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61-398: PageRank works by counting the number and quality of links to a page to determine a rough estimate of how important the website is. The underlying assumption is that more important websites are likely to receive more links from other websites. 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

122-499: A cognitive model for concepts, the centrality algorithm. A search engine called " RankDex " from IDD Information Services, designed by Robin Li in 1996, developed a strategy for site-scoring and page-ranking. Li referred to his search mechanism as "link analysis," which involved ranking the popularity of a web site based on how many other sites had linked to it. RankDex, the first search engine with page-ranking and site-scoring algorithms,

183-452: A probability distribution used to represent the likelihood that a person randomly clicking on links will arrive at any particular page. PageRank can be calculated for collections of documents of any size. It is assumed in several research papers that the distribution is evenly divided among all documents in the collection at the beginning of the computational process. The PageRank computations require several passes, called "iterations", through

244-492: A stochastic matrix (for more details see the computation section below). Thus this is a variant of the eigenvector centrality measure used commonly in network analysis . Because of the large eigengap of the modified adjacency matrix above, the values of the PageRank eigenvector can be approximated to within a high degree of accuracy within only a few iterations. Google's founders, in their original paper, reported that

305-777: A child, inspired by luminaries like Gauss , he wanted to become a mathematician . Motwani went to St Columba's School, New Delhi . He completed his B.Tech. in Computer Science from the Indian Institute of Technology Kanpur in Kanpur , Uttar Pradesh in 1983 and got his Ph.D. in Computer Science from the University of California, Berkeley in Berkeley , California , United States in 1988, under

366-412: A hub is assortative when it tends to connect to other hubs. A disassortative hub avoids connecting to other hubs. If hubs have connections with the expected random probabilities, they are said to be neutral. There are three methods to quantify degree correlations. The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network. This allows for

427-533: A page counts as a vote of support. The PageRank of a page is defined recursively and depends on the number and PageRank metric of all pages that link to it (" incoming links "). A page that is linked to by many pages with high PageRank receives a high rank itself. Numerous academic papers concerning PageRank have been published since Page and Brin's original paper. In practice, the PageRank concept may be vulnerable to manipulation. Research has been conducted into identifying falsely influenced PageRank rankings. The goal

488-500: A series of funnels connected by tubes. Here, the amount of water from the original source is infinite. Also, any funnels that have been exposed to the water continue to experience the water even as it passes into successive funnels. The non-conserved model is the most suitable for explaining the transmission of most infectious diseases , neural excitation, information and rumors, etc. The question of how to immunize efficiently scale free networks which represent realistic networks such as

549-403: A significant use in military intelligence, for uncovering insurgent networks of both hierarchical and leaderless nature. With the recent explosion of publicly available high throughput biological data , the analysis of molecular networks has gained significant interest. The type of analysis in this context is closely related to social network analysis, but often focusing on local patterns in

610-469: A small universe of four web pages: A , B , C , and D . Links from a page to itself are ignored. Multiple outbound links from one page to another page are treated as a single link. PageRank is initialized to the same value for all pages. In the original form of PageRank, the sum of PageRank over all pages was the total number of pages on the web at that time, so each page in this example would have an initial value of 1. However, later versions of PageRank, and

671-445: Is 1 - d . Various studies have tested different damping factors, but it is generally assumed that the damping factor will be set around 0.85. The damping factor is subtracted from 1 (and in some variations of the algorithm, the result is divided by the number of documents ( N ) in the collection) and this term is then added to the product of the damping factor and the sum of the incoming PageRank scores. That is, So any page's PageRank

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732-782: Is defined as Network theory#Link analysis In mathematics , computer science and network science , network theory is a part of graph theory . It defines networks as graphs where the vertices or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components. Network theory has applications in many disciplines, including statistical physics , particle physics , computer science, electrical engineering , biology , archaeology , linguistics , economics , finance , operations research , climatology , ecology , public health , sociology , psychology , and neuroscience . Applications of network theory include logistical networks,

793-559: Is derived in large part from the PageRanks of other pages. The damping factor adjusts the derived value downward. The original paper, however, gave the following formula, which has led to some confusion: The difference between them is that the PageRank values in the first formula sum to one, while in the second formula each PageRank is multiplied by N and the sum becomes N . A statement in Page and Brin's paper that "the sum of all PageRanks

854-591: Is increasingly employed by banks and insurance agencies in fraud detection, by telecommunication operators in telecommunication network analysis, by medical sector in epidemiology and pharmacology , in law enforcement investigations , by search engines for relevance rating (and conversely by the spammers for spamdexing and by business owners for search engine optimization ), and everywhere else where relationships between many objects have to be analyzed. Links are also derived from similarity of time behavior in both nodes. Examples include climate networks where

915-451: Is interested in dynamics on networks or the robustness of a network to node/link removal, often the dynamical importance of a node is the most relevant centrality measure. These concepts are used to characterize the linking preferences of hubs in a network. Hubs are nodes which have a large number of links. Some hubs tend to link to other hubs while others avoid connecting to hubs and prefer to connect to nodes with low connectivity. We say

976-488: Is one" and claims by other Google employees support the first variant of the formula above. Page and Brin confused the two formulas in their most popular paper "The Anatomy of a Large-Scale Hypertextual Web Search Engine", where they mistakenly claimed that the latter formula formed a probability distribution over web pages. Google recalculates PageRank scores each time it crawls the Web and rebuilds its index. As Google increases

1037-487: Is referred to as the PageRank of E and denoted by P R ( E ) . {\displaystyle PR(E).} A PageRank results from a mathematical algorithm based on the webgraph , created by all World Wide Web pages as nodes and hyperlinks as edges, taking into consideration authority hubs such as cnn.com or mayoclinic.org . The rank value indicates an importance of a particular page. A hyperlink to

1098-563: Is the best known. As of September 24, 2019, all patents associated with PageRank have expired. 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

1159-451: Is the damping factor, or in matrix notation where R i ( t ) = P R ( p i ; t ) {\displaystyle \mathbf {R} _{i}(t)=PR(p_{i};t)} and 1 {\displaystyle \mathbf {1} } is the column vector of length N {\displaystyle N} containing only ones. The matrix M {\displaystyle {\mathcal {M}}}

1220-400: Is the ratio between number of links outbound from page j to page i to the total number of outbound links of page j. The adjacency function is 0 if page p j {\displaystyle p_{j}} does not link to p i {\displaystyle p_{i}} , and normalized such that, for each j i.e. the elements of each column sum up to 1, so the matrix is

1281-416: Is the size of the network. As a result of Markov theory , it can be shown that the PageRank of a page is the probability of arriving at that page after a large number of clicks. This happens to equal t − 1 {\displaystyle t^{-1}} where t {\displaystyle t} is the expectation of the number of clicks (or random jumps) required to get from

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1342-442: Is the total number of pages. The PageRank values are the entries of the dominant right eigenvector of the modified adjacency matrix rescaled so that each column adds up to one. This makes PageRank a particularly elegant metric: the eigenvector is where R is the solution of the equation where the adjacency function ℓ ( p i , p j ) {\displaystyle \ell (p_{i},p_{j})}

1403-681: Is to find an effective means of ignoring links from documents with falsely influenced PageRank. Other link-based ranking algorithms for Web pages include the HITS algorithm invented by Jon Kleinberg (used by Teoma and now Ask.com ), the IBM CLEVER project , the TrustRank algorithm, the Hummingbird algorithm, and the SALSA algorithm . The eigenvalue problem behind PageRank's algorithm

1464-932: The World Wide Web , Internet , gene regulatory networks , metabolic networks, social networks , epistemological networks, etc.; see List of network theory topics for more examples. Euler 's solution of the Seven Bridges of Königsberg problem is considered to be the first true proof in the theory of networks. Network problems that involve finding an optimal way of doing something are studied as combinatorial optimization . Examples include network flow , shortest path problem , transport problem , transshipment problem , location problem , matching problem , assignment problem , packing problem , routing problem , critical path analysis , and program evaluation and review technique . The analysis of electric power systems could be conducted using network theory from two main points of view: Social network analysis examines

1525-628: The diffusion of innovations , news and rumors. Similarly, it has been used to examine the spread of both diseases and health-related behaviors . It has also been applied to the study of markets , where it has been used to examine the role of trust in exchange relationships and of social mechanisms in setting prices. It has been used to study recruitment into political movements , armed groups, and other social organizations. It has also been used to conceptualize scientific disagreements as well as academic prestige. More recently, network analysis (and its close cousin traffic analysis ) has gained

1586-439: The eigenvectors of the adjacency matrix corresponding to a network, to determine nodes that tend to be frequently visited. Formally established measures of centrality are degree centrality , closeness centrality , betweenness centrality , eigenvector centrality , subgraph centrality , and Katz centrality . The purpose or objective of analysis generally determines the type of centrality measure to be used. For example, if one

1647-443: The power iteration method or the power method. The basic mathematical operations performed are identical. At t = 0 {\displaystyle t=0} , an initial probability distribution is assumed, usually where N is the total number of pages, and p i ; 0 {\displaystyle p_{i};0} is page i at time 0. At each time step, the computation, as detailed above, yields where d

1708-570: The Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks since for this case p c {\displaystyle pc} is relatively high and fewer nodes are needed to be immunized. However, in most realistic networks the global structure is not available and the largest degree nodes are unknown. Rajeev Motwani Rajeev Motwani ( Hindi : राजीव मोटवानी , 24 March 1962 – 5 June 2009)

1769-472: The PageRank algorithm for a network consisting of 322 million links (in-edges and out-edges) converges to within a tolerable limit in 52 iterations. The convergence in a network of half the above size took approximately 45 iterations. Through this data, they concluded the algorithm can be scaled very well and that the scaling factor for extremely large networks would be roughly linear in log ⁡ n {\displaystyle \log n} , where n

1830-488: The addresses of suspects and victims, the telephone numbers they have dialed, and financial transactions that they have partaken in during a given timeframe, and the familial relationships between these subjects as a part of police investigation. Link analysis here provides the crucial relationships and associations between very many objects of different types that are not apparent from isolated pieces of information. Computer-assisted or fully automatic computer-based link analysis

1891-487: The analysis of time series by network measures. Applications range from detection of regime changes over characterizing dynamics to synchronization analysis. Many real networks are embedded in space. Examples include, transportation and other infrastructure networks, brain neural networks. Several models for spatial networks have been developed. Content in a complex network can spread via two major methods: conserved spread and non-conserved spread. In conserved spread,

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1952-404: The collection to adjust approximate PageRank values to more closely reflect the theoretical true value. A probability is expressed as a numeric value between 0 and 1. A 0.5 probability is commonly expressed as a "50% chance" of something happening. Hence, a document with a PageRank of 0.5 means there is a 50% chance that a person clicking on a random link will be directed to said document. Assume

2013-549: The collection. Their PageRank scores are therefore divided evenly among all other pages. In other words, to be fair with pages that are not sinks, these random transitions are added to all nodes in the Web. This residual probability, d , is usually set to 0.85, estimated from the frequency that an average surfer uses his or her browser's bookmark feature. So, the equation is as follows: where p 1 , p 2 , . . . , p N {\displaystyle p_{1},p_{2},...,p_{N}} are

2074-430: The completion of this iteration, page A will have a PageRank of approximately 0.458. In other words, the PageRank conferred by an outbound link is equal to the document's own PageRank score divided by the number of outbound links L( ) . In the general case, the PageRank value for any page u can be expressed as: i.e. the PageRank value for a page u is dependent on the PageRank values for each page v contained in

2135-399: The connections between nodes, respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, the amount of content changes as it enters and passes through a complex network. The model of non-conserved spread can best be represented by a continuously running faucet running through

2196-505: The development of the field of network medicine . Recent examples of application of network theory in biology include applications to understanding the cell cycle as well as a quantitative framework for developmental processes. The automatic parsing of textual corpora has enabled the extraction of actors and their relational networks on a vast scale. The resulting narrative networks , which can contain thousands of nodes, are then analyzed by using tools from Network theory to identify

2257-435: The development of the page-rank algorithm. Sergey Brin had the idea that information on the web could be ordered in a hierarchy by "link popularity": a page ranks higher as there are more links to it. The system was developed with the help of Scott Hassan and Alan Steremberg, both of whom were cited by Page and Brin as being critical to the development of Google. Rajeev Motwani and Terry Winograd co-authored with Page and Brin

2318-680: The first employee, Craig Silverstein . He was an author of two widely used theoretical computer science textbooks: Randomized Algorithms with Prabhakar Raghavan and Introduction to Automata Theory, Languages, and Computation with John Hopcroft and Jeffrey Ullman . He was an avid angel investor and helped fund a number of startups to emerge from Stanford. He sat on boards including Google, Kaboodle, Mimosa Systems (acquired by Iron Mountain Incorporated ), Adchemy, Baynote , Vuclip , NeoPath Networks (acquired by Cisco Systems in 2007), Tapulous and Stanford Student Enterprises . He

2379-525: The first paper about the project, describing PageRank and the initial prototype of the Google search engine , published in 1998. Shortly after, Page and Brin founded Google Inc. , the company behind the Google search engine. While just one of many factors that determine the ranking of Google search results, PageRank continues to provide the basis for all of Google's web-search tools. The name "PageRank" plays on

2440-493: The key actors, the key communities or parties, and general properties such as robustness or structural stability of the overall network, or centrality of certain nodes. This automates the approach introduced by Quantitative Narrative Analysis, whereby subject-verb-object triplets are identified with pairs of actors linked by an action, or pairs formed by actor-object. Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining

2501-520: The links between two locations (nodes) are determined, for example, by the similarity of the rainfall or temperature fluctuations in both sites. Several Web search ranking algorithms use link-based centrality metrics, including Google 's PageRank , Kleinberg's HITS algorithm , the CheiRank and TrustRank algorithms. Link analysis is also conducted in information science and communication science in order to understand and extract information from

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2562-413: The name of developer Larry Page, as well as of the concept of a web page . The word is a trademark of Google, and the PageRank process has been patented ( U.S. patent 6,285,999 ). However, the patent is assigned to Stanford University and not to Google. Google has exclusive license rights on the patent from Stanford University. The university received 1.8 million shares of Google in exchange for use of

2623-484: The network. For example, network motifs are small subgraphs that are over-represented in the network. Similarly, activity motifs are patterns in the attributes of nodes and edges in the network that are over-represented given the network structure. Using networks to analyze patterns in biological systems, such as food-webs, allows us to visualize the nature and strength of interactions between species. The analysis of biological networks with respect to diseases has led to

2684-538: The next iteration, for a total of 0.75. Suppose instead that page B had a link to pages C and A , page C had a link to page A , and page D had links to all three pages. Thus, upon the first iteration, page B would transfer half of its existing value (0.125) to page A and the other half (0.125) to page C . Page C would transfer all of its existing value (0.25) to the only page it links to, A . Since D had three outbound links, it would transfer one third of its existing value, or approximately 0.083, to A . At

2745-409: The number of documents in its collection, the initial approximation of PageRank decreases for all documents. The formula uses a model of a random surfer who reaches their target site after several clicks, then switches to a random page. The PageRank value of a page reflects the chance that the random surfer will land on that page by clicking on a link. It can be understood as a Markov chain in which

2806-532: The page back to itself. One main disadvantage of PageRank is that it favors older pages. A new page, even a very good one, will not have many links unless it is part of an existing site (a site being a densely connected set of pages, such as Misplaced Pages ). Several strategies have been proposed to accelerate the computation of PageRank. Various strategies to manipulate PageRank have been employed in concerted efforts to improve search results rankings and monetize advertising links. These strategies have severely impacted

2867-435: The pages under consideration, M ( p i ) {\displaystyle M(p_{i})} is the set of pages that link to p i {\displaystyle p_{i}} , L ( p j ) {\displaystyle L(p_{j})} is the number of outbound links on page p j {\displaystyle p_{j}} , and N {\displaystyle N}

2928-552: The patent; it sold the shares in 2005 for US$ 336 million. PageRank was influenced by citation analysis , early developed by Eugene Garfield in the 1950s at the University of Pennsylvania, and by Hyper Search , developed by Massimo Marchiori at the University of Padua . In the same year PageRank was introduced (1998), Jon Kleinberg published his work on HITS . Google's founders cite Garfield, Marchiori, and Kleinberg in their original papers. The PageRank algorithm outputs

2989-528: The reliability of the PageRank concept, which purports to determine which documents are actually highly valued by the Web community. Since December 2007, when it started actively penalizing sites selling paid text links, Google has combatted link farms and other schemes designed to artificially inflate PageRank. How Google identifies link farms and other PageRank manipulation tools is among Google's trade secrets . PageRank can be computed either iteratively or algebraically. The iterative method can be viewed as

3050-424: The remainder of this section, assume a probability distribution between 0 and 1. Hence the initial value for each page in this example is 0.25. The PageRank transferred from a given page to the targets of its outbound links upon the next iteration is divided equally among all outbound links. If the only links in the system were from pages B , C , and D to A , each link would transfer 0.25 PageRank to A upon

3111-400: The set B u (the set containing all pages linking to page u ), divided by the number L ( v ) of links from page v . The PageRank theory holds that an imaginary surfer who is randomly clicking on links will eventually stop clicking. The probability, at any step, that the person will continue following links is a damping factor d . The probability that they instead jump to any random page

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3172-433: The states are pages, and the transitions are the links between pages – all of which are all equally probable. If a page has no links to other pages, it becomes a sink and therefore terminates the random surfing process. If the random surfer arrives at a sink page, it picks another URL at random and continues surfing again. When calculating PageRank, pages with no outbound links are assumed to link out to all other pages in

3233-427: The structure of collections of web pages. For example, the analysis might be of the interlinking between politicians' websites or blogs. Another use is for classifying pages according to their mention in other pages. Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like sociology . For example, eigenvector centrality uses

3294-499: The structure of relationships between social entities. These entities are often persons, but may also be groups , organizations , nation states , web sites , or scholarly publications . Since the 1970s, the empirical study of networks has played a central role in social science, and many of the mathematical and statistical tools used for studying networks have been first developed in sociology . Amongst many other applications, social network analysis has been used to understand

3355-566: The supervision of Richard M. Karp . Motwani joined Stanford soon after U.C. Berkeley. He founded the Mining Data at Stanford project (MIDAS), an umbrella organization for several groups looking into new and innovative data management concepts. His research included data privacy , web search , robotics , and computational drug design . He is also one of the originators of the Locality-sensitive hashing algorithm. Motwani

3416-409: The total amount of content that enters a complex network remains constant as it passes through. The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured into a series of funnels connected by tubes. Here, the pitcher represents the original source and the water is the content being spread. The funnels and connecting tubing represent the nodes and

3477-587: Was active in the Business Association of Stanford Entrepreneurial Students (BASES). He was a winner of the Gödel Prize in 2001 for his work on the PCP theorem and its applications to hardness of approximation . Motwani was found dead in his pool in the backyard of his Atherton , San Mateo County , California home on 5 June 2009. The San Mateo County coroner, Robert Foucrault , ruled

3538-574: Was an Indian-American professor of computer science at Stanford University whose research focused on theoretical computer science . He was a special advisor to Sequoia Capital . He was a winner of the Gödel Prize in 2001. Rajeev Motwani was born in Jammu , Jammu and Kashmir , India , on 24 March 1962, and grew up in New Delhi . His father was in the Indian Army . He had two brothers. As

3599-466: Was independently rediscovered and reused in many scoring problems. In 1895, Edmund Landau suggested using it for determining the winner of a chess tournament. The eigenvalue problem was also suggested in 1976 by Gabriel Pinski and Francis Narin, who worked on scientometrics ranking scientific journals, in 1977 by Thomas Saaty in his concept of Analytic Hierarchy Process which weighted alternative choices, and in 1995 by Bradley Love and Steven Sloman as

3660-647: Was launched in 1996. Li filed a patent for the technology in RankDex in 1997; it was granted in 1999. He later used it when he founded Baidu in China in 2000. Google founder Larry Page referenced Li's work as a citation in some of his U.S. patents for PageRank. Larry Page and Sergey Brin developed PageRank at Stanford University in 1996 as part of a research project about a new kind of search engine. An interview with Héctor García-Molina , Stanford Computer Science professor and advisor to Sergey, provides background into

3721-435: Was one of the co-authors (with Larry Page and Sergey Brin , and Terry Winograd ) of an influential early paper on the PageRank algorithm . He also co-authored another seminal search paper What Can You Do With A Web In Your Pocket with those same authors. PageRank was the basis for search techniques of Google (founded by Page and Brin), and Motwani advised or taught many of Google's developers and researchers, including

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