Extrapolated splitting methods for multilinear PageRank computations

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.

World Wide Web (WWW) is a platform that explores a wide range of information used for the development of web applications. Some examples of these applications include social network analysis, personalized item recommendations, and web page classification and ranking. Among these applications, search engines and web page ranking are particularly important as they consistently index and store billions of web pages on the internet. The main objective of this paper is to create an innovative framework for the classification and re-ranking of web pages using intelligent techniques. The framework is structured into two key phases: classification and re-ranking-based retrieval. In the initial classification phase, a series of pre-processing steps are implemented, including the elimination of HTML tags, punctuation, stop words, and the application of stemming. After pre-processing, a word-to-vector conversion is performed, followed by feature extraction utilizing Principal Component Analysis (PCA). This sequence of actions leads to optimal feature selection, which is vital for the precise classification of web pages. Given the multitude of features present in web pages that can compromise classification accuracy, this study employs a novel meta-heuristic algorithm, the Opposition Based-Tunicate Swarm Algorithm (O-TSA), to facilitate optimal feature selection. The refined features are subsequently processed through the Enhanced Convolutional-Recurrent Neural Network (E-CRNN), enhanced by O-TSA, resulting in the effective classification of diverse web page categories. In the second phase, the re-ranking process is executed using O-TSA, which establishes the objective function based on a similarity function (correlation) for URL matching, leading to optimal re-ranking of web.

This paper addresses resource sharing in a multiclass service center. The flow of service requests from each class of customers are regulated by a class manager who attempts to maximize the class net benefit (throughput reward less delay cost). Relaxed asynchronous algorithms are proposed for obtaining a Nash equilibrium among these competing classes in which each manager iteratively updates his throughput strategy in response to local (and possibly delayed) information on the strategies of the others. The novelty of these flow-control algorithms are in the relaxation, whereby each manager employs a strategy update that is a convex combination of his previous strategy and his best-reply strategy to current information. Alternatively, this relaxation can be viewed as an exponential smoothing of all previous best-reply strategies. For a particular number of classes with specified cost/reward parameters, relaxation conditions for asymptotic convergence to the unique interior Nash equilibrium are presented. For more than two classes, it is shown that relaxation not only accelerates convergence (at least in a special case), but also is a necessary condition for convergence. Due to an equivalent functional form, these results can be directly translated to a network of users employing a power criterion, a common objective in the communications literature.

This paper addresses resource sharing in a multiclass service center. The flow of service requests from each class of customers are regulated by a class manager who attempts to maximize the class net benefit (throughput reward less delay cost). Relaxed asynchronous algorithms are proposed for obtaining a Nash equilibrium among these competing classes in which each manager iteratively updates his throughput strategy in response to local (and possibly delayed) information on the strategies of the others. The novelty of these flow-control algorithms are in the relaxation, whereby each manager employs a strategy update that is a convex combination of his previous strategy and his best-reply strategy to current information. Alternatively, this relaxation can be viewed as an exponential smoothing of all previous best-reply strategies. For a particular number of classes with specified cost/ reward parameters, relaxation conditions for asymptotic convergence to the unique interior Nash equilibrium are presented. For more than two classes, it is shown that relaxation not only accelerates convergence (at least in a special case), but also is a necessary condition for convergence. Due to an equivalent functional form, these results can be directly translated to a network of users employing a power criterion, a common objective in the communications literature.

The focused crawler of a special-purpose search engine aims to selectively seek out pages that are relevant to a pre-defined set of topics, rather than to exploit all regions of the Web. The PageRank algorithm is often used in ranking web pages, and it is also used in URL ordering for focused crawler. It estimates the page's authority by taking into account the link structure of the Web. However, it assigns each outlink the same weight and is independent of topics, resulting in topic-drift. In this paper, we propose an improved PageRank algorithm, which we called To-PageRank, and then we present a crawling strategy using the To-PageRank algorithm combining with the topic similarity of the hyperlink metadata. The experiment in focused crawler shows that the new improved crawling strategy has better performance than the Breath-first and PageRank algorithms.

<span lang="EN-US">Search engine optimization (SEO) is an important internet marketing strategy and process that facilitates maximizing an intended website’s visibility with search engine results. It is widely employed nowadays to improve traffic volume or quality from search engines to a particular website. Even though a significant number of publications imply the essential aspects of SEO, only a few provide generalized ideas to deal with the complex structure of the web. Also, the critical issues of content quality, site popularity, keyword density, and publicity factors were not much considered in the traditional ranking algorithms during SEO processes. This has negatively influenced the retrieval rate in the existing SEO techniques, and consequently, inadequate search results were obtained through search engines. Hence, the study considers web page ranking as a theoretical basis for the research and addresses these limitations in the existing system. It further improves SEO performance by introducing a unique web-page ranking strategic design to gain higher page rank results. The results of the investigational study show that the proposed system effectively contributes towards SEO with an improved page ranking strategy and also provides higher accuracy in calculating the importance score of web pages which is comparable with popular ranking algorithms such as hyperlink-induced topic search (HITS) and PageRank.</span>

Attack graphs have been proposed as useful tools for analyzing security vulnerabilities in network systems. Even when they are produced efficiently, the size and complexity of attack graphs often prevent a human from fully comprehending the information conveyed. A distillation of this overwhelming amount of information is crucial to aid network administrators in efficiently allocating scarce human and financial resources. This paper introduces AssetRank, a generalization of Google’s PageRank algorithm which ranks web pages in web graphs. AssetRank addresses the unique semantics of dependency attack graphs and incorporates vulnerability data from public databases to compute metrics for the graph vertices (representing attacker privileges and vulnerabilities) which reveal their importance in attacks against the system. The results of applying the algorithm on a number of network scenarios show that the numeric ranks computed are consistent with the intuitive importance that the privileges and vulnerabilities have to an attacker. The vertex ranks can be used to prioritize countermeasures, help a human reader to better comprehend security problems, and provide input to further security analysis tools.Keywordsattack graphsecurity metricPageRankeigenvector

Web Crawlers ‘browse’ the World Wide Web (WWW) on behalf of search engine, to collect web pages from numerous collections of billions of documents. Metacrawler is similar to that of a meta search engine that combines the top web search results from popular search engines. World Wide Web is growing rapidly. This possesses great challenges to general purpose crawlers. This paper introduces an architectural framework of a Metacrawler. This crawler enables the user to retrieve information that is relevant to the topic from more than one traditional web search engines. The crawler works in such a way that it fetches only the pages that are relevant to the topic. The PageRank algorithm is often used in ranking web pages. But, the ranking causes the problem of topic-drift. So, modified PageRank algorithm is used to rank the retrieved web pages in such a way that it reduces this problem. The clustering method is used to combine the search results so that the user can easily select web pages from the clustered results based upon the requirement. Experimental results show the effectiveness of the Metacrawler.

Network theory concepts form the core of algorithms that are designed to uncover valuable insights from various datasets. Especially, network centrality measures such as Eigenvector centrality, Katz centrality, PageRank centrality etc., are used in retrieving top-K viral information propagators in social networks,while web page ranking in efficient information retrieval, etc. In this paper, we propose a novel method for identifying top-K viral information propagators from a reduced search space. Our algorithm computes the Katz centrality and Local average centrality values of each node and tests the values against two threshold (constraints) values. Only those nodes, which satisfy these constraints, form the search space for top-K propagators. Our proposed algorithm is tested against four datasets and the results show that the proposed algorithm is capable of reducing the number of nodes in search space at least by 70%. We also considered the parameter (alpha and beta) dependency of Katz centrality values in our experiments and established a relationship between the alpha values, number of nodes in search space and network characteristics. Later, we compare the top-K results of our approach against the top-K results of degree centrality.

One of the key components which ensures the acceptance of web search service is the web page ranker - a component which is said to have been the main contributing factor to the early successes of Google. It is well established that a machine learning method such as the Graph Neural Network (GNN) is able to learn and estimate Google's page ranking algorithm. This paper shows that the GNN can successfully learn many other Web page ranking methods e.g. TrustRank, HITS and OPIC. Experimental results show that GNN may be suitable to learn any arbitrary web page ranking scheme, and hence, may be more flexible than any other existing web page ranking scheme. The significance of this observation lies in the fact that it is possible to learn ranking schemes for which no algorithmic solution exists or is known.

The Web page rank algorithm is always regarded as the core of the search engine. Firstly, this article analyzes the traditional and classical rank algorithms briefly. Then, it proposes a new rank algorithm, which is called A-PageRank. In this algorithm, the PageRank value of the source page is distributed to its link-out pages according to the topic similarity. Lastly, a new method which uses both the similarity and divergence to weigh the match degree between one web page and one user query is adopted in order to increase the precision and recall rate of the search engine.

The patch test and the general convergence criteria of the finite element method

This paper explores the behavior and convergence properties of fuzzy and multifuzzy sequences of fuzzy numbers, with a focus on their 𝛼-level convergence, 𝛼-cut boundedness, and algebraic operations. We extend the concept to multi-fuzzy sequences of dimension 𝑟, where each component sequence consists of fuzzy numbers. Through a series of illustrative examples, we analyze convergence behavior under varying 𝛼-levels, highlighting cases where convergence occurs only at 𝛼 = 1 or fails altogether. Furthermore, we demonstrate that the sum, difference, and product of two convergent multi-fuzzy sequences also converge under the same 𝛼-level. The paper further analyzes sequences of trapezoidal fuzzy numbers—both normalized and non-normalized— identifying necessary conditions for convergence and presenting counterexamples where convergence fails, even when sufficient conditions are true. These results contribute to a deeper understanding of fuzzy sequence behavior and set a foundation for further studies in multi-fuzzy sequence analysis and multi-fuzzy calculus.

The geometric learning algorithm (GLA) was proposed as an application of the affine projection algorithm (APA) for an adaptive filter to a perceptron. In the GLA, the connection weight vector w(n) is updated vertically towards the orthogonal complement of /spl kappa/ patterns. The GLA demonstrates some typical behavior when the learning rate /spl lambda/ is 2, which means that w(n) and w(n+1) are with respect to the complement. Therefore, in this paper, the GLA with /spl lambda/=2 is discriminated as a symmetric learning algorithm (SLA) and the convergence properties of the SLA are analyzed. The convergence condition among the order /spl kappa/ of the SLA, the number P of patterns and the dimension N of patterns is analyzed theoretically. It is proved that /spl kappa/<N is the necessary condition for convergence when P/spl ges/2 N. The relation between /spl kappa/ and the learning speed is analyzed theoretically. It becomes clear that the maximum learning speed on average can be obtained when /spl kappa/=N/2. These properties are supported by computer simulations. Furthermore, the goodness of the solution by the SLA is investigated through computer simulation. That is, there exists little difference in the goodness of solution by changing the order k.

This paper investigates accelerated gossip algorithms for distributed computations in networks where shift-registers are utilized at each node. By using tools from matrix analysis, we prove the existence of the desired acceleration and establish the fastest rate of convergence in expectation for two-register symmetric gossip. Some classes of networks with regular graph topologies are studied in detail to validate the analytical results by comparison with existing empirical data. We also analyze convergence of second moment and provide a necessary condition for convergence in multi-register symmetric gossip. The proposed approach has the potential to be applied to the more challenging open problem of asymmetric gossip.