PRSim: Sublinear Time SimRank Computation on Large Power-Law Graphs

{\it SimRank} is a classic measure of the similarities of nodes in a graph. Given a node $u$ in graph $G =(V, E)$, a {\em single-source SimRank query} returns the SimRank similarities $s(u, v)$ between node $u$ and each node $v \in V$. This type of queries has numerous applications in web search and social networks analysis, such as link prediction, web mining, and spam detection. Existing methods for single-source SimRank queries, however, incur query cost at least linear to the number of nodes $n$, which renders them inapplicable for real-time and interactive analysis. { This paper proposes \prsim, an algorithm that exploits the structure of graphs to efficiently answer single-source SimRank queries. \prsim uses an index of size $O(m)$, where $m$ is the number of edges in the graph, and guarantees a query time that depends on the {\em reverse PageRank} distribution of the input graph. In particular, we prove that \prsim runs in sub-linear time if the degree distribution of the input graph follows the power-law distribution, a property possessed by many real-world graphs. Based on the theoretical analysis, we show that the empirical query time of all existing SimRank algorithms also depends on the reverse PageRank distribution of the graph.} Finally, we present the first experimental study that evaluates the absolute errors of various SimRank algorithms on large graphs, and we show that \prsim outperforms the state of the art in terms of query time, accuracy, index size, and scalability.Comment: ACM SIGMOD 201

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

Personalized PageRank on Evolving Graphs with an Incremental Index-Update Scheme

{\em Personalized PageRank (PPR)} stands as a fundamental proximity measure in graph mining. Since computing an exact SSPPR query answer is prohibitive, most existing solutions turn to approximate queries with guarantees. The state-of-the-art solutions for approximate SSPPR queries are index-based and mainly focus on static graphs, while real-world graphs are usually dynamically changing. However, existing index-update schemes can not achieve a sub-linear update time. Motivated by this, we present an efficient indexing scheme to maintain indexed random walks in expected $O(1)$ time after each graph update. To reduce the space consumption, we further propose a new sampling scheme to remove the auxiliary data structure for vertices while still supporting $O(1)$ index update cost on evolving graphs. Extensive experiments show that our update scheme achieves orders of magnitude speed-up on update performance over existing index-based dynamic schemes without sacrificing the query efficiency

COMPUTING APPROXIMATE CUSTOMIZED RANKING

As the amount of information grows and as users become more sophisticated, ranking techniques become important building blocks to meet user needs when answering queries. PageRank is one of the most successful link-based ranking methods, which iteratively computes the importance scores for web pages based on the importance scores of incoming pages. Due to its success, PageRank has been applied in a number of applications that require customization. We address the scalability challenges for two types of customized ranking. The first challenge is to compute the ranking of a subgraph. Various Web applications focus on identifying a subgraph, such as focused crawlers and localized search engines. The second challenge is to compute online personalized ranking. Personalized search improves the quality of search results for each user. The user needs are represented by a personalized set of pages or personalized link importance in an entity relationship graph. This requires an efficient online computation. To solve the subgraph ranking problem efficiently, we estimate the ranking scores for a subgraph. We propose a framework of an exact solution (IdealRank) and an approximate solution (ApproxRank) for computing ranking on a subgraph. Both IdealRank and ApproxRank represent the set of external pages with an external node $\Lambda$ and modify the PageRank-style transition matrix with respect to $\Lambda$. The IdealRank algorithm assumes that the scores of external pages are known. We prove that the IdealRank scores for pages in the subgraph converge to the true PageRank scores. Since the PageRank-style scores of external pages may not typically be available, we propose the ApproxRank algorithm to estimate scores for the subgraph. We analyze the $L_1$ distance between IdealRank scores and ApproxRank scores of the subgraph and show that it is within a constant factor of the $L_1$ distance of the external pages. We demonstrate with real and synthetic data that ApproxRank provides a good approximation to PageRank for a variety of subgraphs. We consider online personalization using ObjectRank; it is an authority flow based ranking for entity relationship graphs. We formalize the concept of an aggregate surfer on a data graph; the surfer's behavior is controlled by multiple personalized rankings. We prove a linearity theorem over these rankings which can be used as a tool to scale this type of personalization. DataApprox uses a repository of precomputed rankings for a given set of link weights assignments. We define DataApprox as an optimization problem; it selects a subset of the precomputed rankings from the repository and produce a weighted combination of these rankings. We analyze the $L_1$ distance between the DataApprox scores and the real authority flow ranking scores and show that DataApprox has a theoretical bound. Our experiments on the DBLP data graph show that DataApprox performs well in practice and allows fast and accurate personalized authority flow ranking