Personalized PageRank Vectors Research Articles

A local updating algorithm for personalized PageRank via Chebyshev polynomials

The personalized PageRank algorithm is one of the most versatile tools for the analysis of networks. In spite of its ubiquity, maintaining personalized PageRank vectors when the underlying network constantly evolves is still a challenging task. To address this limitation, this work proposes a novel distributed algorithm to locally update personalized PageRank vectors when the graph topology changes. The proposed algorithm is based on the use of Chebyshev polynomials and a novel update equation that encompasses a large family of PageRank-based methods. In particular, the algorithm has the following advantages: (i) it has faster convergence speed than state-of-the-art alternatives for local personalized PageRank updating; and (ii) it can update the solution of recent extensions of personalized PageRank that rely on complex dynamical processes for which no updating algorithms have been developed. Experiments in a real-world temporal network of an autonomous system validate the effectiveness of the proposed algorithm.

Containing epidemics in a local cluster via antidote distribution and partial quarantine.

The study of spreading phenomena in networks, in particular the spread of disease, has attracted considerable interest in the network science research community. In this paper, we show that the outbreak of an epidemic can be effectively contained and suppressed in a small subnetwork by a combination of antidote distribution and partial quarantine. We improve over existing antidote distribution schemes based on personalized PageRank in two ways. First, we replace the constraint on the topology of this subnetwork described by Chung etal. [Internet Math. 6, 237 (2009)1542-795110.1080/15427951.2009.10129184] that a large fraction of the value of the personalized PageRank vector must be contained in the local cluster, with a partial quarantine scheme. Second, we derive a different lower bound on the amount of antidote. We show that, under our antidote distribution scheme, the probability of the infection spreading to the whole network is bounded, and the infection inside the subnetwork will disappear after a period that is proportional to the logarithm of the number of initially infected nodes. We demonstrate the effectiveness of our strategy with numerical simulations of epidemics on benchmark networks. We also test our strategy on two examples of epidemics in real-world networks. Our strategy is dependent only on the rate of infection, the rate of recovery, and the topology around the initially infected nodes, and is independent of the rest of the network.

A Structural Result for Personalized PageRank and its Algorithmic Consequences

Many systems, such as the Internet, social networks, and the power grid, can be represented as graphs. When analyzing graphs, it is often useful to compute scores describing the relative importance or distance between nodes. One example is Personalized PageRank (PPR), which assigns to each node v a vector whose i-th entry describes the importance of the i-th node from the perspective of v. PPR has proven useful in many applications, such as recommending who users should follow on social networks (if this i-th entry is large, v may be interested in following the i-th user). Unfortunately, computing n PPR vectors exactly for a graph of n nodes has complexity O(n^3), which is infeasible for many graphs of interest. In this work, we devise a scheme to estimate all n PPR vectors with bounded l_1 error and complexity O(n c ), where c < 2 depends on the degrees of the graph at hand, the desired error tolerance, and a parameter that defines PPR. This improves upon existing methods, the best of which have complexity O(n 2 łog n) in our setting. Our complexity guarantee holds with high probability, for certain choices of the PPR parameter, and for a certain class of random graphs (roughly speaking, the sparse directed configuration model with heavy-tailed in-degrees); our accuracy guarantee holds with probability 1 and for arbitrary graphs and PPR parameters. The complexity result arises as a consequence of our main (structural) result, which shows that the dimensionality of the set of PPR vectors scales sublinearly in n with high probability, for the same class of random graphs and for a notion of dimensionality similar to matrix rank. It is this coupling of the PPR vectors for the nodes on a common underlying graph that allows for estimating them faster. Hence, at a high level, our scheme is analogous to (but distinct from) low-rank matrix approximation. We also note that our scheme is similar to one that was proposed in [Jeh and Widom 2003] but lacked accuracy and complexity guarantees, so another contribution of our paper is to address this gap in the literature.

PPR-partitioning: a distributed graph partitioning algorithm based on the personalized PageRank vectors in vertex-centric systems

Relations among data items can be modeled with graphs in most of big data sets such as social networks’ data. This modeling creates big graphs with many vertices and edges. Balanced k-way graph partitioning is a common problem with big graphs. It has many applications in several fields. There are many approximate solutions for this problem; however, most of them do not have enough scalability for big graph partitioning and cannot be executed in a distributed manner. Vertex-centric model has been introduced recently as a scalable distributed processing method for big graphs. There are a few methods for graph partitioning based on this model. Existing approaches only consider one-step neighbors of vertices for graph partitioning and do not consider neighbors with higher steps. In this paper, a distributed method is introduced based on vertex-centric model for balanced k-way graph partitioning. This method applies the personalized PageRank vectors of vertices and partitions to decide how vertices are joined partitions. This method has been implemented in the Giraph system. The proposed method has been evaluated with several synthetic and real graphs. Experimental results have shown that this method has scalability for partitioning big graphs. It was also found that this method produces partitions with higher quality compared to the state-of-the-art stream-based methods and distributed methods based on vertex-centric programming model. Its result is close to the results of Metis method.

CUFID-query: accurate network querying through random walk based network flow estimation

BackgroundFunctional modules in biological networks consist of numerous biomolecules and their complicated interactions. Recent studies have shown that biomolecules in a functional module tend to have similar interaction patterns and that such modules are often conserved across biological networks of different species. As a result, such conserved functional modules can be identified through comparative analysis of biological networks.ResultsIn this work, we propose a novel network querying algorithm based on the CUFID (Comparative network analysis Using the steady-state network Flow to IDentify orthologous proteins) framework combined with an efficient seed-and-extension approach. The proposed algorithm, CUFID-query, can accurately detect conserved functional modules as small subnetworks in the target network that are expected to perform similar functions to the given query functional module. The CUFID framework was recently developed for probabilistic pairwise global comparison of biological networks, and it has been applied to pairwise global network alignment, where the framework was shown to yield accurate network alignment results. In the proposed CUFID-query algorithm, we adopt the CUFID framework and extend it for local network alignment, specifically to solve network querying problems. First, in the seed selection phase, the proposed method utilizes the CUFID framework to compare the query and the target networks and to predict the probabilistic node-to-node correspondence between the networks. Next, the algorithm selects and greedily extends the seed in the target network by iteratively adding nodes that have frequent interactions with other nodes in the seed network, in a way that the conductance of the extended network is maximally reduced. Finally, CUFID-query removes irrelevant nodes from the querying results based on the personalized PageRank vector for the induced network that includes the fully extended network and its neighboring nodes.ConclusionsThrough extensive performance evaluation based on biological networks with known functional modules, we show that CUFID-query outperforms the existing state-of-the-art algorithms in terms of prediction accuracy and biological significance of the predictions.

Supporting inter-topic entity search for biomedical Linked Data based on heterogeneous relationships

ObjectiveThe keyword-based entity search restricts search space based on the preference of search. When given keywords and preferences are not related to the same biomedical topic, existing biomedical Linked Data search engines fail to deliver satisfactory results. This research aims to tackle this issue by supporting an inter-topic search—improving search with inputs, keywords and preferences, under different topics. MethodsThis study developed an effective algorithm in which the relations between biomedical entities were used in tandem with a keyword-based entity search, Siren. The algorithm, PERank, which is an adaptation of Personalized PageRank (PPR), uses a pair of input: (1) search preferences, and (2) entities from a keyword-based entity search with a keyword query, to formalize the search results on-the-fly based on the index of the precomputed Individual Personalized PageRank Vectors (IPPVs). ResultsOur experiments were performed over ten linked life datasets for two query sets, one with keyword-preference topic correspondence (intra-topic search), and the other without (inter-topic search). The experiments showed that the proposed method achieved better search results, for example a 14% increase in precision for the inter-topic search than the baseline keyword-based search engine. ConclusionThe proposed method improved the keyword-based biomedical entity search by supporting the inter-topic search without affecting the intra-topic search based on the relations between different entities.

Finding low-conductance sets with dense interactions (FLCD) for better protein complex prediction

BackgroundIntuitively, proteins in the same protein complexes should highly interact with each other but rarely interact with the other proteins in protein-protein interaction (PPI) networks. Surprisingly, many existing computational algorithms do not directly detect protein complexes based on both of these topological properties. Most of them, depending on mathematical definitions of either “modularity” or “conductance”, have their own limitations: Modularity has the inherent resolution problem ignoring small protein complexes; and conductance characterizes the separability of complexes but fails to capture the interaction density within complexes.ResultsIn this paper, we propose a two-step algorithm FLCD (Finding Low-Conductance sets with Dense interactions) to predict overlapping protein complexes with the desired topological structure, which is densely connected inside and well separated from the rest of the networks. First, FLCD detects well-separated subnetworks based on approximating a potential low-conductance set through a personalized PageRank vector from a protein and then solving a mixed integer programming (MIP) problem to find the minimum-conductance set within the identified low-conductance set. At the second step, the densely connected parts in those subnetworks are discovered as the protein complexes by solving another MIP problem that aims to find the dense subnetwork in the minimum-conductance set.ConclusionExperiments on four large-scale yeast PPI networks from different public databases demonstrate that the complexes predicted by FLCD have better correspondence with the yeast protein complex gold standards than other three state-of-the-art algorithms (ClusterONE, LinkComm, and SR-MCL). Additionally, results of FLCD show higher biological relevance with respect to Gene Ontology (GO) terms by GO enrichment analysis.

A local dynamic method for tracking communities and their evolution in dynamic networks

The analysis of communities and their evolutionary behaviors in dynamic networks is a challenging topic. Although a growing body of work on this topic is emerging, there are few methods which can reveal and track meaningful communities over time and can also deal with large networks efficiently. In this paper, we propose a method to track dynamic communities and their evolutionary behaviors. The main idea behind our method is to discover dynamic communities by exploring the local views of nodes that change. Moreover, based on the discovered dynamic communities, the global community structure can be derived by updating the historical community structure and the evolutionary behaviors of communities can also be tracked. To discover the dynamic communities, we apply the technique of approximate personalized PageRank vector; to track the evolutionary behaviors of the communities, we introduce a partial evolutionary graph. We compare the proposed method with several existing methods by performing experiments on nine synthetic networks and one real network. The experimental results show that the proposed method performs well on discovering communities as well as tracking their evolution in dynamic networks, and spends much less running time than the existing methods.

Scheduled approximation for Personalized PageRank with Utility-based Hub Selection

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.

Monte Carlo Based Personalized PageRank on Dynamic Networks

In large-scale networks, the structure of the underlying network changes frequently, and thus the power iteration method for Personalized PageRank computation cannot deal with this kind of dynamic network efficiently. In this paper, we design a Monte Carlo-based incremental method for Personalized PageRank computation. In a dynamic network, first, we do a random walk starting from each node and save the performed walks into a fingerprint database; second, we update the fingerprint database in a fixed time interval with our proposed update algorithm; finally, when a query is issued by a user, we estimate the Personalized PageRank vector by our proposed approximation algorithm. Experiments on real-world networks show that our method can handle multichanges of the underlying network at a time and is more efficient than related work, so it can be used in real incremental Personalized PageRank-based applications.

Incremental and accuracy-aware personalized pagerank through scheduled approximation

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.

Sampling from complex networks with high community structures

In this paper, we propose a novel link-tracing sampling algorithm, based on the concepts from PageRank vectors, to sample from networks with high community structures. Our method has two phases; (1) Sampling the closest nodes to the initial nodes by approximating personalized PageRank vectors and (2) Jumping to a new community by using PageRank vectors and unknown neighbors. Empirical studies on several synthetic and real-world networks show that the proposed method improves the performance of network sampling compared to the popular link-based sampling methods in terms of accuracy and visited communities.

Finding and Visualizing Graph Clusters Using PageRank Optimization

We give algorithms for finding graph clusters and drawing graphs, highlighting local community structure within the context of a larger network. For a given graph _G_, we use the personalized PageRank vectors to determine a set of clusters, by optimizing the jumping parameter α subject to several cluster variance measures in order to capture the graph structure according to PageRank. We then give a graph visualization algorithm for the clusters using PageRank-based coordinates. Several drawings of real-world data are given, illustrating the partition and local community structure.

A local spectral method for graphs: with applications to improving graph partitions and exploring data graphs locally

The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local regions of interest, and the second eigenvector will typically fail to provide information fine-tuned to each local region. In this paper, we introduce a locally-biased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semi-supervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearly-linear time. In addition, we show that this locally-biased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful for identifying and refining clusters locally, as it allows us to focus on a local region of interest in a semi-supervised manner. Finally, we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally-biased sparse cuts around an input vertex seed set in social and information networks.

Leadership groups on Social Network Sites based on Personalized PageRank

In this paper we present a new framework to identify leaders in a Social Network Site using the Personalized PageRank vector. The methodology is based on the concept of Leadership group recently introduced by one of the authors. We show how to analyze the structure of the Leadership group as a function of a single parameter. Zachary’s network and a Facebook university network are used to illustrate the applicability of the model.

Local Partitioning for Directed Graphs Using PageRank

A local partitioning algorithm finds a set with small conductance near a specified seed vertex. In this paper, we present a generalization of a local partitioning algorithm for undirected graphs to strongly connected directed graphs. In particular, we prove that by computing a personalized PageRank vector in a directed graph, starting from a single seed vertex within a set _S_ that has conductance at most α, and by performing a sweep over that vector, we can obtain a set of vertices _S_′ with conductance <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129297_o_ilf0001.gif"></inline-graphic>. Here, the conductance function Φ_<em>M</em> is defined in terms of the stationary distribution of a random walk in the directed graph. In addition, we describe how this algorithm may be applied to the PageRank Markov chain of an arbitrary directed graph, which provides a way to partition directed graphs that are not strongly connected.

A Survey on PageRank Computing

This survey reviews the research related to PageRank computing. Components of a PageRank vector serve as authority weights for web pages independent of their textual content, solely based on the hyperlink structure of the web. PageRank is typically used as a web search ranking component. This defines the importance of the model and the data structures that underly PageRank processing. Computing even a single PageRank is a difficult computational task. Computing many PageRanks is a much more complex challenge. Recently, significant effort has been invested in building sets of personalized PageRank vectors. PageRank is also used in many diverse applications other than ranking. We are interested in the theoretical foundations of the PageRank formulation, in the acceleration of PageRank computing, in the effects of particular aspects of web graph structure on the optimal organization of computations, and in PageRank stability. We also review alternative models that lead to authority indices similar to PageRank and the role of such indices in applications other than web search. We also discuss linkbased search personalization and outline some aspects of PageRank infrastructure from associated measures of convergence to link preprocessing.