SimRank Computation Research Articles

ClipSim: A GPU-friendly Parallel Framework for Single-Source SimRank with Accuracy Guarantee

SimRank is an important metric to measure the topological similarity between two nodes in a graph. In particular, single-source and top-k SimRank has numerous applications in recommendation systems, network analysis, and web mining, etc. Mathematically, given a vertex, the computation of single-machine and single-source SimRank mainly lies in matrix-matrix operations. However, it is almost impossible to directly compute on large graphs. Thus, existing works yield to two main operations: a series of random walks, and sparse matrix and dense vector multiplication operations. This brings about high computation cost for SimRank on large graphs. In real-world applications, there is always the query time and accuracy trade-off, which hinders the computation of high-precision SimRank on large-scale graphs. To handle this problem, this paper proposesClipSim, the first GPU-friendly parallel framework that accelerates the single-source SimRank on GPU with accuracy guarantee. We design a novel data structure and GPU-friendly parallel algorithms for efficient computation of all the operations of SimRank on GPU. Moreover, our theoretical derivation enables ClipSim to largely reduce the number of random walks required for each node, while maintaining the same theoretical accuracy as the state-of-the-art algorithm, ExactSim. We conduct extensive experiments on real-world and synthetic datasets to demonstrate the accuracy and efficiency of ClipSim. The results show that compared with ExactSim, ClipSim obtains single-source SimRank vectors with the same accuracy and up to 160× faster computation time.

Fast and Accurate SimRank Computation via Forward Local Push and its Parallelization

Measuring similarity among data objects is important in data analysis and mining. SimRank is a popular link-based similarity measurement among nodes in a graph. To compute the all-pairs SimRank matrix accurately, iterative methods are usually used. For static graphs, current iterative solutions are not efficient enough, both in time and space, due to the unnecessary cost and storage by the nature of iterative updating. For dynamic graphs, all current incremental solutions for updating the SimRank matrix are based on an approximated SimRank definition, and thus have no accuracy guarantee. In this paper, we propose a novel local push based algorithm for computing and tracking all-pairs SimRank. Furthermore, we develop an iterative parallel two-step framework for local push to take advantage of modern hardwares with multicore CPUs. We show that our algorithms outperform the state-of-the-art methods.

A matrix sampling approach for efficient SimRank computation

Evaluating similarities between node pairs in a graph is an important task for data analytics and mining. Among various similarity measures proposed in recent years, SimRank is regarded as one of the most influential measures. However, the computation of SimRank is very expensive especially for large graphs. Although pruning technique and random walk based methods were proposed to accelerate the computation, the accuracy of SimRank score is still very low. In this paper, we propose a novel matrix random sampling approach to accelerate computation speed and reduce memory cost. The matrix random sampling technique not only guarantees the sparsity of the involved matrices, but also enhances the precision of estimated SimRank scores. Moreover, we design a fast sparse matrix–matrix multiplication technique which makes the time complexity of single-source query free of the graph size. We further exploit the Steepest Decent technique to accelerate the speed of convergence. The experimental results show our proposed algorithms outperform the state-of-the-art SimRank algorithms.

Realtime index-free single source SimRank processing on web-scale graphs

Given a graph G and a node u ∈ G, a single source SimRank query evaluates the similarity between u and every node v ∈ G. Existing approaches to single source SimRank computation incur either long query response time, or expensive pre-computation, which needs to be performed again whenever the graph G changes. Consequently, to our knowledge none of them is ideal for scenarios in which (i) query processing must be done in realtime, and (ii) the underlying graph G is massive, with frequent updates. Motivated by this, we propose SimPush, a novel algorithm that answers single source SimRank queries without any pre-computation, and achieves significantly higher query speed than even the fastest known index-based solutions. Further, SimPush provides rigorous result quality guarantees, and its high performance does not rely on any strong assumption of the graph. Specifically, compared to existing methods, SimPush employs a radically different algorithmic design that focuses on (i) identifying a small number of nodes relevant to the query, and subsequently (ii) computing statistics and performing residue push from these nodes only. We prove the correctness of SimPush, analyze its time complexity, and compare its asymptotic performance with that of existing methods. Meanwhile, we evaluate the practical performance of SimPush through extensive experiments on 9 real datasets. The results demonstrate that SimPush consistently outperforms all existing solutions, often by over an order of magnitude. In particular, on a commodity machine, SimPush answers a single source SimRank query on a web graph containing over 133 million nodes and 5.4 billion edges in under 62 milliseconds, with 0.00035 empirical error, while the fastest index-based competitor needs 1.18 seconds.

Speeding up SimRank computations by polynomial preconditioners

SimRank is popularly used for evaluating similarities between nodes in a graph. Though some recent work has transformed the SimRank into a linear system which can be solved by using conjugate gradient (CG) algorithm. However, the diagonal preconditioner used in the algorithm still left some room for improvement. There is no work showing what should be an optimal preconditioner for the linear system. For such a sake, in this paper, we theoretically study the preconditioner problem and propose a polynomial preconditioner which can improve the computation speed significantly. In our investigation, we also modify the original linear system to adapt to all situations of the graph. We further prove that the condition number of a polynomial preconditioned matrix is bounded in a narrow interval. It implies that the new preconditioner can effectively accelerate the convergence rate of the CG algorithm. The experimental results show the proposed algorithm outperforms the state-of-the-art algorithms for the all-pairs SimRank computation.

Efficient Similarity Search for Sets over Graphs

Measuring similarities among different nodes is important in graph analysis tasks, such as link prediction, and recommendation. Among different similarity measures, SimRank is one of the most popular and promising ones, and has received a lot of research attention. While most current studies focus on single-pair, single-source/top-k, and all-pairs SimRank computation, few of them have studied finding similar pairs given a set of node pairs, which has attractive applications in personalized search and recommendation tasks. In this paper, we present Carmo, an efficient algorithm for retrieving the top-k similarities from an arbitrary set of pairs. In addition, we introduce two types of indexes to boost the efficiency of Carmo: one is hub-based, the other is tree-based. We show the effectiveness and efficiency of our proposed methods by extensive experiments.

UniWalk: Unidirectional Random Walk Based Scalable SimRank Computation over Large Graph

SimRank is an important measure of vertex-pair similarity according to the structure of graphs. Although progress has been achieved, existing methods still face challenges to handle large graphs. Besides huge index construction and maintenance cost, existing methods may require considerable search space and time overheads in the online SimRank query. In this paper, we design a Monte Carlo based method, UniWalk, to enable the fast top- $k$ SimRank computation over large undirected graphs. UniWalk directly locates the top- $k$ similar vertices for any single source vertex $u$ via $R$ sampling paths originating from $u$ , which avoids selecting candidate vertex set $\mathcal{C}$ and the following $O(|\mathcal{C}|R)$ bidirectional sampling paths. We also devise a path enumeration strategy to improve the SimRank precision by using path probabilities instead of path frequencies when sampling, a space-efficient method to reduce intermediate results, and a path-sharing strategy to lower the redundant path sampling cost for multiple source vertices. Furthermore, we extend UniWalk to existing distributed graph processing frameworks to improve its scalability. We conduct extensive experiments to illustrate that UniWalk has high scalability, and outperforms the state-of-the-art methods by orders of magnitude.

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 O({r}^{4}n^2) time and O({r}^{2}n^2) 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 {varvec{Delta }}{} mathbf{S} in response to every link update is expressible as a rank-one Sylvester matrix equation. This provides an incremental method requiring O(Kn^2) time and O(n^2) memory in the worst case to update n^2 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 {varvec{Delta }}{} mathbf{S} to eliminate unnecessary retrieval. This reduces the time of the incremental SimRank to O(K(m+|{textsf {AFF}}|)), where m is the number of edges in the old graph, and |{textsf {AFF}}| (le n^2) is the size of “affected areas” in {varvec{Delta }}{} mathbf{S}, and in practice, |{textsf {AFF}}| ll n^2. (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 O(Kn+m) memory, without the need to store all (n^2) 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.

SimRank on Uncertain Graphs

SimRank is a similarity measure between vertices in a graph. Recently, many algorithms have been proposed to efficiently evaluate SimRank similarities. However, the existing algorithms either overlook uncertainty in graph structures or depends on an unreasonable assumption. In this paper, we study SimRank on uncertain graphs. Following the random-walk-based formulation of SimRank on deterministic graphs and the possible world model of uncertain graphs, we first define random walks on uncertain graphs and show that our definition of random walks satisfies Markov’s property. We formulate our SimRank measure based on random walks on uncertain graphs. We discover a critical difference between random walks on uncertain graphs and random walks on deterministic graphs, which makes all existing SimRank computation algorithms on deterministic graphs inapplicable to uncertain graphs. For SimRank computation, we consider computing both single-pair SimRank and single-source top- $K$ SimRank. We propose three algorithms, namely the sampling algorithm with high efficiency, the two-phase algorithm with comparable efficiency and higher accuracy, and a speeding-up algorithm with much higher efficiency. Meanwhile, we present an optimized algorithm for efficient computing the single-source top- $K$ SimRank. The experimental results verify the effectiveness of our SimRank measure and the efficiency of the proposed SimRank computation algorithms.

READS

Similarity among entities in graphs plays a key role in data analysis and mining. SimRank is a widely used and popular measurement to evaluate the similarity among the vertices. In real-life applications, graphs do not only grow in size, requiring fast and precise SimRank computation for large graphs, but also change and evolve continuously over time, demanding an efficient maintenance process to handle dynamic updates. In this paper, we propose a random walk based indexing scheme to compute SimRank efficiently and accurately over large dynamic graphs. We show that our algorithm outperforms the state-of-the-art static and dynamic SimRank algorithms.

Walking in the cloud

Despite its popularity, SimRank is computationally costly, in both time and space. In particular, its recursive nature poses a great challenge in using modern distributed computing power, and also prevents querying similarities individually. Existing solutions suffer greatly from these practical issues. In this paper, we break such dependency for maximum efficiency possible. Our method consists of offline and online phases. In offline phase, a length- n indexing vector is derived by solving a linear system in parallel. At online query time, the similarities are computed instantly from the index vector. Throughout, the Monte Carlo method is used to maximally reduce time and space. Our algorithm, called CloudWalker, is highly parallelizable, with only linear time and space. Remarkably, it responses to both single-pair and single-source queries in constant time. CloudWalker is orders of magnitude more efficient and scalable than existing solutions for large-scale problems. Implemented on Spark with 10 machines and tested on the web-scale clue-web graph with 1 billion nodes and 43 billion edges, it takes 110 hours for offline indexing, 64 seconds for a single-pair query, and 188 seconds for a single-source query. To the best of our knowledge, our work is the first to report results on clue-web, which is 10x larger than the largest graph ever reported for SimRank computation.

Efficient link-based similarity search in web networks

Similarity search in web networks, aiming to find entities similar to the given entity, is one of the core tasks in network analysis. With the proliferation of web applications, including web search and recommendation system, SimRank has been a well-known measure for evaluating entity similarity in a network. However, the existing work computes SimRank iteratively over a huge similarity matrix, which is expensive in terms of time and space cost and cannot efficiently support similarity search over large networks. In this paper, we propose a link-based similarity search method, WebSim, towards efficiently finding similar entities in web networks. WebSim defines the similarity between entities as the 2-hop similarity of SimRank. To reduce computation cost, we divide the similarity search process into two stages: off-line stage and on-line stage. In the off-line stage, the 1-hop similarities are computed, and an optimized algorithm is designed to reduce the unnecessary accumulation operations on zero similarities. In the on-line stage, the 2-hop similarities are computed, and a pruning algorithm is developed to support fast query processing through searching similar entries from a partial sums index derived from the 1-hop similarities. The index items that are lower than a given threshold are skipped to reduce the searching space. Compared to the iterative SimRank computation, the time and space cost of similarity computation is significantly reduced, since WebSim maintains only the similarity matrix of 1-hop that is much smaller than that of multi-hop. Experiments through comparison with SimRank and its optimized algorithms demonstrate that WebSim has on average a 99.83% reduction in the time cost and a 92.12% reduction in the space cost of similarity computation, and achieves on average 99.98% NDCG.

Fast All-Pairs SimRank Assessment on Large Graphs and Bipartite Domains

SimRank is a powerful model for assessing vertex-pair similarities in a graph. It follows the concept that two vertices are similar if they are referenced by similar vertices. The prior work [18] exploits partial sums memoization to compute SimRank in $O(Kmn)$ time on a graph of $n$ vertices and $m$ edges, for $K$ iterations. However, computations among different partial sums may have redundancy. Besides, to guarantee a given accuracy $\epsilon$ , the existing SimRank needs $K=\lceil \log _C \,\epsilon \rceil$ iterations, where $C$ is a damping factor, but the geometric rate of convergence is slow if a high accuracy is expected. In this paper, (1) a novel clustering strategy is proposed to eliminate duplicate computations occurring in partial sums, and an efficient algorithm is then devised to accelerate SimRank computation to $O(K d^{\prime } n^2)$ time, where $d^{\prime }$ is typically much smaller than $\frac{m}{n}$ . (2) A new differential SimRank equation is proposed, which can represent the SimRank matrix as an exponential sum of transition matrices, as opposed to the geometric sum of the conventional counterpart. This leads to a further speedup in the convergence rate of SimRank iterations. (3) In bipartite domains, a novel finer-grained partial max clustering method is developed to speed up the computation of the Minimax SimRank variation from $O(Kmn)$ to $O(Km^{\prime }n)$ time, where $m^{\prime } ({\le} m)$ is the number of edges in a reduced graph after edge clustering, which can be typically much smaller than $m$ . Using real and synthetic data, we empirically verify that (1) our approach of partial sums sharing outperforms the best known algorithm by up to one order of magnitude; (2) the revised notion of SimRank further achieves a 5X speedup on large graphs while also fairly preserving the relative order of original SimRank scores; (3) our finer-grained partial max memoization for the Minimax SimRank variation in bipartite domains is 5X-12X faster than the baselines.

Using Graphics Processors for High Performance SimRank Computation

Recently there has been a lot of interest in graph-based analysis. One of the most important aspects of graph-based analysis is to measure similarity between nodes in a graph. SimRank is a simple and influential measure of this kind, based on a solid graph theoretical model. However, existing methods on SimRank computation suffer from two limitations: 1) the computing cost can be very high in practice; and 2) they can only be applied on static graphs. In this paper, we exploit the inherent parallelism and high memory bandwidth of graphics processing units (GPU) to accelerate the computation of SimRank on large graphs. Furthermore, based on the observation that SimRank is essentially a first-order Markov Chain, we propose to utilize the iterative aggregation techniques for uncoupling Markov chains to compute SimRank scores in parallel for large graphs. The iterative aggregation method can be applied on dynamic graphs. Moreover, it can handle not only the link-updating problem but also the node-updating problem. We give the corresponding theoretical justification and analysis, propose three optimization strategies to further improve the computation efficiency, and extend the proposed algorithm to dynamic graphs. Extensive experiments on synthetic and real data sets verify that the proposed methods are efficient and effective.

Special issue: best papers of APWeb 2010

This issue of World Wide Web includes revised and expanded versions of two papers from the 12th Asia-Pacific Web Conference (APWeb), which took place in Busan, Korea on 6–8 April, 2010. The papers were selected as best papers based on their overall technical quality. They were selected by the program committee co-chairs, Wook-Shin Han, Divesh Srivastava, and Ge Yu. The first paper, “A Space and Time Efficient Algorithm for SimRank Computation,” by Weiren Yu, Wenjie Zhang, Qing Zhang, Xuemin Lin, and Jiajin Le proposes novel optimization techniques to accelerate SimRank Computation. In addition, the authors develop a reordering scheme to achieve both faster convergence rate and I/O efficiency. They also perform extensive experiments using many data sets to demonstrate the efficiency of the proposed techniques. The second paper, “Quick Identification of Near-Duplicate Video Sequences with Cut Signature,” by Qing Xie, Zi Huang, Heng Tao Shen, Xiaofang Zhou, and Chaoyi Pang is about fast identification of duplicate video sequences. The authors consider the issue of how to efficiently and effectively detect near-duplicate video sequences over continuous video streams, and propose a novel compact signature called CutSig to approximate the video structure. They exploit CutSig to efficiently filter candidates. They empirically validate the effectiveness and efficiency of the proposed method using real video streams. World Wide Web (2012) 15:325 DOI 10.1007/s11280-011-0140-6

A space and time efficient algorithm for SimRank computation

SimRank has become an important similarity measure to rank web documents based on a graph model on hyperlinks. The existing approaches for conducting SimRank computation adopt an iteration paradigm. The most efficient deterministic technique yields \(O\left(n^3\right)\) worst-case time per iteration with the space requirement \(O\left(n^2\right)\), where n is the number of nodes (web documents). In this paper, we propose novel optimization techniques such that each iteration takes \(O \left(\min \left\{ n \cdot m , n^r \right\}\right)\) time and \(O \left( n + m \right)\) space, where m is the number of edges in a web-graph model and r ≤ log2 7. In addition, we extend the similarity transition matrix to prevent random surfers getting stuck, and devise a pruning technique to eliminate impractical similarities for each iteration. Moreover, we also develop a reordering technique combined with an over-relaxation method, not only speeding up the convergence rate of the existing techniques, but achieving I/O efficiency as well. We conduct extensive experiments on both synthetic and real data sets to demonstrate the efficiency and effectiveness of our iteration techniques.

Accuracy estimate and optimization techniques for SimRank computation

The measure of similarity between objects is a very useful tool in many areas of computer science, including information retrieval. SimRank is a simple and intuitive measure of this kind, based on ...

Accuracy estimate and optimization techniques for SimRank computation

The measure of similarity between objects is a very useful tool in many areas of computer science, including information retrieval. SimRank is a simple and intuitive measure of this kind, based on graph-theoretic model. SimRank is typically computed iteratively, in the spirit of PageRank. However, existing work on SimRank lacks accuracy estimation of iterative computation and has discouraging time complexity. In this paper we present a technique to estimate the accuracy of computing SimRank iteratively. This technique provides a way to find out the number of iterations required to achieve a desired accuracy when computing SimRank. We also present optimization techniques that improve the computational complexity of the iterative algorithm from O ( n 4 ) to O ( n 3 ) in the worst case. We also introduce a threshold sieving heuristic and its accuracy estimation that further improves the efficiency of the method. As a practical illustration of our techniques we computed SimRank scores on a subset of English Wikipedia corpus, consisting of the complete set of articles and category links.