A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> As Personalized PageRank has been widely leveraged for ranking on a graph, the efficient computation of Personalized PageRank Vector (PPV) becomes a prominent issue. In this paper, we propose FastPPV, an approximate PPV computation algorithm that is incremental and accuracy-aware. Our approach hinges on a novel paradigm of scheduled approximation: the computation is partitioned and scheduled for processing in an "organized" way, such that we can gradually improve our PPV estimation in an incremental manner, and quantify the accuracy of our approximation at query time. Guided by this principle, we develop an efficient hub based realization, where we adopt the metric of hub-length to partition and schedule random walk tours so that the approximation error reduces exponentially over iterations. Furthermore, as tours are segmented by hubs, the shared substructures between different tours (around the same hub) can be reused to speed up query processing both within and across iterations. Finally, we evaluate FastPPV over two real-world graphs, and show that it not only significantly outperforms two state-of-the-art baselines in both online and offline phrases, but also scale well on larger graphs. In particular, we are able to achieve near-constant time online query processing irrespective of graph size. <s> BIB001 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> Random Walk with Restart (RWR) has become an appealing measure of node proximities in emerging applications \eg recommender systems and automatic image captioning. In practice, a real graph is typically large, and is frequently updated with small changes. It is often cost-inhibitive to recompute proximities from scratch via \emph{batch} algorithms when the graph is updated. This paper focuses on the incremental computations of RWR in a dynamic graph, whose edges often change over time. The prior attempt of RWR [1] deploys \kdash to find top-$k$ highest proximity nodes for a given query, which involves a strategy to incrementally \emph{estimate} upper proximity bounds. However, due to its aim to prune needless calculation, such an incremental strategy is \emph{approximate}: in $O(1)$ time for each node. The main contribution of this paper is to devise an \emph{exact} and fast incremental algorithm of RWR for edge updates. Our solution, \IRWR\!, can incrementally compute any node proximity in $O(1)$ time for each edge update without loss of exactness. The empirical evaluations show the high efficiency and exactness of \IRWR for computing proximities on dynamic networks against its batch counterparts. <s> BIB002 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> We propose a new algorithm, FAST-PPR, for computing personalized PageRank: given start node s and target node t in a directed graph, and given a threshold δ, it computes the Personalized PageRank π_s(t) from s to t, guaranteeing that the relative error is small as long πs(t) > δ. Existing algorithms for this problem have a running-time of Ω(1/δ in comparison, FAST-PPR has a provable average running-time guarantee of O(√d/δ) (where d is the average in-degree of the graph). This is a significant improvement, since δ is often O(1/n) (where n is the number of nodes) for applications. We also complement the algorithm with an Ω(1/√δ) lower bound for PageRank estimation, showing that the dependence on δ cannot be improved. We perform a detailed empirical study on numerous massive graphs, showing that FAST-PPR dramatically outperforms existing algorithms. For example, on the 2010 Twitter graph with 1.5 billion edges, for target nodes sampled by popularity, FAST-PPR has a 20 factor speedup over the state of the art. Furthermore, an enhanced version of FAST-PPR has a 160 factor speedup on the Twitter graph, and is at least 20 times faster on all our candidate graphs. <s> BIB003 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> As Personalized PageRank has been widely leveraged for ranking on a graph, the efficient computation of Personalized PageRank Vector (PPV) becomes a prominent issue. In this paper, we propose FastPPV, an approximate PPV computation algorithm that is incremental and accuracy-aware. Our approach hinges on a novel paradigm of scheduled approximation: the computation is partitioned and scheduled for processing in an "organized" way, such that we can gradually improve our PPV estimation in an incremental manner and quantify the accuracy of our approximation at query time. Guided by this principle, we develop an efficient hub-based realization, where we adopt the metric of hub length to partition and schedule random walk tours so that the approximation error reduces exponentially over iterations. In addition, as tours are segmented by hubs, the shared substructures between different tours (around the same hub) can be reused to speed up query processing both within and across iterations. Given the key roles played by the hubs, we further investigate the problem of hub selection. In particular, we develop a conceptual model to select hubs based on the two desirable properties of hubs--sharing and discriminating, and present several different strategies to realize the conceptual model. Finally, we evaluate FastPPV over two real-world graphs, and show that it not only significantly outperforms two state-of-the-art baselines in both online and offline phrases, but also scales well on larger graphs. In particular, we are able to achieve near-constant time online query processing irrespective of graph size. <s> BIB004 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> Real-world networks, such as the World Wide Web and online social networks, are very large and are evolving rapidly. Thus tracking personalized PageRank in such evolving networks is an important challenge in network analysis and graph mining. ::: In this paper, we propose an efficient online algorithm for tracking personalized PageRank in an evolving network. The proposed algorithm tracks personalized PageRank accurately (i.e., within a given accuracy ε > 0). Moreover it can update the personalized PageRank scores in amortized O(1/ε) iterations for each graph modification. In addition, when m edges are randomly and sequentially inserted, the total number of iterations is expected to be O(log m/ε). ::: We evaluated our algorithm in real-world networks. In average case, for each edge insertion and deletion, our algorithm updated the personalized PageRank in 3us in a web graph with 105M vertices and 3.7B edges, and 20ms in a social network with 42M vertices and 1.5B edges. By comparing existing state-of-the-arts algorithms, our algorithm is 2--290 times faster with an equal accuracy. <s> BIB005 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> We propose and analyze two algorithms for maintaining approximate Personalized PageRank (PPR) vectors on a dynamic graph, where edges are added or deleted. Our algorithms are natural dynamic versions of two known local variations of power iteration. One, Forward Push, propagates probability mass forwards along edges from a source node, while the other, Reverse Push, propagates local changes backwards along edges from a target. In both variations, we maintain an invariant between two vectors, and when an edge is updated, our algorithm first modifies the vectors to restore the invariant, then performs any needed local push operations to restore accuracy. For Reverse Push, we prove that for an arbitrary directed graph in a random edge model, or for an arbitrary undirected graph, given a uniformly random target node t, the cost to maintain a PPR vector to t of additive error e as k edges are updated is O(k + d/e, where d is the average degree of the graph. This is O(1) work per update, plus the cost of computing a reverse vector once on a static graph. For Forward Push, we show that on an arbitrary undirected graph, given a uniformly random start node s, the cost to maintain a PPR vector from s of degree-normalized error e as k edges are updated is O(k + 1/e, which is again O(1) per update plus the cost of computing a PPR vector once on a static graph. <s> BIB006 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> How can we measure similarity between nodes quickly and accurately on large graphs? Random walk with restart (RWR) provides a good measure, and has been used in various data mining applications including ranking, recommendation, link prediction and community detection. However, existing methods for computing RWR do not scale to large graphs containing billions of edges; iterative methods are slow in query time, and preprocessing methods require too much memory. In this paper, we propose BePI, a fast, memory-efficient, and scalable method for computing RWR on billion-scale graphs. BePI exploits the best properties from both preprocessing methods and iterative methods. BePI uses a block elimination approach, which is a preprocessing method, to enable fast query time. Also, BePI uses a preconditioned iterative method to decrease memory requirement. The performance of BePI is further improved by decreasing non-zeros of the matrix for the iterative method. Through extensive experiments, we show that BePI processes 100 times larger graphs, and requires up to 130 times less memory space than other preprocessing methods. In the query phase, BePI computes RWR scores up to 9 times faster than existing methods. <s> BIB007 </s> A Survey on Personalized PageRank Computation Algorithms <s> G. RECENT ADVANCES <s> Given a time-evolving graph, how can we track similarity between nodes in a fast and accurate way, with theoretical guarantees on the convergence and the error? Random Walk with Restart (RWR) is a popular measure to estimate the similarity between nodes and has been exploited in numerous applications. Many real-world graphs are dynamic with frequent insertion/deletion of edges; thus, tracking RWR scores on dynamic graphs in an efficient way has aroused much interest among data mining researchers. Recently, dynamic RWR models based on the propagation of scores across a given graph have been proposed, and have succeeded in outperforming previous other approaches to compute RWR dynamically. However, those models fail to guarantee exactness and convergence time for updating RWR in a generalized form. In this paper, we propose OSP, a fast and accurate algorithm for computing dynamic RWR with insertion/deletion of nodes/edges in a directed/undirected graph. When the graph is updated, OSP first calculates offset scores around the modified edges, propagates the offset scores across the updated graph, and then merges them with the current RWR scores to get updated RWR scores. We prove the exactness of OSP and introduce OSP-T, a version of OSP which regulates a trade-off between accuracy and computation time by using error tolerance ?. Given restart probability c, OSP-T guarantees to return RWR scores with O(?/c) error in O(log(?/2)/log(1-c)) iterations. Through extensive experiments, we show that OSP tracks RWR exactly up to 4605x faster than existing static RWR method on dynamic graphs, and OSP-T requires up to 15x less time with 730x lower L1 norm error and 3.3x lower rank error than other state-of-the-art dynamic RWR methods. <s> BIB008
The efficient computation of PPR is still one of major ongoing issues of massive graph processing, and many remarkable studies have been published recently. BePI BIB007 is a study that combines the preprocessing method and the iterative method based on the BEAR. BePI addresses the challenges that are faced by previous approaches by combining the best of both the preprocessing and iterative methods. BePI uses a block elimination approach, which is a preprocessing method, to achieve a fast query time. BePI incorporates an iterative method within the block elimination to decrease the memory requirements by avoiding expensive matrix inversions. BePI takes the advantages of both the preprocessing methods and iterative methods. Consequently, BePI achieves a better scalability and a faster query time than other methods. FastPPV (different from the Fast-PPR BIB003 that was introduced in section 4.E) is also a remarkable innovation for this problem BIB004 , BIB001 . The study presented a scheduled approximation strategy to approximate PPR vectors. Specifically, they developed a hub based scheduling scheme and a structured aggregation model. They also explored the issue of hub selection. Their experiment results show that their method outperforms HubRank and Monte-Carlo sampling methods. Multiple methods focusing on dealing with dynamic graphs have been suggested BIB002 , BIB006 - BIB005 . They update the precomputed PPR vectors when the edges are added or deleted, and they also can be categorized into major approaches like other works. IRWR BIB002 updates PPR vectors based on direct solving formula, and LayFwdUpdate BIB006 , and OSP BIB008 are dynamic variants of iterative methods. TrackingPPR BIB005 works on the bookmark coloring scheme.