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| May 2003 | Almaden Research Center, San Jose, CA © 2003 IBM Corporation IMA Tutorial (part II): Measurement and modeling of the web and related data sets Andrew Tomkins IBM Almaden Research Center May 5, 2003

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- Web Measurement- Self similarity on the web- Extraction of information from large graphs- A word on evolution

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  • 1. IMA Tutorial (part II): Measurement and modeling of the web and related data sets Andrew Tomkins IBM Almaden Research Center May 5, 2003 Title slide

2. Setup

  • This hour: data analysis on the web
  • Next hour: probabilistic generative models, particularly focused on models that generate distributions that are power laws in the limit

3. Context

  • Data Analysison the web
  • as a hyperlinked corpus
  • Note: Many areas of document analysis are highly relevant to the web, and should not be ignored (but will be):
    • Supervised/unsupervised classification (Jon combinatorial side)
    • Machine learning (Jon a little)
    • Information retrieval (Jon dimensionality reduction)
    • Information extraction
    • NLP
    • Discourse analysis
    • Relationship induction
    • etc

4. Focus Areas

  • Web Measurement
  • Self similarity on the web
  • Extraction of information from large graphs
  • A word on evolution

5. One view of the Internet: Inter-Domain Connectivity

  • Core: maximal clique of high-degree nodes
  • Shells: nodes in 1-neighborhood of core, or of previous shell, with degree > 1
  • Legs: 1-degree nodes

Core Shells: 1 2 3 [Tauro, Palmer, Siganos, Faloutsos, 2001 Global Internet] 6. Another view of the web: the hyperlink graph

  • Each static html page = a node
  • Each hyperlink = a directed edge
  • Currently ~10 10nodes (mostly junk), 10 11edges

7. Getting started structure at the hyperlink level

  • Measure properties of the link structure of the web.
  • Study a sample of the web that contains a reasonable fraction of the entire web.
  • Apply tools from graph theory to understand the structure.

[Broder, Kumar, Maghoul, Raghavan, Rajagopalan, Stata, Tomkins, Wiener, 2001] 8. Terminology

  • SCC strongly connected component
  • WCC weakly connected component connected component in the underlyingundirectedgraph

9. Data

  • Altavista crawls, up to 500M pages
  • Ran strong and weak connected component algorithms
  • Ran random directed breadth-first searches from 1000 starting nodes, both forwards and backwards along links

10. Breadth-first search from random starts

  • How many vertices are reachable from a random vertex?

11. A Picture of (~200M) pages. 12. Some distance measurements

  • Pr[ ureachable fromv ] ~ 1/4
  • Max distance between 2 SCC nodes: 28
  • Max distance between 2 nodes (if there is a path) > 900
  • Avg distance between 2 SCC nodes: 16

13. Facts (about the crawl).

  • Indegree and Outdegree distributions satisfy the power law. Consistent over time and scale.

The distribution of indegrees on the web is given by a Power Law --- Heavy-tailed distribution, with many high-indegree pages (eg, Yahoo) 14. Analysis of power law Pr [ page haskinlinks ]=~k -2.1 Pr [ page has >kinlinks ]=~1/ k Pr [ page haskoutlinks ]=~k -2.7 Corollary: 15. Component sizes.

  • Component sizes are distributed by the power law.

16. Other observed power laws in the web

  • Depths of URLs
  • Sizes of sites
  • Eigenvalues of adjacency matrix of hyperlink graph [Mihail and Papadimitriou shed some light here]
  • Many different traffic measures
  • Linkage between hosts and domains
  • Many of the above measures on particular subsets of the graph

[Faloutsos, Faloutsos, Faloutsos 99] [Bharat, Chang, Henzinger, Ruhl 02] 17. More Characterization: Self-Similarity 18. Ways to Slice the Web

  • Domain (*.it)
  • Host (www.ibm.com)
  • Geography (pages with a geographical reference in the Western US)
  • Content
    • Keyword: Math, subdivided by Math Geometry
    • Keyword: MP3, subdivided by MP3 Napster

We call these slices Thematically Unified Communities, or TUCs 19. Self-Similarity on the Web

  • Pervasive: holds for all reasonable characteristics
  • Robust: holds for all reasonable slices
  • Theorem:
    • TUCs share properties with the web at large
    • TUCs are linked by a navigational backbone

20. In particular

  • All TUCs have:
    • Power laws for degree, SCC, and WCC distributions
    • Similar exponents for power laws
    • Similar bow tie structure
    • Large number of dense subgraphs

21. Is this surprising?

  • YES (for downsampling general graphs).Example:
  • This graph has 1 SCC containing all nodes
  • Remove any nonzero fraction of edges graph hasncomponents of size 1
  • Generally: random subset of size n 1/2in a graph with O( n ) edges will have only constant number of edges

22. A structural explanation

  • Each TUC has a bow tie how do they relate?

23. The Navigational Backbone Each TUC contains a large SCC that is well-connected to the SCCs of other TUCs 24. Information Extraction from Large Graphs 25. Overview WWW Distill KB1 KB2 KB3 Goal:Create higher-level "knowledge bases" of web information for further processing. [Kumar, Raghavan, Rajagopalan, Tomkins 1999] 26. Many approaches to this problem

  • Databases over the web:
    • Web SQL, Lore, ParaSite, etc
  • Data mining
    • A priori, Query flocks, etc
  • Information foraging
  • Community extraction
    • [Lawrence et al]
  • Authority-based search
    • HITS, and variants

27. General approach

  • Its hard (though getting easier) to analyze the content of all pages on the web
  • Its easier (though still hard) to analyze the graph
  • How successfully can we extract useful semantic knowledge (ie, community structure) from links alone?

28. Web Communities Fishing Outdoor Magazine Bill's Fishing Resources Linux Linux Links LDP Different communities appear to have very different structure. 29. Web Communities Fishing Outdoor Magazine Bill's Fishing Resources Linux Linux Links LDP But both contain a common footprint: two pages () that both Point to three other pages in common () 30. Communities and cores Example K 2,3 Definition:A "core" K ij consists ofileft nodes, jright nodes, and all left->right edges. Critical facts: 1. Almost all communities contain a core [expected] 2. Almost all cores betoken a community [unexpected] 31. Other footprint structures Newsgroup thread Web ring Corporate partnership Intranet fragment 32. Subgraph enumeration

  • Goal:Given a graph-theoretic "footprint" for structures of interest, find ALL occurrences of these footprints.

33. Enumerating cores a a belongs to a K 2,3 if and only if some node points to b1, b2, b3. b2 b1 b3 Inclusion/Exclusion Pruning Clean data by removing: mirrors (true and approximate) empty pages, too-popular pages, nepotistic pages Preprocessing When no more pruning is possible, finish using database techniques Postprocessing 34. Results for cores 3 5 7 9 0 20 40 60 80 100 Thousands i=3 i=4 i=5 i=6 Number of cores found by Elimination/Generation 3 5 7 9 0 20 40 60 80 Thousands i=3 i=4 Number of cores found during postprocessing 35. The cores are interesting (1) Implicit communities are defined by cores. (2) There are an order ofmagnitude more of these.(10 5+ ) (3) Can grow the core to the community using further processing. Explicit communities.

  • Yahoo!, Excite, Infoseek
  • webrings
  • news groups
  • mailing lists

Implicit communities

  • japanese elementary schools
  • turkish student associations
  • oil spills off the coast of japan
  • australian fire brigades

36. Elementary Schools in Japan

  • The American School in Japan
  • The Link Page
  • scwZz[y[W
  • Kids' Space
  • swZ
  • {wwZ
  • KEIMEI GAKUEN Home Page ( Japanese )
  • Shiranuma Home Page
  • fuzoku-es.fukui-u.ac.jp
  • welcome to Miasa E&J school
  • _ElswZy
  • http://www...p/~m_maru/index.html
  • fukui haruyama-es HomePage
  • Torisu primary school
  • goo
  • Yakumo Elementary,Hokkaido,Japan
  • FUZOKU Home Page
  • Kamishibun Elementary School...
  • schools
  • LINK Page-13
  • {wZ
  • awZz[y[W
  • 100 Schools Home Pages (English)
  • K-12 from Japan 10/...rnet and Education )
  • http://www...iglobe.ne.jp/~IKESAN
  • lfjwZUNPg
  • wZ
  • Koulutus ja oppilaitokset
  • TOYODA HOMEPAGE
  • Education
  • Cay's Homepage(Japanese)
  • ywZz[y[W
  • UNIVERSITY
  • JwZ DRAGON97-TOP
  • wZTNPgz[y[W

37. So

  • Possible to extract order-of-magnitude more communities than currently known.
  • Few (4%) of these appear coincidental.
  • Entirely automatic extraction.
  • Open question:how to use implicit communities?

38. A word on evolution 39. A word on evolution

  • Phenomenon to characterize: A topic in a temporal stream occurs in a burst of activity
  • Model source as multi-state
  • Each state has certain emission properties
  • Traversal between states is controlled by a Markov Model
  • Determine most likely underlying state sequence over time, given observable output

[Kleinberg02] 40. Example Time Ive been thinking about your idea with the asparagus Uh huh I think I see Uh huh Yeah, thats what Im saying So then I said Hey, lets give it a try And anyway she said maybe, okay? Most likely hidden sequence: 0.005 1 2 0.01 State 1: Output rate: very low State 2: Output rate: very high Pr[2] ~ 10 Pr[2] ~ 10 Pr[2] ~ 7 Pr[2] ~ 2 Pr[2] ~ 5 Pr[2] ~ 2 Pr[2] ~ 5 Pr[1] ~ 2 Pr[1] ~ 1 Pr[1] ~ 2 Pr[1] ~ 10 Pr[1] ~ 5 Pr[1] ~ 10 Pr[1] ~ 1 2 2 2 1 1 1 1 41. More bursts

  • Infinite chain of increasingly high-output states
  • Allows hierarchical bursts
  • Example 1: email messages
  • Example 2: conference titles

42. Integrating bursts and graph analysis Wired magazine publishes an article on weblogs that impacts the tech community Newsweek magazine publishes an article that reaches the population at large, responding to emergence, and triggering mainstream adoption [KNRT03] Number of communities identified automatically as exhibiting bursty behavior measure of cohesiveness of the blogspace Number of blog pages that belong to a community Number of blog communities 43. IMA Tutorial (part III): Generative and probabilistic models of data May 5, 2003 Title slide 44. Probabilistic generative models

  • Observation: These distributions have the same form:
    • Fraction of laptops that fail catastrophically during tutorials, by city
    • Fraction of pairs of shoes that spontaneously de-sole during periods of stress, by city
  • Conclusion: The distribution arises because the same stochastic process is at work, and this process can be understood beyond the context of each example

45. Models for Power Laws

  • Power laws arise in many different areas of human endeavor, the hallmark of human activity
  • (they also occur in nature)
  • Can we find the underlying process (processes?) that accounts for this prevalence?

46. An Introduction to the Power Law

  • Definition: a distribution is said to have a power law if Pr[ X >= x ] cx
  • Normally: 0< =W ] has some form
    • Number of words with count >=Whas some form
    • The frequency of the word with rankrhas some form
  • The first two forms are clearly identical.
  • What about the third?

51. Equivalence of rank versus value formulation

  • Given: number of words occurringttimes ~t
  • Approach:
    • Consider single most frequent word, with countT
    • Characterize word occurringttimes in terms ofT
    • Approximate rank of words occurringttimes by counting number of words occurring at each more frequent count.
  • Conclusion: Rank- jword occurs (c j+ d) times (power law)
  • But... high ranks correspond to low values must keep straight the head and the tail

[Bookstein90, Adamic99] 52. Early modeling work

  • The characterization of power laws is a limiting statement
  • Early modeling work showed approaches that provide the correct form of the tail in the limit
  • Later work introduced the rate of convergence of a process to its limiting distribution

53. A model of Simon

  • Following Simon [1955], described in terms of word frequences
  • Consider a book being written.Initially, the book contains a single word, the.
  • At timet , the book containstwords.The process of Simon generates thet+1 stword based on the current book.

54. Constructing a book: snapshot at timet When in the course of human events, it becomes necessary Current word frequencies:Letf(i,t)be the number of words of countiat timet Count Word Rank 11,325 4,791 3 2 1 ... ... 5 necessary 1 neccesary ... 300 from 600 of 1000 the 55. The Generative Model

  • Assumptions:
    • Constant probability that a neologism will be introduced at any timestep
    • Probability of re-using a word of countiis proportional toif(i,t) , that is, number of occurrences of countiwords.
  • Algorithm:
    • With probabilitya new word is introduced into the text
    • With remaining probability, a word with countiis introduced with probability proportional toif(i,t)

56. Constructing a book: snapshot at timet Current word frequencies:Letf(i,t)be the number of words of countiat timet Pr[the] = (1- ) 1000 / K Pr[of] = (1- ) 600 / K Pr[some count-1 word] = (1- ) 1 *f(1,t)/ K K = if(i,t) Count Word Rank 11,325 4,791 3 2 1 ... ... 5 necessary 1 neccesary ... 300 from 600 of 1000 the 57. Whats going on? One unique word (which occurs 1 or more times) 1 2 3 4 5 6 Each word in bucketioccursitimes in the current document . 58. Whats going on? 1 With probabilitya new word is introduced into the text 2 3 4 5 6 59. Whats going on? 1 4 How many times do words in this bucket occur? With probability 1- an existing word is reused 2 3 5 6 60. Whats going on? 2 3 4 Size of bucket 3 at timet+1depends only on sizes of buckets 2 and 3 at timet ? ? Must show: fraction of balls in 3 rdbucket approaches some limiting value 61. Models for power laws in the web graph

  • Retelling the Simon model: preferential attachment
    • Barabasi et al
    • Kumar et al
  • Other models for the web graph:
    • [Aiello, Chung, Lu], [Huberman et al]

62. Why create such a model?

  • Evaluate algorithms and heuristics
  • Get insight into page creation
  • Estimate hard-to-sample parameters
  • Help understand web structure
  • Cost modeling for query optimization
  • To find surprises means we must understand what is typical .

63. Random graph models G(n,p) Web indeg > 1000 k23's 4-cliques 0 0 0 100000 125000 many Traditional random graphs [Bollobas 85] are not like the web! Is there a better model? 64. Desiderata for a graph model

  • Succinct description
  • Insight into page creation
  • No a priori set of "topics", but...
  • ... topics should emerge naturally
  • Reflect structural phenomena
  • Dynamic page arrivals
  • Should mirror web's "rich get richer" property, and manifest link correlation.

65. Page creation on the web

  • Some page creators will link to other sites without regard to existing topics, but
  • Most page creators will be drawn to pages covering existing topics they care about, and will link to pages within these topics

Model idea:new pages add links by "copying" them from existing pages 66. Generally, would require

  • Separate processes for:
    • Node creation
    • Node deletion
    • Edge creation
    • Edge deletion

67. A specific model

  • Nodes are created in a sequence of discrete time steps
    • e.g. at each time step, a new node is created withd 1) out-links
  • Probabilistic copying
      • links go to random nodes with probability
      • copyd links from a random node with probability 1-

68. Example New node arrives With probability , it links to a uniformly-chosen page 69. Example To copy, it first chooses a page uniformly Then chooses a uniform out-edge from that page Then links to the destination of that edge ("copies" the edge) Under copying, your rate of getting new inlinks is proportional to your in-degree. With probability (1- ), it decides to copy a link. 70. Degree sequences in this model Pr[page haskinlinks]=~k Heavy-tailed inverse polynomial degree sequences. Pages like netscape and yahoo exist. Many cores, cliques, and other dense subgraphs ( = 1/11 matches web) -(2- ) (1- ) 71. Model extensions

  • Component size distributions.
  • More complex copying.
  • Tighter lower tail bounds.
  • More structure results.

72. A model of Mandelbrot

  • Key idea: Generate frequencies of English words to maximize information transferred per unit cost
  • Approach:
    • Say wordioccurs with probabilityp(i)
    • Set the transmission cost of wordito be log( i)
    • Average information per word: p(i) log(p(i))
    • Cost of a word with probabilityp(j): log (j)
    • Average cost per word: p(j) log(j)
    • Choose probabilitiesp(i)to maximize information/cost
  • Result:p(j) = c j

73. Discussion of Mandelbrots model

  • Trade-offs between communication cost ( log(p(j) ) and information.
  • Are there other tradeoff-based models that drive similar properties?

74. Heuristically Optimized Trade-offs

  • Goal: construction of trees (note: models to generate trees with power law behavior were first proposed in [Yule26])
  • Idea: New nodes must trade off connecting to nearby nodes, and connecting to central nodes.
  • Model:
    • Points arrive uniformly within the unit square
    • New point arrives, and computes two measures for candidate connection pointsj
      • d(j) : distance from new node to existing nodej(nearness)
      • h(j) : distance from nodejto root of tree (centrality)
    • New destination chosen to minimize d(j) + h(j)
  • Result: for a wide variety of values of , distribution of degrees has a power law

[Fabrikant, Koutsoupias, Papadimitriou 2002] 75. Monkeys on Typewriters

  • Consider a creation model divorced form concerns of information and cost
  • Model:
    • Monkey types randomly, hits space bar with probabilityq , character chosen uniformly with remaining probability
  • Result:
    • Rankjword occurs with probabilityqj log(1-q)-1= c j

76. Other Distributions

  • Power law means a clean characterization of a particular property on distribution upper tails
  • Often used to mean heavy tailed, meaning bounded away from an exponentially decaying distribution
  • There are other forms of heavy-tailed distributions
  • A commonly-occurring example: lognormal distribution

77. Quick characterization of lognormal distributions

  • Let X be a normally-distributed random variable
  • Let Y = ln X
  • Then Y is lognormal
  • Properties:
    • Often occur in situations of multiplicative growth
    • Prop2
  • Concern: There is a growing sequence of papers dating back several decades questioning whether certain observed values are best described by power law or lognormal (or other) distributions.

78. One final direction

  • The Central Limit Theorem tells us how sums of independent random variables behave in the limit
  • Example: lnX j= lnX 0+ lnF j
  • X j well-approximated by a lognormal variable
  • Thus, lognormal variables arise in situations of multiplicative growth
  • Examples in biology, ecology, economics,
  • Example: [Huberman et al]: growth of web sites
  • Similarly: the product The same result applies to the product of lognormal variables
  • Each of these generative models is evolutionary
  • What is the role of time?