This document summarizes the PageRank algorithm used by Google to rank webpages. It describes PageRank as modeling a random web surfer who randomly clicks on links. PageRank can be defined as the principal eigenvector of the Google matrix, which is constructed from the webpage adjacency matrix and accounts for dangling pages. The power method is used to efficiently approximate PageRank values since directly computing eigenvectors is infeasible for web-scale graphs. The document also discusses how changing the α value in the Google matrix construction affects convergence rates.
The document introduces the PageRank algorithm. It explains that PageRank ranks pages based on the principle that pages with more inlinks are considered more important. It discusses how PageRank addresses issues like cyclic links and pages with no outlinks. It also provides the formula used to calculate PageRank through an iterative process and describes how Google uses over 200 factors beyond PageRank alone to determine search rankings.
The document discusses how Markov chains are used as the methodology behind PageRank to rank web pages on the internet. It provides an overview of key concepts, including defining Markov chains and stochastic processes. It explains the idea behind PageRank, treating each web page as a journal and measuring importance based on the number of citations/links to other pages. The PageRank algorithm models web surfing as a Markov chain and the steady-state probabilities of the chain indicate the importance of each page.
Done reread sketchinglandscapesofpagefarmsnpcomplete(2)James Arnold
The document proposes and analyzes the concept of "page farms", which are sets of pages that contribute to the PageRank score of a target page. It finds that computing page farms is NP-hard, so it develops a greedy algorithm to extract approximate page farms. It then analyzes the page farms of over 3 million randomly sampled web pages, finding that the distributions of page farm sizes follow a power law distribution and reflect the importance of pages.
Finding important nodes in social networks based on modified pagerankcsandit
Important nodes are individuals who have huge influence on social network. Finding important
nodes in social networks is of great significance for research on the structure of the social
networks. Based on the core idea of Pagerank, a new ranking method is proposed by
considering the link similarity between the nodes. The key concept of the method is the use of
the link vector which records the contact times between nodes. Then the link similarity is
computed based on the vectors through the similarity function. The proposed method
incorporates the link similarity into original Pagerank. The experiment results show that the
proposed method can get better performance.
Implementing page rank algorithm using hadoop map reduceFarzan Hajian
The document describes how to implement PageRank, an algorithm for ranking the importance of web pages, using Hadoop MapReduce. PageRank is calculated iteratively by treating each web page as a "random surfer" that follows links with certain probabilities based on the page's own importance ranking. The MapReduce implementation involves multiple stages where mappers distribute PageRank values to outbound links and reducers calculate new PageRank values based on the formula. The process iterates until PageRank values converge within a set threshold.
This document provides an overview of PageRank, the algorithm used by Google to rank websites in search results. PageRank is a way to measure a page's importance by analyzing the number and quality of links to it, with more incoming links and links from important pages improving its rank. The algorithm calculates PageRank recursively based on the PageRank scores of incoming pages divided by the number of outgoing links. It also includes a damping factor to account for random surfing. The Google Toolbar displays an approximate PageRank value from 0 to 10 to indicate a page's importance relative to other sites.
This document discusses the PageRank algorithm used by Google to rank web pages. It begins with an introduction to how search engines work and the importance of PageRank. It then provides an example network of 12 web pages connected by links. It explains how to represent this as a matrix and use a damping factor to calculate the PageRank values for each page. The document concludes with business recommendations around optimizing link structure and content to improve PageRank.
This is the class project of Stanford Univ. online course Mining Massive Datasets. The mission is to write a Hadoop program that calculates Pagerank for 2002 Google Programming Contest Web Graph Data.
The document introduces the PageRank algorithm. It explains that PageRank ranks pages based on the principle that pages with more inlinks are considered more important. It discusses how PageRank addresses issues like cyclic links and pages with no outlinks. It also provides the formula used to calculate PageRank through an iterative process and describes how Google uses over 200 factors beyond PageRank alone to determine search rankings.
The document discusses how Markov chains are used as the methodology behind PageRank to rank web pages on the internet. It provides an overview of key concepts, including defining Markov chains and stochastic processes. It explains the idea behind PageRank, treating each web page as a journal and measuring importance based on the number of citations/links to other pages. The PageRank algorithm models web surfing as a Markov chain and the steady-state probabilities of the chain indicate the importance of each page.
Done reread sketchinglandscapesofpagefarmsnpcomplete(2)James Arnold
The document proposes and analyzes the concept of "page farms", which are sets of pages that contribute to the PageRank score of a target page. It finds that computing page farms is NP-hard, so it develops a greedy algorithm to extract approximate page farms. It then analyzes the page farms of over 3 million randomly sampled web pages, finding that the distributions of page farm sizes follow a power law distribution and reflect the importance of pages.
Finding important nodes in social networks based on modified pagerankcsandit
Important nodes are individuals who have huge influence on social network. Finding important
nodes in social networks is of great significance for research on the structure of the social
networks. Based on the core idea of Pagerank, a new ranking method is proposed by
considering the link similarity between the nodes. The key concept of the method is the use of
the link vector which records the contact times between nodes. Then the link similarity is
computed based on the vectors through the similarity function. The proposed method
incorporates the link similarity into original Pagerank. The experiment results show that the
proposed method can get better performance.
Implementing page rank algorithm using hadoop map reduceFarzan Hajian
The document describes how to implement PageRank, an algorithm for ranking the importance of web pages, using Hadoop MapReduce. PageRank is calculated iteratively by treating each web page as a "random surfer" that follows links with certain probabilities based on the page's own importance ranking. The MapReduce implementation involves multiple stages where mappers distribute PageRank values to outbound links and reducers calculate new PageRank values based on the formula. The process iterates until PageRank values converge within a set threshold.
This document provides an overview of PageRank, the algorithm used by Google to rank websites in search results. PageRank is a way to measure a page's importance by analyzing the number and quality of links to it, with more incoming links and links from important pages improving its rank. The algorithm calculates PageRank recursively based on the PageRank scores of incoming pages divided by the number of outgoing links. It also includes a damping factor to account for random surfing. The Google Toolbar displays an approximate PageRank value from 0 to 10 to indicate a page's importance relative to other sites.
This document discusses the PageRank algorithm used by Google to rank web pages. It begins with an introduction to how search engines work and the importance of PageRank. It then provides an example network of 12 web pages connected by links. It explains how to represent this as a matrix and use a damping factor to calculate the PageRank values for each page. The document concludes with business recommendations around optimizing link structure and content to improve PageRank.
This is the class project of Stanford Univ. online course Mining Massive Datasets. The mission is to write a Hadoop program that calculates Pagerank for 2002 Google Programming Contest Web Graph Data.
The Google Pagerank algorithm - How does it work?Kundan Bhaduri
This document discusses how Google uses Markov chains and the PageRank algorithm to rank web pages. It begins by explaining Markov chains and how they can model random user behavior on the web. It then describes how Google implemented PageRank as a non-absorbing Markov chain to calculate the probability of a random user reaching any given page. The document outlines issues with applying this to the large-scale web, and proposes techniques like the power method to efficiently approximate PageRank values for the trillion-page internet graph. Finally, it provides an example of how links between related high-authority sites can increase the PageRank of a given page.
This document provides an overview of the PageRank algorithm. It begins with background on PageRank and its development by Brin and Page. It then introduces the concepts behind PageRank, including how it uses the link structure of webpages to determine importance. The core PageRank algorithm is explained, modeling the web as a graph and calculating page importance based on both the number and quality of inbound links. Iterative methods like power iteration are described for approximating solutions. Examples are given to illustrate PageRank calculations over multiple iterations. Implementation details, applications, advantages/disadvantages are also discussed at a high level. Pseudocode is included.
Distinguishing the Roles of a Rabbi and CantorLeib Tropper
Leib Tropper is a rabbi who has spent several years teaching in Israel. Prior to teaching abroad and establishing the Kol Yaakov Torah Center in Monsey, New York, Leib Tropper studied at the Rabbi Jacob Joseph School and the Talmudical Yeshiva of Philadelphia.
Volcanoes form when molten rock flows from fissures in the Earth's surface caused by movement of tectonic plates. Volcanoes are classified as active, extinct, or dormant based on eruption likelihood. Active volcanoes frequently erupt, like Mount Etna and Mount Fujiyama. Extinct volcanoes were once active but will not erupt again, such as Kilimanjaro. Dormant volcanoes erupted in the past but are not currently active, though they could become active again, for example Vesuvius in Italy.
Sistem Informasi Sekolah Terpadu (SIST) yang ditawarkan berbasis komputer dan menggunakan barcode untuk memudahkan administrasi sekolah seperti database siswa, keuangan, nilai, perpustakaan, absensi, BK, dan tabungan siswa. SIST ini diharapkan dapat memberikan contoh penerapan TI di sekolah serta mempermudah dan mempercepat proses administrasi.
This document discusses providing excellent customer service that exceeds expectations. It states that their agents are experienced communicators who can provide accurate information to customers in their own language. They aim to deliver top-quality customer support through their experienced agents and latest technology to ensure excellent customer care is the starting point. Contact information is provided to learn more.
This trail document describes a 4 kilometer long walking path near Murcia, Spain called the Abarán Waterwheels Trail. The trail has a low difficulty level and is open to disabled visitors, taking approximately 2.5 hours to complete. A map of the trail route is provided.
This document outlines the roles and responsibilities of the Community Initiative Task Force (CITF) coaching team in providing training and technical assistance to GASPS providers. The primary roles of CITF coaches are to partner with regional prevention specialists (RPS) to guide GASPS providers through the Strategic Prevention Framework process and ensure timely guidance. Coaches provide training to increase knowledge of SPF and technical assistance to help providers understand and adopt SPF principles. The document describes the support elements in place, including RPS, CITF coaches, and the Electronic Control Coordination Operations system to submit questions and requests for assistance.
We often forget the power which words have and use them in indiscriminately. Here are some words which if used properly can motivate us and the listeners as well.
The document discusses how the opening two minutes of films are used to introduce important characters and elements through techniques like shocking moments, speech, and setting establishment. It also notes that technical filmmaking codes like sound, editing, camera work and mise en scene are used to set the setting, characters, storyline, theme, and pace. Research deconstructing four action films found their soundtracks commonly built tension, camerawork was often shaky, jump cuts and continuity editing were used, and mise en scene reflected the scene.
Social can learn from tradicional customer serviceCentrecom
Traditional customer service is adept at directing queries to the right team or individual. Social platforms can also use automation by analysing keywords and phrases
This document provides an overview of Japanese culture, including its geography, traditional styles, arts, and ceremonies. Japan consists of over 3,000 islands, with the largest being Honshu (which includes the capital Tokyo) and Hokkaido, Kyushu, and Shikoku. Traditional Japanese clothing like kimonos and yukatas are still worn during special occasions. Japanese arts include martial arts like judo and sumo, as well as games and painting. Colorful festivals are an important part of Japanese culture and reflect ancient religious beliefs, with ceremonies like the doll festival and festival for children.
The document proposes a narrative and mood board for a music video for the song "Staring At The Stars" by Passenger. It suggests that the protagonist has a fight with his girlfriend and storms off, meeting up with friends by a campfire where they enjoy drinks together, realizing friendship may be better than love. It provides inspiration videos and discusses capturing the protagonist's journey through scenes of nature contrasted with trash. Target audiences are described as any age who enjoy relaxed music. Logistics address using a car for transport and safety around the campfire.
This document contains summaries of 13 episodes from "Happiness Index" by BK Shivani. Some key points discussed across the episodes include:
- Physical and emotional health are necessary for a smooth life journey. Emotional instability is often the root cause of physical problems.
- We have a choice in how we respond to situations - to empower or vulnerabalize ourselves. Our inner state should not be altered by outer situations.
- Happiness comes from within, not from material possessions or dependence on others. Our thoughts create our feelings, attitudes, habits, personality and ultimately our destiny.
- The quality of thoughts we generate influence our happiness. We can choose to think positively even in difficult situations rather than
This document shows 10 pairs of "Before" and "After" images, suggesting it is demonstrating the photo editing capabilities of Photoshop through examples of edits made to images. The "Before" images are being compared side-by-side with the "After" images that have been edited or modified using Photoshop's photo editing tools and features.
This document discusses the evolution of annual reports from their origins in the 19th century to modern practices. It covers:
- How annual reports began as a requirement for gold mining companies in Australia in the 1850s-60s to share information with investors. This set a precedent for other regions to mandate annual reports.
- How annual reports evolved from basic financial statements to include more strategic, marketing, and branding communications over the 20th century. Companies explored using graphic design, themes, and distribution to attract new investors.
- How some Indian IT companies in the 1990s-2000s used annual reports to communicate broader messages around social responsibility, governance, and risk management beyond basic financial data to strengthen relationships with stakeholders
UNHI Creative Works Symposium Session: Deconstructing a Cpyright LicenseUNHInnovation
UNHInnovation Creative Works Symposium session: Deconstructing a Copyright License slide deck
Presenters: Kim A.W. Peaslee, Upton & Hatfield and Tim Benoît-Ledoux, UNHInnovation
This document discusses techniques for detecting link farms, which are groups of web pages that link to each other to artificially boost their PageRank scores. It provides background on PageRank and how link farms can manipulate it. The proposed method calculates both PageRank and a new "GapRank" score for pages, and identifies pages as part of a link farm if they have identical PageRank and GapRank values. The method is demonstrated on a sample dataset, where pages with duplicate PageRank scores are found and shown to also have identical GapRank, identifying them as a link farm that is then removed from the dataset. This improves the PageRank algorithm's ability to rank pages accurately.
This document discusses techniques for detecting link farms, which are groups of web pages that link to each other to artificially boost their PageRank scores. It provides background on PageRank and how link farms can manipulate it. The proposed method calculates both PageRank and a new "GapRank" score for pages, and identifies pages as part of a link farm if they have identical PageRank and GapRank values. The method is demonstrated on a sample dataset, where pages with duplicate PageRank scores are found and shown to also have identical GapRank, identifying them as a link farm that is then removed from the dataset. This improves the PageRank algorithm's ability to rank pages accurately.
The Google Pagerank algorithm - How does it work?Kundan Bhaduri
This document discusses how Google uses Markov chains and the PageRank algorithm to rank web pages. It begins by explaining Markov chains and how they can model random user behavior on the web. It then describes how Google implemented PageRank as a non-absorbing Markov chain to calculate the probability of a random user reaching any given page. The document outlines issues with applying this to the large-scale web, and proposes techniques like the power method to efficiently approximate PageRank values for the trillion-page internet graph. Finally, it provides an example of how links between related high-authority sites can increase the PageRank of a given page.
This document provides an overview of the PageRank algorithm. It begins with background on PageRank and its development by Brin and Page. It then introduces the concepts behind PageRank, including how it uses the link structure of webpages to determine importance. The core PageRank algorithm is explained, modeling the web as a graph and calculating page importance based on both the number and quality of inbound links. Iterative methods like power iteration are described for approximating solutions. Examples are given to illustrate PageRank calculations over multiple iterations. Implementation details, applications, advantages/disadvantages are also discussed at a high level. Pseudocode is included.
Distinguishing the Roles of a Rabbi and CantorLeib Tropper
Leib Tropper is a rabbi who has spent several years teaching in Israel. Prior to teaching abroad and establishing the Kol Yaakov Torah Center in Monsey, New York, Leib Tropper studied at the Rabbi Jacob Joseph School and the Talmudical Yeshiva of Philadelphia.
Volcanoes form when molten rock flows from fissures in the Earth's surface caused by movement of tectonic plates. Volcanoes are classified as active, extinct, or dormant based on eruption likelihood. Active volcanoes frequently erupt, like Mount Etna and Mount Fujiyama. Extinct volcanoes were once active but will not erupt again, such as Kilimanjaro. Dormant volcanoes erupted in the past but are not currently active, though they could become active again, for example Vesuvius in Italy.
Sistem Informasi Sekolah Terpadu (SIST) yang ditawarkan berbasis komputer dan menggunakan barcode untuk memudahkan administrasi sekolah seperti database siswa, keuangan, nilai, perpustakaan, absensi, BK, dan tabungan siswa. SIST ini diharapkan dapat memberikan contoh penerapan TI di sekolah serta mempermudah dan mempercepat proses administrasi.
This document discusses providing excellent customer service that exceeds expectations. It states that their agents are experienced communicators who can provide accurate information to customers in their own language. They aim to deliver top-quality customer support through their experienced agents and latest technology to ensure excellent customer care is the starting point. Contact information is provided to learn more.
This trail document describes a 4 kilometer long walking path near Murcia, Spain called the Abarán Waterwheels Trail. The trail has a low difficulty level and is open to disabled visitors, taking approximately 2.5 hours to complete. A map of the trail route is provided.
This document outlines the roles and responsibilities of the Community Initiative Task Force (CITF) coaching team in providing training and technical assistance to GASPS providers. The primary roles of CITF coaches are to partner with regional prevention specialists (RPS) to guide GASPS providers through the Strategic Prevention Framework process and ensure timely guidance. Coaches provide training to increase knowledge of SPF and technical assistance to help providers understand and adopt SPF principles. The document describes the support elements in place, including RPS, CITF coaches, and the Electronic Control Coordination Operations system to submit questions and requests for assistance.
We often forget the power which words have and use them in indiscriminately. Here are some words which if used properly can motivate us and the listeners as well.
The document discusses how the opening two minutes of films are used to introduce important characters and elements through techniques like shocking moments, speech, and setting establishment. It also notes that technical filmmaking codes like sound, editing, camera work and mise en scene are used to set the setting, characters, storyline, theme, and pace. Research deconstructing four action films found their soundtracks commonly built tension, camerawork was often shaky, jump cuts and continuity editing were used, and mise en scene reflected the scene.
Social can learn from tradicional customer serviceCentrecom
Traditional customer service is adept at directing queries to the right team or individual. Social platforms can also use automation by analysing keywords and phrases
This document provides an overview of Japanese culture, including its geography, traditional styles, arts, and ceremonies. Japan consists of over 3,000 islands, with the largest being Honshu (which includes the capital Tokyo) and Hokkaido, Kyushu, and Shikoku. Traditional Japanese clothing like kimonos and yukatas are still worn during special occasions. Japanese arts include martial arts like judo and sumo, as well as games and painting. Colorful festivals are an important part of Japanese culture and reflect ancient religious beliefs, with ceremonies like the doll festival and festival for children.
The document proposes a narrative and mood board for a music video for the song "Staring At The Stars" by Passenger. It suggests that the protagonist has a fight with his girlfriend and storms off, meeting up with friends by a campfire where they enjoy drinks together, realizing friendship may be better than love. It provides inspiration videos and discusses capturing the protagonist's journey through scenes of nature contrasted with trash. Target audiences are described as any age who enjoy relaxed music. Logistics address using a car for transport and safety around the campfire.
This document contains summaries of 13 episodes from "Happiness Index" by BK Shivani. Some key points discussed across the episodes include:
- Physical and emotional health are necessary for a smooth life journey. Emotional instability is often the root cause of physical problems.
- We have a choice in how we respond to situations - to empower or vulnerabalize ourselves. Our inner state should not be altered by outer situations.
- Happiness comes from within, not from material possessions or dependence on others. Our thoughts create our feelings, attitudes, habits, personality and ultimately our destiny.
- The quality of thoughts we generate influence our happiness. We can choose to think positively even in difficult situations rather than
This document shows 10 pairs of "Before" and "After" images, suggesting it is demonstrating the photo editing capabilities of Photoshop through examples of edits made to images. The "Before" images are being compared side-by-side with the "After" images that have been edited or modified using Photoshop's photo editing tools and features.
This document discusses the evolution of annual reports from their origins in the 19th century to modern practices. It covers:
- How annual reports began as a requirement for gold mining companies in Australia in the 1850s-60s to share information with investors. This set a precedent for other regions to mandate annual reports.
- How annual reports evolved from basic financial statements to include more strategic, marketing, and branding communications over the 20th century. Companies explored using graphic design, themes, and distribution to attract new investors.
- How some Indian IT companies in the 1990s-2000s used annual reports to communicate broader messages around social responsibility, governance, and risk management beyond basic financial data to strengthen relationships with stakeholders
UNHI Creative Works Symposium Session: Deconstructing a Cpyright LicenseUNHInnovation
UNHInnovation Creative Works Symposium session: Deconstructing a Copyright License slide deck
Presenters: Kim A.W. Peaslee, Upton & Hatfield and Tim Benoît-Ledoux, UNHInnovation
This document discusses techniques for detecting link farms, which are groups of web pages that link to each other to artificially boost their PageRank scores. It provides background on PageRank and how link farms can manipulate it. The proposed method calculates both PageRank and a new "GapRank" score for pages, and identifies pages as part of a link farm if they have identical PageRank and GapRank values. The method is demonstrated on a sample dataset, where pages with duplicate PageRank scores are found and shown to also have identical GapRank, identifying them as a link farm that is then removed from the dataset. This improves the PageRank algorithm's ability to rank pages accurately.
This document discusses techniques for detecting link farms, which are groups of web pages that link to each other to artificially boost their PageRank scores. It provides background on PageRank and how link farms can manipulate it. The proposed method calculates both PageRank and a new "GapRank" score for pages, and identifies pages as part of a link farm if they have identical PageRank and GapRank values. The method is demonstrated on a sample dataset, where pages with duplicate PageRank scores are found and shown to also have identical GapRank, identifying them as a link farm that is then removed from the dataset. This improves the PageRank algorithm's ability to rank pages accurately.
The document discusses applications of Markov chains, including PageRank and random walks. It provides details on:
- PageRank, which was developed by Larry Page and Sergey Brin to rank web pages based on the link structure of the web. It models the random surfing of a user on the web as a Markov chain.
- The PageRank algorithm assigns initial uniform probabilities to web pages and then iteratively updates the probabilities based on the links between pages until it converges. This stationary distribution provides the ranking of pages.
- Computing PageRank on the entire web graph is slow, so Google estimates it by running the random walk for a finite number of steps to approximate the stationary distribution.
A Generalization of the PageRank Algorithm : NOTESSubhajit Sahu
This paper discusses a method of Generalizing PageRank algorithm for different types of networks. Rank of each vertex is considered to be dependent upon both the in- and out-edges. Each edge can also have differing importance. This solves the problem of dead ends and spider traps without the need of taxation (?).
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Abstract— PageRank is a well-known algorithm that has been used to understand the structure of the Web. In its classical formulation the algorithm considers only forward looking paths in its analysis- a typical web scenario. We propose a generalization of the PageRank algorithm based on both out-links and in-links. This generalization enables the elimination network anomalies- and increases the applicability of the algorithm to an array of new applications in networked data. Through experimental results we illustrate that the proposed generalized PageRank minimizes the effect of network anomalies, and results in more realistic representation of the network.
Keywords- Search Engine; PageRank; Web Structure; Web Mining; Spider-Trap; dead-end; Taxation;Web spamming
PageRank is a method for ranking web pages based on the link structure of the web. It assigns a ranking to each page based on both the number and quality of links from other pages. The quality of a page is determined in part by the quality of pages linking to it, as indicated by their own PageRank. PageRank handles both inbound and outbound links to measure a page's importance and spread rankings iteratively through the link structure. It helps search engines better understand the vastness of the web and order results.
PageRank is a method for ranking web pages based on the link structure of the web. It assigns a ranking to each page based on both the number and quality of links from other pages. The quality of a page is determined in part by the quality of pages linking to it, as indicated by those page's PageRank scores. PageRank handles both inbound and outbound links to measure a page's relative importance within the web as a whole. It helps search engines better understand the vast heterogeneity of the web and provide more relevant results to users.
The PageRank algorithm calculates the importance of web pages based on the structure of incoming links. It models a random web surfer that randomly clicks on links, and also occasionally jumps to a random page. Pages are given more importance if they are linked to by other important pages. The algorithm represents this as a Markov chain and computes the PageRank scores through an iterative process until convergence. It has the advantages of being resistant to spam and efficiently pre-computing scores independently of user queries.
Done reread sketchinglandscapesofpagefarmsnpcomplete(3)James Arnold
The document proposes and analyzes the concept of "page farms", which are sets of pages that contribute to the PageRank score of a target page. It finds that computing page farms is NP-hard, so it develops a greedy algorithm to extract approximate page farms. It then analyzes the page farms of over 3 million randomly sampled web pages, finding that the distributions of page farm sizes follow a power law distribution and reflect the importance of pages.
Done reread sketchinglandscapesofpagefarmsnpcompleteJames Arnold
The document proposes and analyzes the concept of "page farms", which are sets of pages that contribute to the PageRank score of a target page. It finds that computing page farms is NP-hard, so it develops a greedy algorithm to extract approximate page farms. It then analyzes the page farms of over 3 million randomly sampled web pages, finding that the distributions of page farm sizes follow a power law distribution and reflect the importance of pages.
Done reread maximizingpagerankviaoutlinksJames Arnold
This document analyzes strategies for maximizing the PageRank of a set of webpages I that a webmaster controls the outlinks for but not the inlinks. It provides the optimal outlink structure, which is for each page in I to link to a single external page. It also analyzes the optimal internal link structure, which is a forward chain of links with every possible backward link when there is a unique leaking node. The overall optimal structure combines these, with a unique outlink from the last node in the chain.
Done reread maximizingpagerankviaoutlinks(3)James Arnold
This document analyzes strategies for maximizing the PageRank scores of a set of webpages that a webmaster controls the outlinks for. It summarizes previous work on how adding or removing links impacts PageRank. The authors provide theorems on the optimal outlink and internal link structure for a website to maximize the sum of PageRanks of pages within the site. The optimal structure consists of a forward chain of internal links with all possible backward links, and a unique outlink from the last page in the chain.
Done reread maximizingpagerankviaoutlinks(2)James Arnold
This document analyzes strategies for maximizing the PageRank of a set of webpages I that a webmaster controls the outlinks for but not the inlinks. It provides the optimal outlink structure, which is for each page in I to link to a single external page. It also analyzes the optimal internal link structure, which is a forward chain of links with every possible backward link when there is a unique leaking node. The overall optimal structure combines these, with a unique outlink from the last node in the chain.
PageRank is a method for ranking web pages based on the graph of the web. It takes advantage of the link structure to determine the importance or "rank" of each page. PageRank assumes that more important pages are likely to receive more links from other pages. It calculates a PageRank score through an iterative process that distributes weight from highly ranked pages to the pages they link to. The algorithm accounts for pages with no outgoing links and pages that link only to each other to avoid "rank sinks".
This document summarizes the PageRank algorithm. It acknowledges those who helped with the project. It then provides an overview of PageRank, explaining that it is an algorithm used by Google search to rank web pages based on the number and quality of links to a page. It discusses life on the web before and after PageRank. It also includes the PageRank formula, limitations of early implementations, and improvements like damping factors that address those limitations. Pseudocode and a Python program for calculating PageRank are provided.
PageRank is a method for ranking web pages based on the link structure of the web. It was developed by Google to help search engines make sense of the vast heterogeneity of the World Web. PageRank works by treating individual web pages as nodes and links between pages as edges, and recursively propagating importance weights through this link structure. It helps address issues like some pages having more backlinks but from less important places compared to pages with fewer but highly ranked backlinks. Dangling links that point to pages with no outgoing links are initially removed to avoid them forming rank sinks before final PageRank calculations are made.
PageRank is a method for ranking web pages based on the link structure of the web. It was developed by Google to help search engines make sense of the vast heterogeneity of the World Web. PageRank works by treating individual pages as nodes and links between pages as edges, then recursively propagating importance weights through this link structure. It helps address issues like some pages having more backlinks due to their popularity rather than their actual importance. The algorithm involves iteratively computing PageRanks until they converge based on a damping factor and the number of outbound links from pages.
PageRank is a method for ranking web pages based on the link structure of the web. It was developed by Google to help search engines make sense of the vast heterogeneity of the World Web. PageRank works by treating individual pages as nodes and links between pages as edges, then recursively propagating importance weights through this link structure. It helps address issues like some pages having many low-quality backlinks versus others having a few highly important backlinks. PageRank defines the importance of a page as a damping factor times the sum of the importance of pages linking to it.
PageRank is a method for ranking web pages based on the link structure of the web. It was developed by Google to help search engines make sense of the vast heterogeneity of the World Web. PageRank works by treating individual pages as nodes and links between pages as edges, then recursively propagating importance weights through this link structure. It helps address issues like some pages having many low-quality backlinks versus others having a few highly important backlinks. PageRank models the probability of a person randomly clicking on links by treating it as a random walk through the link graph.
PageRank is a method for ranking web pages based on the link structure of the web. It was developed by Google to help search engines make sense of the vast heterogeneity of the World Web. PageRank works by treating individual pages as nodes and links between pages as edges, then recursively propagating importance weights through this link structure. It helps address issues like some pages having more backlinks due to their popularity rather than their actual importance. The algorithm involves iteratively computing PageRanks until they converge based on a damping factor and the number of outbound links from pages.
PageRank is a method for ranking web pages based on the link structure of the web. It was developed by Google to help search engines make sense of the vast heterogeneity of the World Web. PageRank works by treating individual web pages as nodes and links between pages as edges, and recursively propagating importance weights through this link structure. It helps address issues like some pages having more backlinks due to their popularity rather than their actual importance. The algorithm involves iteratively computing the PageRank scores until they converge based on the link structure and a damping factor.
Similar to Random web surfer pagerank algorithm (20)
5th LF Energy Power Grid Model Meet-up SlidesDanBrown980551
5th Power Grid Model Meet-up
It is with great pleasure that we extend to you an invitation to the 5th Power Grid Model Meet-up, scheduled for 6th June 2024. This event will adopt a hybrid format, allowing participants to join us either through an online Mircosoft Teams session or in person at TU/e located at Den Dolech 2, Eindhoven, Netherlands. The meet-up will be hosted by Eindhoven University of Technology (TU/e), a research university specializing in engineering science & technology.
Power Grid Model
The global energy transition is placing new and unprecedented demands on Distribution System Operators (DSOs). Alongside upgrades to grid capacity, processes such as digitization, capacity optimization, and congestion management are becoming vital for delivering reliable services.
Power Grid Model is an open source project from Linux Foundation Energy and provides a calculation engine that is increasingly essential for DSOs. It offers a standards-based foundation enabling real-time power systems analysis, simulations of electrical power grids, and sophisticated what-if analysis. In addition, it enables in-depth studies and analysis of the electrical power grid’s behavior and performance. This comprehensive model incorporates essential factors such as power generation capacity, electrical losses, voltage levels, power flows, and system stability.
Power Grid Model is currently being applied in a wide variety of use cases, including grid planning, expansion, reliability, and congestion studies. It can also help in analyzing the impact of renewable energy integration, assessing the effects of disturbances or faults, and developing strategies for grid control and optimization.
What to expect
For the upcoming meetup we are organizing, we have an exciting lineup of activities planned:
-Insightful presentations covering two practical applications of the Power Grid Model.
-An update on the latest advancements in Power Grid -Model technology during the first and second quarters of 2024.
-An interactive brainstorming session to discuss and propose new feature requests.
-An opportunity to connect with fellow Power Grid Model enthusiasts and users.
zkStudyClub - LatticeFold: A Lattice-based Folding Scheme and its Application...Alex Pruden
Folding is a recent technique for building efficient recursive SNARKs. Several elegant folding protocols have been proposed, such as Nova, Supernova, Hypernova, Protostar, and others. However, all of them rely on an additively homomorphic commitment scheme based on discrete log, and are therefore not post-quantum secure. In this work we present LatticeFold, the first lattice-based folding protocol based on the Module SIS problem. This folding protocol naturally leads to an efficient recursive lattice-based SNARK and an efficient PCD scheme. LatticeFold supports folding low-degree relations, such as R1CS, as well as high-degree relations, such as CCS. The key challenge is to construct a secure folding protocol that works with the Ajtai commitment scheme. The difficulty, is ensuring that extracted witnesses are low norm through many rounds of folding. We present a novel technique using the sumcheck protocol to ensure that extracted witnesses are always low norm no matter how many rounds of folding are used. Our evaluation of the final proof system suggests that it is as performant as Hypernova, while providing post-quantum security.
Paper Link: https://eprint.iacr.org/2024/257
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
Skybuffer SAM4U tool for SAP license adoptionTatiana Kojar
Manage and optimize your license adoption and consumption with SAM4U, an SAP free customer software asset management tool.
SAM4U, an SAP complimentary software asset management tool for customers, delivers a detailed and well-structured overview of license inventory and usage with a user-friendly interface. We offer a hosted, cost-effective, and performance-optimized SAM4U setup in the Skybuffer Cloud environment. You retain ownership of the system and data, while we manage the ABAP 7.58 infrastructure, ensuring fixed Total Cost of Ownership (TCO) and exceptional services through the SAP Fiori interface.
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
A Comprehensive Guide to DeFi Development Services in 2024Intelisync
DeFi represents a paradigm shift in the financial industry. Instead of relying on traditional, centralized institutions like banks, DeFi leverages blockchain technology to create a decentralized network of financial services. This means that financial transactions can occur directly between parties, without intermediaries, using smart contracts on platforms like Ethereum.
In 2024, we are witnessing an explosion of new DeFi projects and protocols, each pushing the boundaries of what’s possible in finance.
In summary, DeFi in 2024 is not just a trend; it’s a revolution that democratizes finance, enhances security and transparency, and fosters continuous innovation. As we proceed through this presentation, we'll explore the various components and services of DeFi in detail, shedding light on how they are transforming the financial landscape.
At Intelisync, we specialize in providing comprehensive DeFi development services tailored to meet the unique needs of our clients. From smart contract development to dApp creation and security audits, we ensure that your DeFi project is built with innovation, security, and scalability in mind. Trust Intelisync to guide you through the intricate landscape of decentralized finance and unlock the full potential of blockchain technology.
Ready to take your DeFi project to the next level? Partner with Intelisync for expert DeFi development services today!
Fueling AI with Great Data with Airbyte WebinarZilliz
This talk will focus on how to collect data from a variety of sources, leveraging this data for RAG and other GenAI use cases, and finally charting your course to productionalization.
Skybuffer AI: Advanced Conversational and Generative AI Solution on SAP Busin...Tatiana Kojar
Skybuffer AI, built on the robust SAP Business Technology Platform (SAP BTP), is the latest and most advanced version of our AI development, reaffirming our commitment to delivering top-tier AI solutions. Skybuffer AI harnesses all the innovative capabilities of the SAP BTP in the AI domain, from Conversational AI to cutting-edge Generative AI and Retrieval-Augmented Generation (RAG). It also helps SAP customers safeguard their investments into SAP Conversational AI and ensure a seamless, one-click transition to SAP Business AI.
With Skybuffer AI, various AI models can be integrated into a single communication channel such as Microsoft Teams. This integration empowers business users with insights drawn from SAP backend systems, enterprise documents, and the expansive knowledge of Generative AI. And the best part of it is that it is all managed through our intuitive no-code Action Server interface, requiring no extensive coding knowledge and making the advanced AI accessible to more users.
Digital Marketing Trends in 2024 | Guide for Staying AheadWask
https://www.wask.co/ebooks/digital-marketing-trends-in-2024
Feeling lost in the digital marketing whirlwind of 2024? Technology is changing, consumer habits are evolving, and staying ahead of the curve feels like a never-ending pursuit. This e-book is your compass. Dive into actionable insights to handle the complexities of modern marketing. From hyper-personalization to the power of user-generated content, learn how to build long-term relationships with your audience and unlock the secrets to success in the ever-shifting digital landscape.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
Programming Foundation Models with DSPy - Meetup Slides
Random web surfer pagerank algorithm
1. International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
36
Random Web Surfer PageRank Algorithm
Navadiya Hareshkumar, Dr. Deepak Garg
Computer Science and Engineering Department
Thapar University, Punjab
ABSTRACT
In this paper analyzes how the Google web search engine
implements the PageRank algorithm to define prominent status to
web pages in a network. It describes the PageRank algorithm as a
Markov process, web page as state of Markov chain, Link
structure of web as Transitions probability matrix of Markov
chains, the solution to an eigenvector equation and Vector
iteration power method.
It mainly focus on how to relate the eigenvalues and eigenvector
of Google matrix to PageRank values to guarantee that there is a
single stationary distribution vector to which the PageRank
algorithm converges and efficiently compute the PageRank for
large sets of web Pages. Finally, it will demonstrate example of
the PageRank algorithm.
Keywords
PageRank, Markov chains, Power method, Google matrix,
stationary distribution vector, Eigen Vector and Values
1. INTRODUCTION
Paper begins with a review of the most basic PageRank model for
determining the importance of a webpage.
Paper provides comprehensive survey of all issues associated with
PageRank, available and recommended solution methods,
changing α value and convergence rate, storage issues and
compares PageRank to an idealized random Web surfer and
shows how to efficiently compute PageRank for large numbers of
pages.
2. PAGERANK
A search query with Google‘s search engine usually returns a very
large number of pages. E.g., a search on ‗University‘ returns 225
million pages. Although the search returns several million pages,
the most relevant pages are usually found within the top ten or
twenty pages in the list of results. How does the search engine
know which pages are the most important?
Google assigns a number to each individual webpage based on the
link structure of the web, expressing its importance. This number
is known as the PageRank and is computed via the PageRank
Algorithm. PageRank has applications in search, browsing, and
traffic estimation.
3. BASIC PAGERANK ALGORITHM
MODEL
An intuitive definition of PageRank algorithm shows how one
might measure the importance of a web page on the Internet.
A webpage U‘s PageRank is calculated base on how many other
WebPages Backlink (inedges) into U. The PageRank of U is the
sum of the PageRanks of each webpage Vi that backlinks to U
divided by the number of WebPages to which Vi links [1]. This
intuitive definition says that: if webpage U is linked to only by
WebPages with low PageRanks, then U may not get more
importance. Further, if U is linked to by a page Vj with a high
PageRank, but Vj links to many other pages, U should not receive
the full weight (benefits) of Vj‘s PageRank.
For example, if a webpage has a link to the Facebook Home
page, it may be just one link but it is a very important one. This
page should be ranked higher than many pages with more links
but from lower important pages. PageRank is an attempt to see
how good an approximation to ―importance‖ can be obtained from
the link structure.
3.1 FORMAL DEFINITION
Now that we have an idea of how PageRank works, we will
formally define the algorithm. Drawing on [2], we define
PageRank as follows:
Let U be a web page.
Let FU be the set of Forward links (outedges) from U.
Let BU be the set of Backlinks into U.
Let NU = |FU| be the number of forward links from U.
Let c be a normalization factor so that ―the total rank of
all web pages is constant‖ [2].
Back Links
Forward Links
Fig 1: A Webpage’s Link Structure
We begin by defining a simple ranking, R which is a slightly
simplified version of PageRank:
R U = c
R(V)
NV
VϵBU
(1)
Note that here c < 1 because there are number of pages with no
forward links and their weight is lost from the system (see Section
4.2). The equation is recursive but it may be computed by starting
with any set of ranks and iterating the computation until it
converges. Fig 2 demonstrates the propagation of rank from one
pair of pages to another.
Webpage U
2. International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
37
50
3
50
3
3
Fig 2: Simplified PageRank Calculation
There is a small problem with this simplified ranking algorithm.
Consider two web pages that point to each other but to no other
page. And suppose there is some web page which points to one of
them. Then, during iteration, this loop will accumulate rank but
never distribute any rank (since there are no outedges). The loop
forms a sort of trap which is called a Rank Sink.
Fig 3: Loop which acts as a Rank Sink
To overcome this problem of rank sinks, we introduce a rank
source:
Let E(U) be ―some vector over the Web pages that
corresponds to a source of rank‖ [2].
Then, the PageRank of page U is given by
R′
U = c
R′
V
NV
VϵBU
+ c E U (2)
This is iterated until the values have converged sufficiently [3].
0.2
0.2 0.2
0.4
Fig4: Simplified PageRank Calculation
Notice that PageRank can be described using equation 2, the
summation method is neither the most interesting nor the most
illustrative of the algorithm‘s properties. So, it is preferred to
rewrite the sum as Matrix Multiplication.
4. RANDOM WEB SURFER MODEL
PageRank also has a second definition based upon the model of a
random web surfer navigating the Internet. In short, the model
states that PageRank models the behavior of someone who ―keeps
clicking on successive links at random‖. However, occasionally
the surfer ―gets bored and jumps to a random page chosen based
on the distribution in E.‖ [2]. so, surfer will not continue in the
loop forever.
Consider the illustration in Fig 5 of a simple Link Structure of
web pages; we can represent this structure using an N × N
adjacency matrix A, where Aij = 1 if there is a link (Edge) from
webpage (Vertex) i to webpage j, and 0 otherwise [4].
Fig 5: A Simple Link Structure
Thus, the 6×6 adjacency matrix for the link structure in Fig 5 is
A =
0 1 0 0 1 0
0 0 1 1 0 0
0 0 0 1 1 1
1 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 0
Notice that the diagonal entries are all zero. We assume that links
from a page to itself are ignored when constructing an adjacency
matrix.
To illustrate the PageRank algorithm based on second definition,
the following variables are defined:
Let N be the total number of web pages in the web.
Let πT
be the 1×N PageRank row vector (Stationary
vector).
Let H the N × N row-normalized adjacency matrix.
(Transition Probability Matrix)
100 53
9 50
∞ ∞ ∞
A
0.4
B
0.2
C
0.4
Webpage A
Webpage E Webpage F
Webpage D
Webpage B
Webpage C
3. International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
38
Then we can describe the PageRank vector at the kth
iteration as
π(k)T
= π(k−1)T
H (3)
So that successive iterations π(k)T
converge to the PageRank
vector πT
[3].
4.1 ROW-NORMALITY
Let to build a Transition Probabilities matrix,
Hi =
Ai
Aik
N
k=1
(4)
So, that each row 𝐴i of A is divided by its row sum. Apply
equation 4 on Adjacency matrix A. so we have, sparse matrix
H =
0 1/2 0 0 1/2 0
0 0 1/2 1/2 0 0
0 0 0 1/3 1/3 1/3
1 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 0
Matrix H whose element Hij is the probability of moving from
state i (page i) to state j (page j) in one time step (In Fig 5, assume
that, starting from any webpage, it is Half the probability to arrive
from page A to page B in one time step).
4.2 DANGLING LINKS
One issue with this model is dangling links. Dangling links are
simply links that point to any page with no outgoing links (In
Fig 5, Link to Webpage F) and Pages with no outlinks are called
dangling nodes (Webpage F). They affect the model because it is
not clear where their weight should be distributed. A page of data,
a page with a postscript graph, a page with jpeg pictures, a pdf
document, a page that has been fetched by a crawler but not yet
explored - these are all examples of possible dangling nodes. In
fact, for some subsets of the web, dangling nodes make up 80% of
the collection‘s pages [5].
4.3 STOCHASTICITY
Using the matrix H is insufficient for the PageRank algorithm,
however, because the iteration using H alone might not converge
properly — ―it may be dependent on the startingvector‖[3]. Here
matrix H is not yet stochastic [3]. A matrix is stochastic when it
is a ―matrix of transition probabilities for a Markov chain,‖ [A
Markov chain is ―collection of random variables‖ whose future
values are independent of their past values [6].] with the property
that ―all elements are non-negative and all its row sums are unity‖
(one) [7]. Thus, to ensure that H is stochastic, we must ensure that
every row sums to one. But here, the sum of the last row of matrix
H is Zero because of Dangling Node (see Section 4.2).
To overcome this problem, pages with no Forward links
(Webpage F) are assigned artificial links or ―teleporters‖ to all
other pages.
Artificial Links
Fig 6: A StochasticLink Structure
Therefore, we define the stochastic S as,
S = H +
a ∗ eT
N
(5)
Where,
a = N×1 Column Vector such that
ai = 1 if Hik = 0N
k=1 (i.e. Page i is Dangling Page)
= 0 otherwise
e = N×1 Column Vector of one‘s [3].
Apply equation 5 on matrix H. so we have stochastic matrix S as,
S =
0 1/2 0 0 1/2 0
0 0 1/2 1/2 0 0
0 0 0 1/3 1/3 1/3
1 0 0 0 0 0
1 0 0 0 0 0
1/6 1/6 1/6 1/6 1/6 1/6
It makes sure that surfer‘s random walk process does not get stuck
and the web pages are the states of the Markov chain.
4.4 THE GOOGLE MATRIX
There is no guarantee that S has a unique stationary distribution
vector (i.e., there might not be a single ―correct‖ PageRank
vector) [3]. For us to guarantee that there is a single stationary
distribution vector πT
to which we can converge, we must ensure
that S is irreducible as well as stochastic [3]. A matrix is
irreducible if and only if its graph is strongly connected [8].
Webpage A
Webpage E Webpage F
Webpage D
Webpage B
Webpage C
4. International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
39
So we define the irreducible row-stochastic matrix G as
G = αS + 1 − α E ; 0 ≤ α ≤ 1 6
E =
e ∗ eT
N
G is the Google matrix, and we can use it to define
π(k)T
= π(k−1)T
G (7)
as the new iterative method for PageRank to replace equation 3.
Google uses the value of 0.85 for α [3] (see Section 6), and we
will use this same value. Thus, apply equation6 on matrix S, our
G is, G =
1/40 9/20 1/40 1/40 9/20 1/40
1/40 1/40 9/20 9/20 1/40 1/40
1/40 1/40 1/40 77/250 77/250 77/250
7/8 1/40 1/40 1/40 1/40 1/40
7/8 1/40 1/40 1/40 1/40 1/40
83/500 83/500 83/500 83/500 83/500 83/500
Since the matrix G is completely dense, and the graph of G now
strongly connected and G is irreducible.
5. THE POWER METHOD
The Google matrix G is currently of size more than eight billion
web pages [9] and therefore the Eigen value computation is not
trivial. To find an approximation of the principal Eigenvector the
Power Methodis used.
We iterate using the Google matrix G by writing equation 7. Let
us take 1×N initial row vectorπ(0)T
=
eT
N
Based on it result found
is, 𝛑(𝟏𝟗)𝐓 =
0.320 0.170 0.107 0.137 0.200 0.064
The PageRank of page i is given by the ith
element of πT
.
Table 1.PageRank of Tiny 6 web pages Link Structure
Starting anywhere within Fig 5 and randomly surfing the web
pages in given network, the likelihood of ending up at any of the
Pages are, respectively, about 32%, 17%, 11%, 14%, 20 % and
6%.
5.1 IMPROVEMENT OF POWER
METHOD
When dealing with large data sets, it is difficult to form a matrix
G and find its dominant eigenvector. It is more efficient to
compute the PageRank vector using the power method, where we
iterate using the sparse matrix H by rewriting equation 7.
π k T
= π k−1 T
G
= π k−1 T
αS + 1 − α E
= π k−1 T
αS + 1 − α
e ∗ eT
N
=απ k−1 T
S + 1 − α π k−1 T e∗eT
N
=απ k−1 T
S + 1 − α
eT
N
=απ k−1 T
( H +
a∗eT
N
) + 1 − α
eT
N
=απ k−1 T
H + απ k−1 T
a + 1 − α
eT
N
(8)
Since π(k−1)T is a probability vector, and thus, π(k−1)T e = 1
Fortunately, since H is sparse, each vector-matrix multiplication
required by the power method can be computed in nnz(H) flops,
where nnz(H) is the number of nonzeros in H and since the
average number of nonzeros per row in H is 3-10, O(nnz(H)) ≈
O(n) [5]. Furthermore, at each iterationthe power method only
requires the storage of one vector, the current iterate. The
founders of Google, Lawrence Page and Sergey Brin, use α = 0.85
and report success using only 50 to 100 power iterations [2].
6. CHANGING 𝛂 VALUE
ANDCONVERGENCE RATE
One of the most obvious places to begin fiddling with the basic
PageRank model is α. Brin and Page, have reported using α =
0.85. One wonders why this choice for α? Might a different
choice produce a very different ranking of retrieved WebPages?
As mentioned in sections The Google Matrix and The Power
Method, there are good reasons for using α = 0.85, one being the
speedy convergence of the power method. With this value for α,
rough estimate of the number of iterations needed to converge to a
tolerance level τ (measured by the residual,π(k)T
− π
(k−1)T
) is
log10τ / log10α. For τ = 10-6
and α = 0.85, one can expect roughly
-6 / log100.85 ≈ 85 iterations until convergence to the PageRank
vector. For τ = 10-8
, about 114 iterations and for τ = 10-10
, about
142 iterations. Brin and Page report success using only 50 to 100
power iterations, implying that τ could range from 10-3
to 10-7
.
Obviously, this choice of α brings faster convergence than higher
values of α.
Compare with α = 0.99, whereby roughly 1833 iterations are
required to achieve a residual less than 10-8
. When working with a
sparse 8 billion by 8 billion matrix, each iteration counts; over a
few hundred power iterations is more than Google is willing to
compute. However, in addition to the computational reasons for
choosing α = 0.85, this choice for α also carries some intuitive
weight: α = 0.85 implies that roughly 5/6 of the time a Web surfer
randomly clicks on hyperlinks (i.e., following the structure of the
Web, as captured by the 𝛂𝐒 part of the equation 6), while 1/6 of
the time this Web surfer will go to the URL line and type the
address of a new page to ―Artificial Link‖ to (as captured by the
Web Page PageRank Relative importance
A 0.320 1
B 0.170 3
C 0.107 5
D 0.137 4
E 0.200 2
F 0.064 6
Total = 1
5. International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
40
𝟏 − 𝛂 𝐄 part of the equation 6). Perhaps this was the original
motivation behind Brin and Page‘s choice of α = 0.85; it produces
an accurate model for Web surfing behavior. Whereas α = 0.99,
not only slows convergence of the power method, but also places
much greater emphasis on the link structure of the Web and much
less on the teleportation tendencies of surfers. The PageRank
vector derived from α = 0.99 can be vastly different from that
obtained using α = 0.85. Perhaps it gives a ―truer‖ PageRanking.
An Experiment with various α‘s show significant variation in
rankings produced by different values of α[10].
7. ALTERNATIVE OF POWER METHOD
The iterative aggregation/disaggregation (IAD) method: is an
improvement of the PageRank algorithm used bythe search engine
Google to compute stationary probabilities of very large Markov
chains.
It is shown that the power method applied to the Google matrix
always converges, and that the convergence rate of theIAD
method is at least as good as that of the power method.
Furthermore, by exploiting the hyperlink structure of the web it
can be shown that the convergence rate of the IAD method
applied to the Google matrix can be made strictly faster than that
of the power method [11].
The adaptive randomized method: for finding
eigenvectorcorresponding to Eigen Value 1 for stochastic matrices
hasbeen proposed. The upper bound of its rate of convergenceis of
non-asymptotic type and has the explicit factor. Moreover, the
bound is valid for the whole class of stochastic matrices and does
not depend on propertiesof the individual matrix. The method can
be applied for PageRank computation with small parameter α.
Further workon acceleration of the method is possible [12].
DYNA-RANK: focuses upon efficiently calculating and updating
Google‘s PageRank vector using ―peer to peer‖ system. The
changes inthe web structure will be handled incrementally
amongstthe peers. DYNA-RANK produces the relative PageRank
oneach peer. DYNA-RANK is proven to take less computation
time and less number of iterations compared to centralized
approach [13].
8. STORAGE ISSUES OF GOOGLE
MATRIX
The size of the Markov matrix makes storage issues nontrivial.
For subsets of the web, the transition probability matrix H may or
may not fit in main memory. When a large matrix exceeds a
machine‘s memory, researchers usually try one of two things:
1. Compress the data needed so that the compressed
representation fits in main memory and then creatively
implement a modified version of PageRank on this
compressed representation.
2. Keep the data in its uncompressed form and develop
I/O-efficient implementations of the computations that
must take place on the large, uncompressed data.
For modern web structure for which the transition probability
matrix H can be stored in main memory, compression of the data
is not essential.
Rather than storing the full matrix or a compressed version of the
matrix, Web-sized implementations of the PageRank model store
the H or A matrix in an adjacency list of the columns of the
matrix [14]. In order to compute the PageRank vector, the
PageRank power method requires vector-matrix multiplications of
𝛑 𝐤−𝟏 𝐓
𝐇 at each iteration k. Therefore, quick access to the
columns of the matrix H (or A) is essential to algorithm speed.
Column i contains the inlink (Backlink) information for page i,
which, for the PageRank system of ranking WebPages, is more
important than outlink (Forwardlink) information contained in the
rows of H or A. For the tiny 6-node web from Fig 5, an
adjacency list representation of the columns of matrix A is:
Table 2.Adjacency list of matrix A
Cleve Moler gives one possible implementation of the power
method applied to an adjacency list, along with sample Matlab
code [15]. When the adjacency list does not fit in main memory
[14], suggest methods for compressing the data. Reference [16]
take the other approach and suggest I/O-efficient implementations
of PageRank. Since the PageRank vector itself is large and
completely dense, containing over 8 billion pages, and must be
consulted in order to process each user query, Reference [16] has
also suggested a technique to compress the PageRank vector. This
encoding of the PageRank vector hopes to keep the ranking
information cached in main memory, thus speeding query
processing.
9. CONCLUSION
In this paper, we have taken on the audacious task of condensing
every page on the World Wide Web into a single number, its
PageRank. PageRank is a global ranking of all web pages,
regardless of their content, based solely on their location in the
Web's link structure. Using PageRank, we are able to order search
results so that more important and central WebPages are given
preference.
The intuition behind Page Rank is that it uses information which
is external to the Web pages themselves – their backlinks, which
provide a kind of peer review. Furthermore, backlinks from
―important‖ pages are more significant than backlinks from
average pages. This is encompassed in the recursive definition of
PageRank (equation 2).
In the proposed algorithm a value of α is usedthat play a very
important role on the analysis of thealgorithm. After the analysis
it is concluded that α must not be selected closer to zero. At α = 1,
the system enters into the ideal state and theranking provided is
insignificant. As per evaluation α must be selected greater than or
equals to 0.5. However, if we consider convergence speed as only
factor toevaluate the performance than the best factor α will be
.85. The proposed algorithm is query independent algorithmand
does not consider query during ranking.As the Web continues its
amazing growth, the need for smarter storage schemes and
evenfaster numerical methods will become more evident. Both are
exciting areas for computer scientists and numerical analysts
interested in information retrieval.
Web Page BackLink From
A D, E
B A
C B
D B, C
E A, C
F C
6. International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
41
As future work, it is desirable to conduct more experiments using
larger and various kinds of datasets in order to further validate our
conclusions in this paper.
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