CS246: Mining Massive Datasets Jure Leskovec, Stanford University.
|
|
- James Long
- 6 years ago
- Views:
Transcription
1 CS246: Mining Massive Datasets Jure Leskovec, Stanford University
2 What is the structure of the Web? How is it organized? 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 2
3 3 What is the structure of the Web? How is it organized? Web as a directed graph
4 4 Two types of directed graphs: DAG Directed Acyclic Graph: Has no cycles: if u can reach v, then v can not reach u Strongly connected: Any node can reach any node via a directed path Any directed graph can be expressed in terms of these two types of graphs
5 Strongly connected component (SCC) is a set of nodes S: Every pair of nodes in S can reach each other There is no larger set containing S with this property Any directed graph is a DAG on its SCCs: Each SCC is a super-node Super-node A links to super-node B if a node in A links to node in B 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 5
6 6 Take a large snapshot of the Web and try to understand how it s SCCs fit as a DAG Computational issues: Say want to find SCC containing specific node v? Observation: Out(v) nodes reachable from v (via out-edges) In(v) nodes reachable from v (via in-edges) SCC containing v: = Out(v, G) In(v, G) = Out(v, G) Out(v, G) where G is G with directions of edges flipped v
7 [Broder et al., 00] 250 million webpages, 1.5 billion links [Altavista] 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 7
8 8 Out-/In- Degree Distribution: p k : fraction of nodes with k out-/in-links Histogram of p k vs. k Normalized count, p k
9 9 Plot the same data on log-log axes: Normalized count, p k log p k = βk α p k = log β α log k
10 [Broder et al., 00] 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 10
11 Random network Power-law network Degree distribution is Binomial, i.e., all nodes have similar degree 2/7/2011 Degrees are Power-law, i.e., heavily skewed Jure Leskovec, Stanford C246: Mining Massive Datasets 11
12 12 Web pages are not equally important vs. Since there is large diversity in the connectivity of the webgraph we can rank the pages by the link structure
13 13 We will cover the following Link Analysis approaches to computing importances of nodes in a graph: Page Rank Hubs and Authorities (HITS) Topic-Specific (Personalized) Page Rank Spam Detection Algorithms
14 14 First try: Page is more important if it has more links In-coming links? Out-going links? Think of in-links as votes: has 23,400 inlinks has 1 inlink Are all in-links are equal? Links from important pages count more Recursive question!
15 15 Each link s vote is proportional to the importance of its source page If page P with importance x has n out-links, each link gets x/n votes Page P s own importance is the sum of the votes on its in-links
16 16 The web in 1839 y/2 y Yahoo y = y /2 + a /2 a = y /2 + m m = a /2 a/2 y/2 Amazon a m a/2 M soft m
17 17 3 equations, 3 unknowns, no constants No unique solution All solutions equivalent modulo scale factor Additional constraint forces uniqueness y+a+m = 1 y = 2/5, a = 2/5, m = 1/5 Gaussian elimination method works for small examples, but we need a better method for large web-size graphs
18 Matrix M has one row and one column for each web page Suppose page j has n out-links If j i, then M ij = 1/n else M ij = 0 M is a column stochastic matrix Columns sum to 1 Suppose r is a vector with one entry per web page: r i is the importance score of page i Call it the rank vector r = 1 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 18
19 19 Suppose page j links to 3 pages, including i j i 1/3 = i M r r
20 20 The flow equations can be written r = M r So the rank vector is an eigenvector of the stochastic web matrix In fact, its first or principal eigenvector, with corresponding eigenvalue 1
21 21 Yahoo Y! A MS Y! ½ ½ 0 A ½ 0 1 MS 0 ½ 0 Amazon M soft y = y /2 + a /2 a = y /2 + m m = a /2 r = Mr y ½ ½ 0 y a = ½ 0 1 a m 0 ½ 0 m
22 22 Simple iterative scheme Suppose there are N web pages Initialize: r 0 = [1/N,.,1/N] T Iterate: r k+1 = M r k Stop when r k+1 - r k 1 < ε x 1 = 1 i N x i is the L1 norm Can use any other vector norm e.g., Euclidean
23 23 Power iteration: Set r i =1/n Y! Y! A MS Y! ½ ½ 0 r i = j M ij r j And iterate A MS A ½ 0 1 MS 0 ½ 0 Example: y a = m 1/3 1/3 1/3 1/3 1/2 1/6 5/12 1/3 1/4 3/8 11/24 1/6... 2/5 2/5 1/5
24 24 Imagine a random web surfer At any time t, surfer is on some page P At time t+1, the surfer follows an outlink from P uniformly at random Ends up on some page Q linked from P Process repeats indefinitely Let p(t) be a vector whose i th component is the probability that the surfer is at page i at time t p(t) is a probability distribution on pages
25 25 Where is the surfer at time t+1? Follows a link uniformly at random p(t+1) = M p(t) Suppose the random walk reaches a state such that p(t+1) = M p(t) = p(t) Then p(t) is called a stationary distribution for the random walk Our rank vector r satisfies r = M r So it is a stationary distribution for the random surfer
26 26 A central result from the theory of random walks (aka Markov processes): For graphs that satisfy certain conditions, the stationary distribution is unique and eventually will be reached no matter what the initial probability distribution at time t = 0.
27 27 Some pages are dead ends (have no out-links) Such pages cause importance to leak out Spider traps (all out links are within the group) A group of pages is a spider trap if there are no links from within the group to the outside of the group Random surfer gets trapped And eventually spider traps absorb all importance
28 28 Power iteration: Set r i =1 Y! r i = j M ij r j And iterate Example: y 1 1 ¾ 5/8 0 a = 1 ½ ½ 3/8 0 m 1 3/2 7/4 2 3 A MS Y! A MS Y! ½ ½ 0 A ½ 0 0 MS 0 ½ 1
29 29 The Google solution for spider traps At each time step, the random surfer has two options: With probability β, follow a link at random With probability 1-β, jump to some page uniformly at random Common values for β are in the range 0.8 to 0.9 Surfer will teleport out of spider trap within a few time steps
30 30 0.2*1/3 Yahoo 1/2 0.8*1/2 1/2 0.8*1/2 0.2*1/3 0.2*1/3 Amazon M soft y y 1/2 a 1/2 m 0 0.8* y 1/2 1/2 0 1/2 1/ / / * y 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 y 7/15 7/15 1/15 a 7/15 1/15 1/15 m 1/15 7/15 13/15
31 31 Yahoo 1/2 1/ / /2 1 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 Amazon M soft y 7/15 7/15 1/15 a 7/15 1/15 1/15 m 1/15 7/15 13/15 y a = m /11 5/11 21/11
32 32 Some pages are dead ends (have no out-links) Such pages cause importance to leak out Power iteration: Set r i =1 r i = j M ij r j And iterate Example: y 1 1 ¾ 5/8 0 a = 1 ½ ½ 3/8 0 m 1 ½ ¼ ¼ 0 A Y! MS Y! A MS Y! ½ ½ 0 A ½ 0 0 MS 0 ½ 0
33 33 Teleports Follow random teleport links with probability 1.0 from dead-ends Adjust matrix accordingly
34 Suppose there are N pages Consider a page j, with set of outlinks O(j) We have M ij = 1/ O(j) when j i and M ij = 0 otherwise The random teleport is equivalent to adding a teleport link from j to every other page with probability (1-β)/N reducing the probability of following each outlink from 1/ O(j) to β/ O(j) Equivalent: tax each page a fraction (1-β) of its score and redistribute evenly 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 34
35 35 Construct the N x N matrix A as follows A ij = β M ij + (1-β)/N Verify that A is a stochastic matrix The page rank vector r is the principal eigenvector of this matrix satisfying r = A r Equivalently, r is the stationary distribution of the random walk with teleports
36 36 Key step is matrix-vector multiplication r new = A r old Easy if we have enough main memory to hold A, r old, r new Say N = 1 billion pages We need 4 bytes for each entry (say) 2 billion entries for vectors, approx 8GB Matrix A has N 2 entries is a large number!
37 37 r = A r, where A ij = β M ij + (1-β)/N r i = 1 j N A ij r j r i = 1 j N [β M ij + (1-β)/N] r j = β 1 j N M ij r j + (1-β)/N 1 j N r j = β 1 j N M ij r j + (1-β)/N, since r = 1 r = βm r + [(1-β)/N] N where [x] N is an N-vector with all entries x
38 38 We can rearrange the PageRank equation r = β M r + [(1-β)/N] N [(1-β)/N] N is an N-vector with all entries (1-β)/N M is a sparse matrix! 10 links per node, approx 10N entries So in each iteration, we need to: Compute r new = β M r old Add a constant value (1-β)/N to each entry in r new
39 39 Encode sparse matrix using only nonzero entries Space proportional roughly to number of links say 10N, or 4*10*1 billion = 40GB still won t fit in memory, but will fit on disk source node degree 0 3 1, 5, , 64, 113, 117, , 23 destination nodes
40 Assume enough RAM to fit r new into memory Store r old and matrix M on disk Initialize all entries of r new to (1-β)/N For each page p (of out-degree n): Read into memory: p, n, dest 1,,dest n, r old (p) for j = 1 n: r new (dest j ) += β r old (p) / n r new src degree destination 0 3 1, 5, , 64, 113, , 23 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 40 r old
41 41 In each iteration, we have to: Read r old and M Write r new back to disk IO Cost = 2 r + M Questions: What if we had enough memory to fit both r new and r old? What if we could not even fit r new in memory? See reading:
42 46 Measures generic popularity of a page Biased against topic-specific authorities Solution: Topic-Specific PageRank (next lecture) Uses a single measure of importance Other models e.g., hubs-and-authorities Solution: Hubs-and-Authorities (next) Susceptible to Link spam Artificial link topographies created in order to boost page rank Solution: TrustRank (next lecture)
43 47 Interesting pages fall into two classes: 1. Authorities are pages containing useful information Newspaper home pages Course home pages Home pages of auto manufacturers 2. Hubs are pages that link to authorities List of newspapers Course bulletin List of US auto manufacturers NYT: 10 Ebay: 3 Yahoo: 3 CNN: 8 WSJ: 9
44 48
45 49
46 50
47 51 A good hub links to many good authorities A good authority is linked from many good hubs Model using two scores for each node: Hub score and Authority score Represented as vectors h and a
48 52 HITS uses adjacency matrix A[i, j] = 1 if page i links to page j, 0 else A T, the transpose of A, is similar to the PageRank matrix M but A T has 1 s where M has fractions Yahoo A = y a m y a m Amazon M soft
49 53 Yahoo A = y a m y a m Amazon M soft
50 54 Notation: Vector a=(a 1,a n ), h=(h 1,h n ) Adjacency matrix (n x n): A ij =1 if i j else A ij =0 Then: h i = So: h = A a Likewise: a = j hi i j a = A T h j A ij a j
51 55 The hub score of page i is proportional to the sum of the authority scores of the pages it links to: h = λ A a Constant λ is a scaling factor, λ = 1/ h i The authority score of page i is proportional to the sum of the hub scores of the pages it is linked from: a = μ A T h Constant μ is scaling factor, μ = 1/ a i
52 56 The HITS algorithm: Initialize h, a to all 1 s Repeat: h = A a Scale h so that its sums to 1.0 a = A T h Scale a so that its sums to 1.0 Until h, a converge (i.e., change very little)
53 A = A T = Yahoo Amazon M soft a(yahoo) a(amazon) a(m soft) = = = / h(yahoo) = 1 h(amazon) = 1 h(m soft) = 1 1 2/3 1/
54 58 Algorithm: Set: a = h = 1 n Repeat: h=m a, a=m T h Normalize Then: a=m T (M a) new h new a Thus: a=(m T M) a h=(m M T ) h a is being updated (in 2 steps): M T (M a) = (M T M) a h is updated (in 2 steps): M (M T h) = (M M T ) h Repeated matrix powering
55 59 Under reasonable assumptions about A, the HITS iterative algorithm converges to vectors h* and a*: h* is the principal eigenvector of matrix AA T a* is the principal eigenvector of matrix A T A
56 60 PageRank and HITS are two solutions to the same problem: What is the value of an in-link from u to v? In the PageRank model, the value of the link depends on the links into u In the HITS model, it depends on the value of the other links out of u The destinies of PageRank and HITS post-1998 were very different
CS246: Mining Massive Datasets Jure Leskovec, Stanford University
CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu 2/7/2012 Jure Leskovec, Stanford C246: Mining Massive Datasets 2 Web pages are not equally important www.joe-schmoe.com
More informationCS345a: Data Mining Jure Leskovec and Anand Rajaraman Stanford University
CS345a: Data Mining Jure Leskovec and Anand Rajaraman Stanford University TheFind.com Large set of products (~6GB compressed) For each product A=ributes Related products Craigslist About 3 weeks of data
More informationData and Algorithms of the Web
Data and Algorithms of the Web Link Analysis Algorithms Page Rank some slides from: Anand Rajaraman, Jeffrey D. Ullman InfoLab (Stanford University) Link Analysis Algorithms Page Rank Hubs and Authorities
More informationSlide source: Mining of Massive Datasets Jure Leskovec, Anand Rajaraman, Jeff Ullman Stanford University.
Slide source: Mining of Massive Datasets Jure Leskovec, Anand Rajaraman, Jeff Ullman Stanford University http://www.mmds.org #1: C4.5 Decision Tree - Classification (61 votes) #2: K-Means - Clustering
More informationLink Mining PageRank. From Stanford C246
Link Mining PageRank From Stanford C246 Broad Question: How to organize the Web? First try: Human curated Web dictionaries Yahoo, DMOZ LookSmart Second try: Web Search Information Retrieval investigates
More informationSlides based on those in:
Spyros Kontogiannis & Christos Zaroliagis Slides based on those in: http://www.mmds.org High dim. data Graph data Infinite data Machine learning Apps Locality sensitive hashing PageRank, SimRank Filtering
More informationThanks to Jure Leskovec, Stanford and Panayiotis Tsaparas, Univ. of Ioannina for slides
Thanks to Jure Leskovec, Stanford and Panayiotis Tsaparas, Univ. of Ioannina for slides Web Search: How to Organize the Web? Ranking Nodes on Graphs Hubs and Authorities PageRank How to Solve PageRank
More informationCS6220: DATA MINING TECHNIQUES
CS6220: DATA MINING TECHNIQUES Mining Graph/Network Data Instructor: Yizhou Sun yzsun@ccs.neu.edu November 16, 2015 Methods to Learn Classification Clustering Frequent Pattern Mining Matrix Data Decision
More informationCS224W: Social and Information Network Analysis Jure Leskovec, Stanford University
CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University http://cs224w.stanford.edu How to organize/navigate it? First try: Human curated Web directories Yahoo, DMOZ, LookSmart
More informationThanks to Jure Leskovec, Stanford and Panayiotis Tsaparas, Univ. of Ioannina for slides
Thanks to Jure Leskovec, Stanford and Panayiotis Tsaparas, Univ. of Ioannina for slides Web Search: How to Organize the Web? Ranking Nodes on Graphs Hubs and Authorities PageRank How to Solve PageRank
More informationCS6220: DATA MINING TECHNIQUES
CS6220: DATA MINING TECHNIQUES Mining Graph/Network Data Instructor: Yizhou Sun yzsun@ccs.neu.edu March 16, 2016 Methods to Learn Classification Clustering Frequent Pattern Mining Matrix Data Decision
More informationCS249: ADVANCED DATA MINING
CS249: ADVANCED DATA MINING Graph and Network Instructor: Yizhou Sun yzsun@cs.ucla.edu May 31, 2017 Methods Learnt Classification Clustering Vector Data Text Data Recommender System Decision Tree; Naïve
More informationGoogle PageRank. Francesco Ricci Faculty of Computer Science Free University of Bozen-Bolzano
Google PageRank Francesco Ricci Faculty of Computer Science Free University of Bozen-Bolzano fricci@unibz.it 1 Content p Linear Algebra p Matrices p Eigenvalues and eigenvectors p Markov chains p Google
More informationIntroduction to Data Mining
Introduction to Data Mining Lecture #9: Link Analysis Seoul National University 1 In This Lecture Motivation for link analysis Pagerank: an important graph ranking algorithm Flow and random walk formulation
More informationJeffrey D. Ullman Stanford University
Jeffrey D. Ullman Stanford University 2 Web pages are important if people visit them a lot. But we can t watch everybody using the Web. A good surrogate for visiting pages is to assume people follow links
More information0.1 Naive formulation of PageRank
PageRank is a ranking system designed to find the best pages on the web. A webpage is considered good if it is endorsed (i.e. linked to) by other good webpages. The more webpages link to it, and the more
More informationJeffrey D. Ullman Stanford University
Jeffrey D. Ullman Stanford University We ve had our first HC cases. Please, please, please, before you do anything that might violate the HC, talk to me or a TA to make sure it is legitimate. It is much
More informationLink Analysis. Stony Brook University CSE545, Fall 2016
Link Analysis Stony Brook University CSE545, Fall 2016 The Web, circa 1998 The Web, circa 1998 The Web, circa 1998 Match keywords, language (information retrieval) Explore directory The Web, circa 1998
More informationOnline Social Networks and Media. Link Analysis and Web Search
Online Social Networks and Media Link Analysis and Web Search How to Organize the Web First try: Human curated Web directories Yahoo, DMOZ, LookSmart How to organize the web Second try: Web Search Information
More informationA Note on Google s PageRank
A Note on Google s PageRank According to Google, google-search on a given topic results in a listing of most relevant web pages related to the topic. Google ranks the importance of webpages according to
More informationDATA MINING LECTURE 13. Link Analysis Ranking PageRank -- Random walks HITS
DATA MINING LECTURE 3 Link Analysis Ranking PageRank -- Random walks HITS How to organize the web First try: Manually curated Web Directories How to organize the web Second try: Web Search Information
More informationStatistical Problem. . We may have an underlying evolving system. (new state) = f(old state, noise) Input data: series of observations X 1, X 2 X t
Markov Chains. Statistical Problem. We may have an underlying evolving system (new state) = f(old state, noise) Input data: series of observations X 1, X 2 X t Consecutive speech feature vectors are related
More informationPageRank algorithm Hubs and Authorities. Data mining. Web Data Mining PageRank, Hubs and Authorities. University of Szeged.
Web Data Mining PageRank, University of Szeged Why ranking web pages is useful? We are starving for knowledge It earns Google a bunch of money. How? How does the Web looks like? Big strongly connected
More informationOnline Social Networks and Media. Link Analysis and Web Search
Online Social Networks and Media Link Analysis and Web Search How to Organize the Web First try: Human curated Web directories Yahoo, DMOZ, LookSmart How to organize the web Second try: Web Search Information
More informationComputing PageRank using Power Extrapolation
Computing PageRank using Power Extrapolation Taher Haveliwala, Sepandar Kamvar, Dan Klein, Chris Manning, and Gene Golub Stanford University Abstract. We present a novel technique for speeding up the computation
More informationLink Analysis. Reference: Introduction to Information Retrieval by C. Manning, P. Raghavan, H. Schutze
Link Analysis Reference: Introduction to Information Retrieval by C. Manning, P. Raghavan, H. Schutze 1 The Web as a Directed Graph Page A Anchor hyperlink Page B Assumption 1: A hyperlink between pages
More informationIR: Information Retrieval
/ 44 IR: Information Retrieval FIB, Master in Innovation and Research in Informatics Slides by Marta Arias, José Luis Balcázar, Ramon Ferrer-i-Cancho, Ricard Gavaldá Department of Computer Science, UPC
More information1998: enter Link Analysis
1998: enter Link Analysis uses hyperlink structure to focus the relevant set combine traditional IR score with popularity score Page and Brin 1998 Kleinberg Web Information Retrieval IR before the Web
More informationInformation Retrieval and Search. Web Linkage Mining. Miłosz Kadziński
Web Linkage Analysis D24 D4 : Web Linkage Mining Miłosz Kadziński Institute of Computing Science Poznan University of Technology, Poland www.cs.put.poznan.pl/mkadzinski/wpi Web mining: Web Mining Discovery
More informationLink Analysis. Leonid E. Zhukov
Link Analysis Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural Analysis and Visualization
More informationLink Analysis Ranking
Link Analysis Ranking How do search engines decide how to rank your query results? Guess why Google ranks the query results the way it does How would you do it? Naïve ranking of query results Given query
More informationIntroduction to Search Engine Technology Introduction to Link Structure Analysis. Ronny Lempel Yahoo Labs, Haifa
Introduction to Search Engine Technology Introduction to Link Structure Analysis Ronny Lempel Yahoo Labs, Haifa Outline Anchor-text indexing Mathematical Background Motivation for link structure analysis
More informationCS 277: Data Mining. Mining Web Link Structure. CS 277: Data Mining Lectures Analyzing Web Link Structure Padhraic Smyth, UC Irvine
CS 277: Data Mining Mining Web Link Structure Class Presentations In-class, Tuesday and Thursday next week 2-person teams: 6 minutes, up to 6 slides, 3 minutes/slides each person 1-person teams 4 minutes,
More informationLab 8: Measuring Graph Centrality - PageRank. Monday, November 5 CompSci 531, Fall 2018
Lab 8: Measuring Graph Centrality - PageRank Monday, November 5 CompSci 531, Fall 2018 Outline Measuring Graph Centrality: Motivation Random Walks, Markov Chains, and Stationarity Distributions Google
More informationLink Analysis & Ranking CS 224W
Link Analysis & Ranking CS 224W 1 How to Organize the Web? How to organize the Web? First try: Human curated Web directories Yahoo, DMOZ, LookSmart Second try: Web Search Information Retrieval attempts
More informationWeb Ranking. Classification (manual, automatic) Link Analysis (today s lesson)
Link Analysis Web Ranking Documents on the web are first ranked according to their relevance vrs the query Additional ranking methods are needed to cope with huge amount of information Additional ranking
More informationHow does Google rank webpages?
Linear Algebra Spring 016 How does Google rank webpages? Dept. of Internet and Multimedia Eng. Konkuk University leehw@konkuk.ac.kr 1 Background on search engines Outline HITS algorithm (Jon Kleinberg)
More informationGoogle Page Rank Project Linear Algebra Summer 2012
Google Page Rank Project Linear Algebra Summer 2012 How does an internet search engine, like Google, work? In this project you will discover how the Page Rank algorithm works to give the most relevant
More informationToday. Next lecture. (Ch 14) Markov chains and hidden Markov models
Today (Ch 14) Markov chains and hidden Markov models Graphical representation Transition probability matrix Propagating state distributions The stationary distribution Next lecture (Ch 14) Markov chains
More informationPageRank. Ryan Tibshirani /36-662: Data Mining. January Optional reading: ESL 14.10
PageRank Ryan Tibshirani 36-462/36-662: Data Mining January 24 2012 Optional reading: ESL 14.10 1 Information retrieval with the web Last time we learned about information retrieval. We learned how to
More information6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search
6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search Daron Acemoglu and Asu Ozdaglar MIT September 30, 2009 1 Networks: Lecture 7 Outline Navigation (or decentralized search)
More informationData Mining and Matrices
Data Mining and Matrices 10 Graphs II Rainer Gemulla, Pauli Miettinen Jul 4, 2013 Link analysis The web as a directed graph Set of web pages with associated textual content Hyperlinks between webpages
More informationData Mining Techniques
Data Mining Techniques CS 622 - Section 2 - Spring 27 Pre-final Review Jan-Willem van de Meent Feedback Feedback https://goo.gl/er7eo8 (also posted on Piazza) Also, please fill out your TRACE evaluations!
More informationLecture 12: Link Analysis for Web Retrieval
Lecture 12: Link Analysis for Web Retrieval Trevor Cohn COMP90042, 2015, Semester 1 What we ll learn in this lecture The web as a graph Page-rank method for deriving the importance of pages Hubs and authorities
More informationUncertainty and Randomization
Uncertainty and Randomization The PageRank Computation in Google Roberto Tempo IEIIT-CNR Politecnico di Torino tempo@polito.it 1993: Robustness of Linear Systems 1993: Robustness of Linear Systems 16 Years
More informationData Mining Recitation Notes Week 3
Data Mining Recitation Notes Week 3 Jack Rae January 28, 2013 1 Information Retrieval Given a set of documents, pull the (k) most similar document(s) to a given query. 1.1 Setup Say we have D documents
More information1 Searching the World Wide Web
Hubs and Authorities in a Hyperlinked Environment 1 Searching the World Wide Web Because diverse users each modify the link structure of the WWW within a relatively small scope by creating web-pages on
More informationPage rank computation HPC course project a.y
Page rank computation HPC course project a.y. 2015-16 Compute efficient and scalable Pagerank MPI, Multithreading, SSE 1 PageRank PageRank is a link analysis algorithm, named after Brin & Page [1], and
More informationConditioning of the Entries in the Stationary Vector of a Google-Type Matrix. Steve Kirkland University of Regina
Conditioning of the Entries in the Stationary Vector of a Google-Type Matrix Steve Kirkland University of Regina June 5, 2006 Motivation: Google s PageRank algorithm finds the stationary vector of a stochastic
More informationWeb Structure Mining Nodes, Links and Influence
Web Structure Mining Nodes, Links and Influence 1 Outline 1. Importance of nodes 1. Centrality 2. Prestige 3. Page Rank 4. Hubs and Authority 5. Metrics comparison 2. Link analysis 3. Influence model 1.
More informationAs it is not necessarily possible to satisfy this equation, we just ask for a solution to the more general equation
Graphs and Networks Page 1 Lecture 2, Ranking 1 Tuesday, September 12, 2006 1:14 PM I. II. I. How search engines work: a. Crawl the web, creating a database b. Answer query somehow, e.g. grep. (ex. Funk
More informationeigenvalues, markov matrices, and the power method
eigenvalues, markov matrices, and the power method Slides by Olson. Some taken loosely from Jeff Jauregui, Some from Semeraro L. Olson Department of Computer Science University of Illinois at Urbana-Champaign
More informationWeb Ranking. Classification (manual, automatic) Link Analysis (today s lesson)
Link Analysis Web Ranking Documents on the web are first ranked according to their relevance vrs the query Additional ranking methods are needed to cope with huge amount of information Additional ranking
More informationData Mining Techniques
Data Mining Techniques CS 6220 - Section 3 - Fall 2016 Lecture 21: Review Jan-Willem van de Meent Schedule Topics for Exam Pre-Midterm Probability Information Theory Linear Regression Classification Clustering
More informationComplex Social System, Elections. Introduction to Network Analysis 1
Complex Social System, Elections Introduction to Network Analysis 1 Complex Social System, Network I person A voted for B A is more central than B if more people voted for A In-degree centrality index
More informationECEN 689 Special Topics in Data Science for Communications Networks
ECEN 689 Special Topics in Data Science for Communications Networks Nick Duffield Department of Electrical & Computer Engineering Texas A&M University Lecture 8 Random Walks, Matrices and PageRank Graphs
More informationLINK ANALYSIS. Dr. Gjergji Kasneci Introduction to Information Retrieval WS
LINK ANALYSIS Dr. Gjergji Kasneci Introduction to Information Retrieval WS 2012-13 1 Outline Intro Basics of probability and information theory Retrieval models Retrieval evaluation Link analysis Models
More informationCS47300: Web Information Search and Management
CS473: Web Information Search and Management Using Graph Structure for Retrieval Prof. Chris Clifton 24 September 218 Material adapted from slides created by Dr. Rong Jin (formerly Michigan State, now
More informationLecture: Local Spectral Methods (1 of 4)
Stat260/CS294: Spectral Graph Methods Lecture 18-03/31/2015 Lecture: Local Spectral Methods (1 of 4) Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still very rough. They provide
More informationLink Analysis Information Retrieval and Data Mining. Prof. Matteo Matteucci
Link Analysis Information Retrieval and Data Mining Prof. Matteo Matteucci Hyperlinks for Indexing and Ranking 2 Page A Hyperlink Page B Intuitions The anchor text might describe the target page B Anchor
More informationIS4200/CS6200 Informa0on Retrieval. PageRank Con+nued. with slides from Hinrich Schütze and Chris6na Lioma
IS4200/CS6200 Informa0on Retrieval PageRank Con+nued with slides from Hinrich Schütze and Chris6na Lioma Exercise: Assump0ons underlying PageRank Assump0on 1: A link on the web is a quality signal the
More informationMath 304 Handout: Linear algebra, graphs, and networks.
Math 30 Handout: Linear algebra, graphs, and networks. December, 006. GRAPHS AND ADJACENCY MATRICES. Definition. A graph is a collection of vertices connected by edges. A directed graph is a graph all
More informationMathematical Properties & Analysis of Google s PageRank
Mathematical Properties & Analysis of Google s PageRank Ilse Ipsen North Carolina State University, USA Joint work with Rebecca M. Wills Cedya p.1 PageRank An objective measure of the citation importance
More informationMultiRank and HAR for Ranking Multi-relational Data, Transition Probability Tensors, and Multi-Stochastic Tensors
MultiRank and HAR for Ranking Multi-relational Data, Transition Probability Tensors, and Multi-Stochastic Tensors Michael K. Ng Centre for Mathematical Imaging and Vision and Department of Mathematics
More informationCS54701 Information Retrieval. Link Analysis. Luo Si. Department of Computer Science Purdue University. Borrowed Slides from Prof.
CS54701 Information Retrieval Link Analysis Luo Si Department of Computer Science Purdue University Borrowed Slides from Prof. Rong Jin (MSU) Citation Analysis Web Structure Web is a graph Each web site
More informationLink Analysis and Web Search
Link Analysis and Web Search Episode 11 Baochun Li Professor Department of Electrical and Computer Engineering University of Toronto Link Analysis and Web Search (Chapter 13, 14) Information networks and
More informationGraph Models The PageRank Algorithm
Graph Models The PageRank Algorithm Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 The PageRank Algorithm I Invented by Larry Page and Sergey Brin around 1998 and
More informationInf 2B: Ranking Queries on the WWW
Inf B: Ranking Queries on the WWW Kyriakos Kalorkoti School of Informatics University of Edinburgh Queries Suppose we have an Inverted Index for a set of webpages. Disclaimer Not really the scenario of
More informationAnalysis of Google s PageRank
Analysis of Google s PageRank Ilse Ipsen North Carolina State University Joint work with Rebecca M. Wills AN05 p.1 PageRank An objective measure of the citation importance of a web page [Brin & Page 1998]
More informationLecture 7 Mathematics behind Internet Search
CCST907 Hidden Order in Daily Life: A Mathematical Perspective Lecture 7 Mathematics behind Internet Search Dr. S. P. Yung (907A) Dr. Z. Hua (907B) Department of Mathematics, HKU Outline Google is the
More informationLecture: Local Spectral Methods (2 of 4) 19 Computing spectral ranking with the push procedure
Stat260/CS294: Spectral Graph Methods Lecture 19-04/02/2015 Lecture: Local Spectral Methods (2 of 4) Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still very rough. They provide
More informationPageRank: The Math-y Version (Or, What To Do When You Can t Tear Up Little Pieces of Paper)
PageRank: The Math-y Version (Or, What To Do When You Can t Tear Up Little Pieces of Paper) In class, we saw this graph, with each node representing people who are following each other on Twitter: Our
More informationCOMPSCI 514: Algorithms for Data Science
COMPSCI 514: Algorithms for Data Science Arya Mazumdar University of Massachusetts at Amherst Fall 2018 Lecture 4 Markov Chain & Pagerank Homework Announcement Show your work in the homework Write the
More informationPr[positive test virus] Pr[virus] Pr[positive test] = Pr[positive test] = Pr[positive test]
146 Probability Pr[virus] = 0.00001 Pr[no virus] = 0.99999 Pr[positive test virus] = 0.99 Pr[positive test no virus] = 0.01 Pr[virus positive test] = Pr[positive test virus] Pr[virus] = 0.99 0.00001 =
More informationINTRODUCTION TO MCMC AND PAGERANK. Eric Vigoda Georgia Tech. Lecture for CS 6505
INTRODUCTION TO MCMC AND PAGERANK Eric Vigoda Georgia Tech Lecture for CS 6505 1 MARKOV CHAIN BASICS 2 ERGODICITY 3 WHAT IS THE STATIONARY DISTRIBUTION? 4 PAGERANK 5 MIXING TIME 6 PREVIEW OF FURTHER TOPICS
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 6: Numerical Linear Algebra: Applications in Machine Learning Cho-Jui Hsieh UC Davis April 27, 2017 Principal Component Analysis Principal
More informationAlgebraic Representation of Networks
Algebraic Representation of Networks 0 1 2 1 1 0 0 1 2 0 0 1 1 1 1 1 Hiroki Sayama sayama@binghamton.edu Describing networks with matrices (1) Adjacency matrix A matrix with rows and columns labeled by
More informationNode and Link Analysis
Node and Link Analysis Leonid E. Zhukov School of Applied Mathematics and Information Science National Research University Higher School of Economics 10.02.2014 Leonid E. Zhukov (HSE) Lecture 5 10.02.2014
More informationCS 188: Artificial Intelligence Spring Announcements
CS 188: Artificial Intelligence Spring 2011 Lecture 18: HMMs and Particle Filtering 4/4/2011 Pieter Abbeel --- UC Berkeley Many slides over this course adapted from Dan Klein, Stuart Russell, Andrew Moore
More informationWiki Definition. Reputation Systems I. Outline. Introduction to Reputations. Yury Lifshits. HITS, PageRank, SALSA, ebay, EigenTrust, VKontakte
Reputation Systems I HITS, PageRank, SALSA, ebay, EigenTrust, VKontakte Yury Lifshits Wiki Definition Reputation is the opinion (more technically, a social evaluation) of the public toward a person, a
More informationAnnouncements. CS 188: Artificial Intelligence Fall Markov Models. Example: Markov Chain. Mini-Forward Algorithm. Example
CS 88: Artificial Intelligence Fall 29 Lecture 9: Hidden Markov Models /3/29 Announcements Written 3 is up! Due on /2 (i.e. under two weeks) Project 4 up very soon! Due on /9 (i.e. a little over two weeks)
More informationApplications. Nonnegative Matrices: Ranking
Applications of Nonnegative Matrices: Ranking and Clustering Amy Langville Mathematics Department College of Charleston Hamilton Institute 8/7/2008 Collaborators Carl Meyer, N. C. State University David
More informationThe Second Eigenvalue of the Google Matrix
The Second Eigenvalue of the Google Matrix Taher H. Haveliwala and Sepandar D. Kamvar Stanford University {taherh,sdkamvar}@cs.stanford.edu Abstract. We determine analytically the modulus of the second
More informationClass President: A Network Approach to Popularity. Due July 18, 2014
Class President: A Network Approach to Popularity Due July 8, 24 Instructions. Due Fri, July 8 at :59 PM 2. Work in groups of up to 3 3. Type up the report, and submit as a pdf on D2L 4. Attach the code
More informationMachine Learning CPSC 340. Tutorial 12
Machine Learning CPSC 340 Tutorial 12 Random Walk on Graph Page Rank Algorithm Label Propagation on Graph Assume a strongly connected graph G = (V, A) Label Propagation on Graph Assume a strongly connected
More informationApplication. Stochastic Matrices and PageRank
Application Stochastic Matrices and PageRank Stochastic Matrices Definition A square matrix A is stochastic if all of its entries are nonnegative, and the sum of the entries of each column is. We say A
More informationCS 188: Artificial Intelligence
CS 188: Artificial Intelligence Markov Models Instructors: Dan Klein and Pieter Abbeel --- University of California, Berkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to
More informationCalculating Web Page Authority Using the PageRank Algorithm
Jacob Miles Prystowsky and Levi Gill Math 45, Fall 2005 1 Introduction 1.1 Abstract In this document, we examine how the Google Internet search engine uses the PageRank algorithm to assign quantitatively
More informationDegree Distribution: The case of Citation Networks
Network Analysis Degree Distribution: The case of Citation Networks Papers (in almost all fields) refer to works done earlier on same/related topics Citations A network can be defined as Each node is
More informationCS 188: Artificial Intelligence Spring 2009
CS 188: Artificial Intelligence Spring 2009 Lecture 21: Hidden Markov Models 4/7/2009 John DeNero UC Berkeley Slides adapted from Dan Klein Announcements Written 3 deadline extended! Posted last Friday
More informationMAE 298, Lecture 8 Feb 4, Web search and decentralized search on small-worlds
MAE 298, Lecture 8 Feb 4, 2008 Web search and decentralized search on small-worlds Search for information Assume some resource of interest is stored at the vertices of a network: Web pages Files in a file-sharing
More informationHow works. or How linear algebra powers the search engine. M. Ram Murty, FRSC Queen s Research Chair Queen s University
How works or How linear algebra powers the search engine M. Ram Murty, FRSC Queen s Research Chair Queen s University From: gomath.com/geometry/ellipse.php Metric mishap causes loss of Mars orbiter
More informationINTRODUCTION TO MCMC AND PAGERANK. Eric Vigoda Georgia Tech. Lecture for CS 6505
INTRODUCTION TO MCMC AND PAGERANK Eric Vigoda Georgia Tech Lecture for CS 6505 1 MARKOV CHAIN BASICS 2 ERGODICITY 3 WHAT IS THE STATIONARY DISTRIBUTION? 4 PAGERANK 5 MIXING TIME 6 PREVIEW OF FURTHER TOPICS
More informationMarkov Models. CS 188: Artificial Intelligence Fall Example. Mini-Forward Algorithm. Stationary Distributions.
CS 88: Artificial Intelligence Fall 27 Lecture 2: HMMs /6/27 Markov Models A Markov model is a chain-structured BN Each node is identically distributed (stationarity) Value of X at a given time is called
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Mark Schmidt University of British Columbia Winter 2019 Last Time: Monte Carlo Methods If we want to approximate expectations of random functions, E[g(x)] = g(x)p(x) or E[g(x)]
More informationConsider the following example of a linear system:
LINEAR SYSTEMS Consider the following example of a linear system: Its unique solution is x + 2x 2 + 3x 3 = 5 x + x 3 = 3 3x + x 2 + 3x 3 = 3 x =, x 2 = 0, x 3 = 2 In general we want to solve n equations
More informationAnnouncements. CS 188: Artificial Intelligence Fall VPI Example. VPI Properties. Reasoning over Time. Markov Models. Lecture 19: HMMs 11/4/2008
CS 88: Artificial Intelligence Fall 28 Lecture 9: HMMs /4/28 Announcements Midterm solutions up, submit regrade requests within a week Midterm course evaluation up on web, please fill out! Dan Klein UC
More informationCS246: Mining Massive Data Sets Winter Only one late period is allowed for this homework (11:59pm 2/14). General Instructions
CS246: Mining Massive Data Sets Winter 2017 Problem Set 2 Due 11:59pm February 9, 2017 Only one late period is allowed for this homework (11:59pm 2/14). General Instructions Submission instructions: These
More informationHidden Markov Models. Hal Daumé III. Computer Science University of Maryland CS 421: Introduction to Artificial Intelligence 19 Apr 2012
Hidden Markov Models Hal Daumé III Computer Science University of Maryland me@hal3.name CS 421: Introduction to Artificial Intelligence 19 Apr 2012 Many slides courtesy of Dan Klein, Stuart Russell, or
More informationCS246: Mining Massive Datasets Jure Leskovec, Stanford University
CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu 2/26/2013 Jure Leskovec, Stanford CS246: Mining Massive Datasets, http://cs246.stanford.edu 2 More algorithms
More information