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CheiRank
The CheiRank is an eigenvector with a maximal real eigenvalue of the Google matrix G^* constructed for a directed network with the inverted directions of links. It is similar to the PageRank vector, which ranks the network nodes in average proportionally to a number of incoming links being the maximal eigenvector of the Google matrix G with a given initial direction of links. Due to inversion of link directions the CheiRank ranks the network nodes in average proportionally to a number of outgoing links. Since each node belongs both to CheiRank and PageRank vectors the ranking of information flow on a directed network becomes two-dimensional. Definition For a given directed network the Google matrix is constructed in the way described in the article Google matrix. The PageRank vector is the eigenvector with the maximal real eigenvalue \lambda=1 . It was introduced in and is discussed in the article PageRank. In a similar way the CheiRank is the eigenvector with the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CheiRank2
The CheiRank is an Eigenvalues and eigenvectors, eigenvector with a maximal real eigenvalue of the Google matrix G^* constructed for a directed network with the inverted directions of links. It is similar to the PageRank vector, which ranks the network nodes in average proportionally to a number of incoming links being the maximal eigenvector of the Google matrix G with a given initial direction of links. Due to inversion of link directions the CheiRank ranks the network nodes in average proportionally to a number of outgoing links. Since each node belongs both to CheiRank and PageRank vectors the ranking of information flow on a directed network becomes Two-dimensional space, two-dimensional. Definition For a given directed network the Google matrix is constructed in the way described in the article Google matrix. The PageRank vector is the eigenvector with the maximal real eigenvalue \lambda=1 . It was introduced in and is discussed in the article PageRank. In a simil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Google Matrix
A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links between pages. The PageRank of each page can then be generated iteratively from the Google matrix using the power method. However, in order for the power method to converge, the matrix must be stochastic, Irreducible (mathematics), irreducible and aperiodic. Adjacency matrix ''A'' and Markov matrix ''S'' In order to generate the Google matrix ''G'', we must first generate an adjacency matrix ''A'' which represents the relations between pages or nodes. Assuming there are ''N'' pages, we can fill out ''A'' by doing the following: # A matrix element A_ is filled with 1 if node j has a link to node i, and 0 otherwise; this is the adjacency matrix of links. # A related matrix ''S'' corresponding to the transitions in a Markov chain#Properties of Markov chains, Markov chain of given network is constructed from ''A'' by dividing the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PageRank
PageRank (PR) is an algorithm used by Google Search to rank web pages in their search engine results. It is named after both the term "web page" and co-founder Larry Page. PageRank is a way of measuring the importance of website pages. According to Google: Currently, PageRank is not the only algorithm used by Google to order search results, but it is the first algorithm that was used by the company, and it is the best known. As of September 24, 2019, all patents associated with PageRank have expired. Description PageRank is a link analysis algorithm and it assigns a numerical weighting to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of "measuring" its relative importance within the set. The algorithm may be applied to any collection of entities with reciprocal quotations and references. The numerical weight that it assigns to any given element ''E'' is referred to as the ''PageRank of E'' and denoted by PR(E). A PageRank resu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Retrieval
Information retrieval (IR) in computing and information science is the task of identifying and retrieving information system resources that are relevant to an Information needs, information need. The information need can be specified in the form of a search query. In the case of document retrieval, queries can be based on full-text search, full-text or other content-based indexing. Information retrieval is the science of searching for information in a document, searching for documents themselves, and also searching for the metadata that describes data, and for databases of texts, images or sounds. Automated information retrieval systems are used to reduce what has been called information overload. An IR system is a software system that provides access to books, journals and other documents; it also stores and manages those documents. Web search engines are the most visible IR applications. Overview An information retrieval process begins when a user enters a query into the sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfer Operator
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system. The transfer operator is sometimes called the Ruelle operator, after David Ruelle, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination of the eigenvalues of the operator. Definition The iterated function to be studied is a map f\colon X\rightarrow X for an arbitrary set X. The transfer operator is defined as an operator \mathcal acting on the space of functions \ as :(\mathcal\Phi)(x) = \sum_ g(y) \Phi(y) where g\colon X\rightarrow\mathbb is an auxiliary valuation function. When f has a Jacobian determinant , J, , then g is usually taken ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chains
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Trade
International trade is the exchange of capital, goods, and services across international borders or territories because there is a need or want of goods or services. (See: World economy.) In most countries, such trade represents a significant share of gross domestic product (GDP). While international trade has existed throughout history (for example Uttarapatha, Silk Road, Amber Road, salt roads), its economic, social, and political importance has been on the rise in recent centuries. Carrying out trade at an international level is a complex process when compared to domestic trade. When trade takes place between two or more states, factors like currency, government policies, economy, judicial system, laws, and markets influence trade. To ease and justify the process of trade between countries of different economic standing in the modern era, some international economic organizations were formed, such as the World Trade Organization. These organizations work towards the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HITS Algorithm
Hyperlink-Induced Topic Search (HITS; also known as hubs and authorities) is a link analysis algorithm that rates Web pages, developed by Jon Kleinberg. The idea behind Hubs and Authorities stemmed from a particular insight into the creation of web pages when the Internet was originally forming; that is, certain web pages, known as hubs, served as large directories that were not actually authoritative in the information that they held, but were used as compilations of a broad catalog of information that led users direct to other authoritative pages. In other words, a good hub represents a page that pointed to many other pages, while a good authority represents a page that is linked by many different hubs. The scheme therefore assigns two scores for each page: its authority, which estimates the value of the content of the page, and its hub value, which estimates the value of its links to other pages. History In journals Many methods have been used to rank the importance of scient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
