph $(G øminus T)$. Experimental evaluations verify the superiority of \DCoSim over evolving graphs, and the fast speedup of \FCoSim on large-scale static graphs against its competitors, without any loss of accuracy