Random Surfing on Multipartite Graphs
|
|
- Sheryl Sherman
- 5 years ago
- Views:
Transcription
1 Random Surfing on Multipartite Graphs Athanasios N. Nikolakopoulos, Antonia Korba and John D. Garofalakis Department of Computer Engineering and Informatics, University of Patras December 07, 2016 IEEE International Conference on Big Data, IEEE BigData 2016
2 Outline 1. Introduction & Motivation 2. Block Teleportation Model: Definition, Algorithm and Theoretical Analysis. Experimental Evaluation 4. Conclusions & Future Work 1
3 Introduction & Motivation
4 Revisiting the Random Surfer Model I PageRank Model G = αh + (1 α)e Primitivity Adjustment of the Row-Normalized Adjacency Matrix H: Damping Factor α Has received much attention (Generalized Damping Functions (Functional Rankings) [1], Multidamping [5],...) Teleportation matrix E Little have been done towards its generalization [8]. 2
5 Revisiting the Random Surfer Model II Problems With Traditional Teleportation Treats nodes in a leveling way Restrictive or even counter-intuitive (eg. structured graphs) Blind to the spectral characteristics of the underlying graph Overview of Our Approach We focus on Multipartite Graphs We modify the traditional teleportation model Different Teleportation behavior per partite set. u 1 u 2 u v 1 v 2 v w 1 w 2
6 Block Teleportation Model: Definition, Algorithm and Theoretical Analysis
7 Model Definition S = ηh + M H diag(a G 1) 1 A G, Each partite set is a Teleportation Block! [M] ij M = { 1 M i, if v j M i, 0, otherwise, where M i the origin partite set of v i. R R }{{} Sparse and Low-Rank Factors Random Surfing Interpretation The Random Surfer: With probability η follows the edges of the graph With probability 1 η he jumps to a node belonging to the same partite set he is currently in. 4
8 Decomposability and Time-Scale Dissociation Theorem (Decomposability) u1 u1 When the value of the teleportation parameter is small enough, the Markov chain corresponding to matrix S is NCD with respect to the partition of the nodes of the initial graph, into different connected components. u2 u u4 v1 v2 v u2 u u4 v1 v2 v S = S + εc, S diag{s(g 1 ),..., S(G L )} u5 u5 n(i) L L S t = Z 11 + λ t 1 Z 1I + λ t }{{} I m Z mi, I I=2 I=1 m=2 Term A }{{}}{{} Term B Term C S t = n(i) L L Z 1I + ( λ) t m Z mi. I I=1 I=1 m=2 }{{}}{{} Term à Term C Short-term Dynamics. Short-term Equilibrium. Long-term Dynamics. Long-term Equilibrium. Computational Implications... in brush strokes! Study each Connected Component in Isolation, and then combine the results. 5
9 BT-Rank Algorithm and Computational Analysis Block-Teleportation Rank Input: H, M R n n, ϵ, scalars η, > 0 such that η + = 1. Output: π 1: Let the initial approximation be π. Set k = 0. (0) 2: Compute π (k+1) π (k) H ϕ π (k) M π (k+1) ηπ (k+1) + ϕ : Normalize π and compute (k+1) r = π (k+1) π (k) 1. 4: If r < ϵ, quit with π (k+1), otherwise k k + 1 and go to step 2. General Cost: log ϵ Θ(nnz(H)) }{{} log λ 2(S) per iteration If χ(g) = 2 we can dig a little deeper! Theorem (Eigenvalue Property ) Assuming G is a connected graph for which χ(g) = 2 holds, the spectrum of the stochastic matrix S is such that η + λ(s). Theorem (Lumpability ) The BT-Rank Markov chain that corresponds to a 2-chromatic graph, is lumpable wrt A = {A 1, A 2}. Theorem (Perron Vector ) When the BT-Rank Markov chain is lumpable wrt to partition A, for the left Perron eigenvector of matrix S it holds π 1 1 A 1 = π 2 1 A 2 6
10 Experimental Evaluation
11 Computational Experiments # of iterations # of iterations # of iterations Jester Digg votes Yahoo!Music MovieLens20M TREC Wikipedia(en) Time(sec) Time(sec) Time(sec) Jester Digg votes Yahoo!Music MovieLens20M TREC Wikipedia(en) BT-Rank BT-Rank(NoLump) PageRank 7
12 Qualitative Experiments: Ranking-Based Recommendation I Our Approach We model the recommender as a tripartite graph Users Movies Genres ( ) Personalization through matrix M diag 1e i, 1ω i, 1ϖ i ω i : the normalized vector of the users ratings over the set of movies. ϖ i : the normalized vector of his mean ratings per genre. 8
13 Qualitative Experiments: Ranking-Based Recommendation II Methodology Randomly sample 1.4% of the ratings probe set P Use each item v j, rated with 5 stars by user u i in P test set T Randomly select another 1000 unrated items of the same user for each item in T Form ranked lists by ordering all the 1001 items MovieLens1M Metrics Recall@N Normalized Discounted Cumulative Gain (NDCG@N) Mean Reciprocal Rank Recall@N NDCG@N BT-Rank Katz FP MFA L CT MRR BT-Rank Katz FP MFA L CT 9
14 Conclusions & Future Work
15 Conclusion & Future Work Synopsis We proposed a simple alternative teleportation component for Random Surfing on Multipartite Graphs. Can be handled efficiently Entails nice theoretical properties Allows for richer modeling Future Directions Propose a Systematic Framework for the definition of teleportation models that match better the underlying graphs For the Web-Graph: NCDawareRank (WSDM 1) 10
16 References R. Baeza-Yates, P. Boldi, and C. Castillo. Generic damping functions for propagating importance in link-based ranking. Internet Math., (4): , P.-J. Courtois. Decomposability: Queueing and Computer System Applications. ACM monograph series. Academic Press, P. Cremonesi, Y. Koren, and R. Turrin. Performance of recommender algorithms on top-n recommendation tasks. In Proceedings of the fourth ACM conference on Recommender systems, RecSys 10, pages ACM, F. Fouss, K. Francoisse, L. Yen, A. Pirotte, and M. Saerens. An experimental investigation of kernels on graphs for collaborative recommendation and semisupervised classification. Neural Netw., 1:5 72, July G. Kollias, E. Gallopoulos, and A. Grama. Surfing the network for ranking by multidamping. IEEE Trans. Knowl. Data Eng., 26(9):22 26, B. Long, X. Wu, Z. M. Zhang, and P. S. Yu. Unsupervised learning on k-partite graphs. In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 06, pages 17 26, New York, NY, USA, ACM. A. Nikolakopoulos and J. Garofalakis. NCDREC: A decomposability inspired framework for top-n recommendation. In Web Intelligence (WI) and Intelligent Agent Technologies (IAT), 2014 IEEE/WIC/ACM International Joint Conferences on, pages , Aug A. N. Nikolakopoulos and J. D. Garofalakis. NCDawareRank: a novel ranking method that exploits the decomposable structure of the web. In Proceedings of the sixth ACM international conference on Web search and data mining, WSDM 1, pages ACM, 201. L. Page, S. Brin, R. Motwani, and T. Winograd. The pagerank citation ranking: Bringing order to the web. Technical Report , Stanford InfoLab, November H. A. Simon and A. Ando. Aggregation of variables in dynamic systems. Econometrica, 29(2):pp , 1961.
17 Questions?
18 Appendix: Eigenvalue Theorem Back Theorem (Eigenvalue Property) Assuming G is a connected graph for which χ(g) = 2 holds, the spectrum of the stochastic matrix S is such that η + λ(s). Proof Sketch. #nodes of 1st partite set { }} { We define a vector v [ #nodes of 2nd partite set {}}{ ] We show that ( 1, v) is an eigenpair of matrix H, and (1, v) is an eigenpair of matrix M. Then, using any nonsingular matrix, Q ( 1 v X ), allows us to perform a similarity transformation Q 1 SQ = Q 1 (ηh + M) Q = = 1 0 ηy 1 HX + y 1 MX = 0 η + ηy 2 HX + y 2 MX (1) 0 0 ηy HX + Y MX that reveals the desired eigenvalue.
19 Appendix: Lumpability Theorem Back Theorem (Lumpability of the BT-Rank Chain) The BT-Rank Markov chain that corresponds to a 2-chromatic graph, is lumpable. Proof Sketch. u1 v1 u2 v2 u v w1 w2 u1 u2 u w1 w2 v1 v2 v S = η 2 η 2 η η η 2 2 η η η η η 2 η 2 η 2 η 2 We have Pr{i A 2} = j A 2 S ij = η for all i A 1. Pr{i A 1} = j A 1 S ij = η for all i A 2.
20 Appendix: Perron Vector Theorem Back Theorem (Eigenvector Property) When the BT-Rank Markov chain is lumpable with respect to partition A = {A 1, A 2}, for the left Perron-Frobenius eigenvector of matrix S it holds π 1 1 A 1 = π 2 1 A 2 Proof Sketch. ( ) π 1A1 0 S 0 1 A2 ( ) π M111 A1 ηh 121 A2 ηh 211 A1 M 221 A2 ( π 1 π ) ( ) 1 A1 η1 A1 2 η1 A2 1 A2 ( ) = π 1A A2 = ( π 1 1 A 1 π 2 1 A 2 ) = ( π 1 1 A 1 π 2 1 A 2 ) and the result follows from the solution of the system.
21 Appendix: Near Complete Decomposability Back Herbert A. Simon Sparsity Hierarchy Decomposability [10]. Nearly Completely Decomposable Systems Interactions: Strong Within Blocks - Weak Between Blocks Successful Applications in Diverse Disciplines Economics, Cognitive Theory, A Management, Biology, Ecology, etc. Computer Science and Engineering: Performance Analysis (Courtois [2]) Web Search (NG1 [8]) C Recommendation and Data Mining (NG14 [7]) B ε ε ε ε ε ε ε ε
Random Surfing on Multipartite Graphs
Random Surfing on Multipartite Graphs Athanasios N. Nikolakopoulos, Antonia Korba and John D. Garofalakis Department of Computer Engineering and Informatics, University of Patras {nikolako,korba,garofala}@ceid.upatras.gr
More informationNCDREC: A Decomposability Inspired Framework for Top-N Recommendation
NCDREC: A Decomposability Inspired Framework for Top-N Recommendation Athanasios N. Nikolakopoulos,2 John D. Garofalakis,2 Computer Engineering and Informatics Department, University of Patras, Greece
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 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 informationFactored Proximity Models for Top-N Recommendations
Factored Proximity Models for Top- Recommendations Athanasios. ikolakopoulos 1,3, Vassilis Kalantzis 2, Efstratios Gallopoulos 1 and John D. Garofalakis 1 Department of Computer Engineering and Informatics
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 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 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 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 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 informationNode Centrality and Ranking on Networks
Node Centrality and Ranking on Networks Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social
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 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 informationUpdating PageRank. Amy Langville Carl Meyer
Updating PageRank Amy Langville Carl Meyer Department of Mathematics North Carolina State University Raleigh, NC SCCM 11/17/2003 Indexing Google Must index key terms on each page Robots crawl the web software
More informationKrylov Subspace Methods to Calculate PageRank
Krylov Subspace Methods to Calculate PageRank B. Vadala-Roth REU Final Presentation August 1st, 2013 How does Google Rank Web Pages? The Web The Graph (A) Ranks of Web pages v = v 1... Dominant Eigenvector
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 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 informationApplications of The Perron-Frobenius Theorem
Applications of The Perron-Frobenius Theorem Nate Iverson The University of Toledo Toledo, Ohio Motivation In a finite discrete linear dynamical system x n+1 = Ax n What are sufficient conditions for x
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 informationSpectral Graph Theory and You: Matrix Tree Theorem and Centrality Metrics
Spectral Graph Theory and You: and Centrality Metrics Jonathan Gootenberg March 11, 2013 1 / 19 Outline of Topics 1 Motivation Basics of Spectral Graph Theory Understanding the characteristic polynomial
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 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 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 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 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 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 informationMarkov Chains and Spectral Clustering
Markov Chains and Spectral Clustering Ning Liu 1,2 and William J. Stewart 1,3 1 Department of Computer Science North Carolina State University, Raleigh, NC 27695-8206, USA. 2 nliu@ncsu.edu, 3 billy@ncsu.edu
More informationCS246: 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 informationTop-N recommendations in the presence of sparsity: An NCD-based approach
Top-N recommendations in the presence of sparsity: An NCD-based approach arxiv:7.43v2 [cs.ir] 7 Jul 2 Athanasios N. Nikolakopoulos a,b, and John D. Garofalakis a,b, a Department of Computer Engineering
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 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 informationUpdatingtheStationary VectorofaMarkovChain. Amy Langville Carl Meyer
UpdatingtheStationary VectorofaMarkovChain Amy Langville Carl Meyer Department of Mathematics North Carolina State University Raleigh, NC NSMC 9/4/2003 Outline Updating and Pagerank Aggregation Partitioning
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 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 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 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 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 informationEigenRec: Generalizing PureSVD for Effective and Efficient Top-N Recommendations
KAIS manuscript No. (will be inserted by the editor) EigenRec: Generalizing PureSVD for Effective and Efficient Top-N Recommendations Athanasios N. Nikolakopoulos Vassilis Kalantzis Efstratios Gallopoulos
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 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 informationPersonalized PageRank with node-dependent restart
Personalized PageRank with node-dependent restart Konstantin Avrachenkov Remco van der Hofstad Marina Sokol May 204 Abstract Personalized PageRank is an algorithm to classify the improtance of web pages
More informationSUPPLEMENTARY MATERIALS TO THE PAPER: ON THE LIMITING BEHAVIOR OF PARAMETER-DEPENDENT NETWORK CENTRALITY MEASURES
SUPPLEMENTARY MATERIALS TO THE PAPER: ON THE LIMITING BEHAVIOR OF PARAMETER-DEPENDENT NETWORK CENTRALITY MEASURES MICHELE BENZI AND CHRISTINE KLYMKO Abstract This document contains details of numerical
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 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 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 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 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 informationA hybrid reordered Arnoldi method to accelerate PageRank computations
A hybrid reordered Arnoldi method to accelerate PageRank computations Danielle Parker Final Presentation Background Modeling the Web The Web The Graph (A) Ranks of Web pages v = v 1... Dominant Eigenvector
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 informationTHE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR
THE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR WEN LI AND MICHAEL K. NG Abstract. In this paper, we study the perturbation bound for the spectral radius of an m th - order n-dimensional
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 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 informationEigenvalue Problems Computation and Applications
Eigenvalue ProblemsComputation and Applications p. 1/36 Eigenvalue Problems Computation and Applications Che-Rung Lee cherung@gmail.com National Tsing Hua University Eigenvalue ProblemsComputation and
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 informationON THE LIMITING PROBABILITY DISTRIBUTION OF A TRANSITION PROBABILITY TENSOR
ON THE LIMITING PROBABILITY DISTRIBUTION OF A TRANSITION PROBABILITY TENSOR WEN LI AND MICHAEL K. NG Abstract. In this paper we propose and develop an iterative method to calculate a limiting probability
More informationFrancois Fouss, Alain Pirotte, Jean-Michel Renders & Marco Saerens. January 31, 2006
A novel way of computing dissimilarities between nodes of a graph, with application to collaborative filtering and subspace projection of the graph nodes Francois Fouss, Alain Pirotte, Jean-Michel Renders
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 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 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 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 informationCSI 445/660 Part 6 (Centrality Measures for Networks) 6 1 / 68
CSI 445/660 Part 6 (Centrality Measures for Networks) 6 1 / 68 References 1 L. Freeman, Centrality in Social Networks: Conceptual Clarification, Social Networks, Vol. 1, 1978/1979, pp. 215 239. 2 S. Wasserman
More informationPseudocode for calculating Eigenfactor TM Score and Article Influence TM Score using data from Thomson-Reuters Journal Citations Reports
Pseudocode for calculating Eigenfactor TM Score and Article Influence TM Score using data from Thomson-Reuters Journal Citations Reports Jevin West and Carl T. Bergstrom November 25, 2008 1 Overview There
More informationComputational Economics and Finance
Computational Economics and Finance Part II: Linear Equations Spring 2016 Outline Back Substitution, LU and other decomposi- Direct methods: tions Error analysis and condition numbers Iterative methods:
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 informationDecoupled Collaborative Ranking
Decoupled Collaborative Ranking Jun Hu, Ping Li April 24, 2017 Jun Hu, Ping Li WWW2017 April 24, 2017 1 / 36 Recommender Systems Recommendation system is an information filtering technique, which provides
More informationThe Push Algorithm for Spectral Ranking
The Push Algorithm for Spectral Ranking Paolo Boldi Sebastiano Vigna March 8, 204 Abstract The push algorithm was proposed first by Jeh and Widom [6] in the context of personalized PageRank computations
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 informationMATH36001 Perron Frobenius Theory 2015
MATH361 Perron Frobenius Theory 215 In addition to saying something useful, the Perron Frobenius theory is elegant. It is a testament to the fact that beautiful mathematics eventually tends to be useful,
More informationLocally-biased analytics
Locally-biased analytics You have BIG data and want to analyze a small part of it: Solution 1: Cut out small part and use traditional methods Challenge: cutting out may be difficult a priori Solution 2:
More informationDistributed Randomized Algorithms for the PageRank Computation Hideaki Ishii, Member, IEEE, and Roberto Tempo, Fellow, IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 9, SEPTEMBER 2010 1987 Distributed Randomized Algorithms for the PageRank Computation Hideaki Ishii, Member, IEEE, and Roberto Tempo, Fellow, IEEE Abstract
More informationMultiple Relational Ranking in Tensor: Theory, Algorithms and Applications
Multiple Relational Ranking in Tensor: Theory, Algorithms and Applications Michael K. Ng Centre for Mathematical Imaging and Vision and Department of Mathematics Hong Kong Baptist University Email: mng@math.hkbu.edu.hk
More informationPersonalized PageRank with Node-dependent Restart
Reprint from Moscow Journal of Combinatorics and Number Theory 204, vol. 4, iss. 4, pp. 5 20, [pp. 387 402] Scale =.7 PS:./fig-eps/Logo-MJCNT.eps Personalized PageRank with Node-dependent Restart Konstantin
More informationFinding central nodes in large networks
Finding central nodes in large networks Nelly Litvak University of Twente Eindhoven University of Technology, The Netherlands Woudschoten Conference 2017 Complex networks Networks: Internet, WWW, social
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 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 informationECS289: Scalable Machine Learning
ECS289: Scalable Machine Learning Cho-Jui Hsieh UC Davis Oct 11, 2016 Paper presentations and final project proposal Send me the names of your group member (2 or 3 students) before October 15 (this Friday)
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 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 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 informationHub, Authority and Relevance Scores in Multi-Relational Data for Query Search
Hub, Authority and Relevance Scores in Multi-Relational Data for Query Search Xutao Li 1 Michael Ng 2 Yunming Ye 1 1 Department of Computer Science, Shenzhen Graduate School, Harbin Institute of Technology,
More informationApplications to network analysis: Eigenvector centrality indices Lecture notes
Applications to network analysis: Eigenvector centrality indices Lecture notes Dario Fasino, University of Udine (Italy) Lecture notes for the second part of the course Nonnegative and spectral matrix
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 informationSome relationships between Kleinberg s hubs and authorities, correspondence analysis, and the Salsa algorithm
Some relationships between Kleinberg s hubs and authorities, correspondence analysis, and the Salsa algorithm François Fouss 1, Jean-Michel Renders 2 & Marco Saerens 1 {saerens,fouss}@isys.ucl.ac.be, jean-michel.renders@xrce.xerox.com
More informationUsing SVD to Recommend Movies
Michael Percy University of California, Santa Cruz Last update: December 12, 2009 Last update: December 12, 2009 1 / Outline 1 Introduction 2 Singular Value Decomposition 3 Experiments 4 Conclusion Last
More informationRaRE: Social Rank Regulated Large-scale Network Embedding
RaRE: Social Rank Regulated Large-scale Network Embedding Authors: Yupeng Gu 1, Yizhou Sun 1, Yanen Li 2, Yang Yang 3 04/26/2018 The Web Conference, 2018 1 University of California, Los Angeles 2 Snapchat
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 informationSome relationships between Kleinberg s hubs and authorities, correspondence analysis, and the Salsa algorithm
Some relationships between Kleinberg s hubs and authorities, correspondence analysis, and the Salsa algorithm François Fouss 1, Jean-Michel Renders 2, Marco Saerens 1 1 ISYS Unit, IAG Université catholique
More informationLinks between Kleinberg s hubs and authorities, correspondence analysis, and Markov chains
Links between Kleinberg s hubs and authorities, correspondence analysis, and Markov chains Francois Fouss, Jean-Michel Renders & Marco Saerens Université Catholique de Louvain and Xerox Research Center
More informationJeffrey D. Ullman Stanford University
Jeffrey D. Ullman Stanford University 2 Often, our data can be represented by an m-by-n matrix. And this matrix can be closely approximated by the product of two matrices that share a small common dimension
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 informationTen good reasons to use the Eigenfactor TM metrics
Ten good reasons to use the Eigenfactor TM metrics Massimo Franceschet Department of Mathematics and Computer Science, University of Udine Via delle Scienze 206 33100 Udine, Italy massimo.franceschet@dimi.uniud.it
More informationCS246: Mining Massive Datasets Jure Leskovec, Stanford University.
CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu What is the structure of the Web? How is it organized? 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive
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 informationThe PageRank Computation in Google: Randomization and Ergodicity
The PageRank Computation in Google: Randomization and Ergodicity Roberto Tempo CNR-IEIIT Consiglio Nazionale delle Ricerche Politecnico di Torino roberto.tempo@polito.it Randomization over Networks - 1
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 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 informationELE 538B: Mathematics of High-Dimensional Data. Spectral methods. Yuxin Chen Princeton University, Fall 2018
ELE 538B: Mathematics of High-Dimensional Data Spectral methods Yuxin Chen Princeton University, Fall 2018 Outline A motivating application: graph clustering Distance and angles between two subspaces Eigen-space
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 and Computation of Google s PageRank
Analysis and Computation of Google s PageRank Ilse Ipsen North Carolina State University, USA Joint work with Rebecca S. Wills ANAW p.1 PageRank An objective measure of the citation importance of a web
More informationA Fast Two-Stage Algorithm for Computing PageRank
A Fast Two-Stage Algorithm for Computing PageRank Chris Pan-Chi Lee Stanford University cpclee@stanford.edu Gene H. Golub Stanford University golub@stanford.edu Stefanos A. Zenios Stanford University stefzen@stanford.edu
More information