On the efficiency of estimating penetrating rank on large graphs

P-Rank (Penetrating Rank) has been suggested as a useful measure of structural similarity that takes account of both incoming and outgoing edges in ubiquitous networks. Existing work often utilizes memoization to compute P-Rank similarity in an iterative fashion, which requires cubic time in the worst case. Besides, previous methods mainly focus on the deterministic computation of P-Rank, but lack the probabilistic framework that scales well for large graphs. In this paper, we propose two efficient algorithms for computing P-Rank on large graphs. The first observation is that a large body of objects in a real graph usually share similar neighborhood structures. By merging such objects with an explicit low-rank factorization, we devise a deterministic algorithm to compute P-Rank in quadratic time. The second observation is that by converting the iterative form of P-Rank into a matrix power series form, we can leverage the random sampling approach to probabilistically compute P-Rank in linear time with provable accuracy guarantees. The empirical results on both real and synthetic datasets show that our approaches achieve high time efficiency with controlled error and outperform the baseline algorithms by at least one order of magnitude

Taming computational complexity: efficient and parallel SimRank optimizations on undirected graphs

SimRank has been considered as one of the promising link-based ranking algorithms to evaluate similarities of web documents in many modern search engines. In this paper, we investigate the optimization problem of SimRank similarity computation on undirected web graphs. We first present a novel algorithm to estimate the SimRank between vertices in O(n3+ Kn2) time, where n is the number of vertices, and K is the number of iterations. In comparison, the most efficient implementation of SimRank algorithm in [1] takes O(K n3 ) time in the worst case. To efficiently handle large-scale computations, we also propose a parallel implementation of the SimRank algorithm on multiple processors. The experimental evaluations on both synthetic and real-life data sets demonstrate the better computational time and parallel efficiency of our proposed techniques

Towards efficient SimRank computation on large networks

SimRank has been a powerful model for assessing the similarity of pairs of vertices in a graph. It is based on the concept that two vertices are similar if they are referenced by similar vertices. Due to its self-referentiality, fast SimRank computation on large graphs poses significant challenges. The state-of-the-art work [17] exploits partial sums memorization for computing SimRank in O(Kmn) time on a graph with n vertices and m edges, where K is the number of iterations. Partial sums memorizing can reduce repeated calculations by caching part of similarity summations for later reuse. However, we observe that computations among different partial sums may have duplicate redundancy. Besides, for a desired accuracy ϵ, the existing SimRank model requires K = [logC ϵ] iterations [17], where C is a damping factor. Nevertheless, such a geometric rate of convergence is slow in practice if a high accuracy is desirable. In this paper, we address these gaps. (1) We propose an adaptive clustering strategy to eliminate partial sums redundancy (i.e., duplicate computations occurring in partial sums), and devise an efficient algorithm for speeding up the computation of SimRank to 0(Kdn2) time, where d is typically much smaller than the average in-degree of a graph. (2) We also present a new notion of SimRank that is based on a differential equation and can be represented 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) Using real and synthetic data, we empirically verify that our approach of partial sums sharing outperforms the best known algorithm by up to one order of magnitude, and that our revised notion of SimRank further achieves a 5X speedup on large graphs while also fairly preserving the relative order of original SimRank scores

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

Search Efficient Binary Network Embedding

Traditional network embedding primarily focuses on learning a dense vector representation for each node, which encodes network structure and/or node content information, such that off-the-shelf machine learning algorithms can be easily applied to the vector-format node representations for network analysis. However, the learned dense vector representations are inefficient for large-scale similarity search, which requires to find the nearest neighbor measured by Euclidean distance in a continuous vector space. In this paper, we propose a search efficient binary network embedding algorithm called BinaryNE to learn a sparse binary code for each node, by simultaneously modeling node context relations and node attribute relations through a three-layer neural network. BinaryNE learns binary node representations efficiently through a stochastic gradient descent based online learning algorithm. The learned binary encoding not only reduces memory usage to represent each node, but also allows fast bit-wise comparisons to support much quicker network node search compared to Euclidean distance or other distance measures. Our experiments and comparisons show that BinaryNE not only delivers more than 23 times faster search speed, but also provides comparable or better search quality than traditional continuous vector based network embedding methods

High quality SimRank-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. [7], 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++ [1], 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 [7] 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 in [7]: “if D is replaced by a scaled identity matrix (1−γ)I, 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 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