Measurement and modeling of the web and related data sets

| 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 TomkinsIBM Almaden Research CenterMay 5, 2003

2 © 2003 IBM Corporation

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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


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Context

Data Analysis on 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


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Focus Areas

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


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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

CoreShells: 12

3

[Tauro, Palmer, Siganos, Faloutsos, 2001 Global Internet]


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Another view of the web: the hyperlink graph

Each static html page = a node Each hyperlink = a directed edge Currently ~1010 nodes (mostly junk), 1011 edges


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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]


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Terminology

SCC – strongly connected component WCC – “weakly connected component” –

connected component in the underlying undirected graph


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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


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Breadth-first search from random starts

How many vertices are reachable from a random vertex?


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A Picture of (~200M) pages.


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Some distance measurements

Pr[u reachable from v] ~ 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


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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)


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Analysis of power law

Pr [ page has k inlinks ] =~ kk -2.1

Pr [ page has > k inlinks ] =~ 1/kk

Pr [ page has k outlinks ] =~ kk-2.7

Corollary:


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Component sizes.

Component sizes are distributed by the power law.


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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]


More Characterization: Self-Similarity


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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


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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”


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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


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Is this surprising? YES (for downsampling general graphs). Example:

This graph has 1 SCC containing all nodes

Remove any nonzero fraction of edges – graph has n components of size 1

Generally: random subset of size n1/2 in a graph with O(n) edges will have only constant number of edges


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A structural explanation

Each TUC has a “bow tie” – how do they relate?


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The Navigational Backbone

Each TUC contains a large SCC that is well-connected to the SCCs of other TUCs


Information Extraction from Large Graphs


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Overview

WWW Distill

KB1

KB2

KB3

Goal: Create higher-level "knowledge bases"

of web information for further processing.

[Kumar, Raghavan, Rajagopalan, Tomkins 1999]


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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


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General approach

It’s hard (though getting easier) to analyze the content of all pages on the web

It’s easier (though still hard) to analyze the graph How successfully can we extract useful semantic

knowledge (ie, community structure) from links alone?


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Web Communities

FishingOutdoor Magazine

Bill's Fishing Resources

Linux Linux Links

LDP

Different communities appear to have very different structure.


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Web Communities

FishingOutdoor Magazine

Bill's Fishing Resources

Linux Linux Links

LDP

But both contain a common “footprint”: two pages ( ) that bothPoint to three other pages in common ( )


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Communities and cores

Example K2,3

Definition: A "core" Kij consists of i left nodes,j right nodes, and all left->right edges.

Critical facts:1. Almost all communities contain a core [expected]2. Almost all cores betoken a community [unexpected]


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Other footprint structures

Newsgroup threadWeb ring

Corporate partnership Intranet fragment


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Subgraph enumeration

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


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Enumerating cores

a

a belongs to a K2,3 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


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Results for cores

3 5 7 90

20

40

60

80

100

Thousands

i=3

i=4

i=5

i=6

Number of cores found by Elimination/Generation

3 5 7 90

20

40

60

80Thousands

i=3

i=4

Number of cores found during postprocessing


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The cores are interesting

(1) Implicit communities are defined by cores.(2) There are an order of magnitude more of these. (105+)(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


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Elementary Schools in Japan

The American School in Japan The Link Page ‰ªès—§ˆä“c¬ŠwZƒz[ƒƒy[ƒW � � � � � � � Kids' Space ˆÀés—§ˆÀé¼•”¬ŠwZ � � � � � � ‹{é‹³ˆç‘åŠw•‘®¬ŠwZ � � � � KEIMEI GAKUEN Home Page ( Japanese

) Shiranuma Home Page fuzoku-es.fukui-u.ac.jp welcome to Miasa E&J school _“ÞìŒ§E‰¡•ls—§’†ì¼¬ŠwZ‚Ìƒy� � � � � � � � 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 “ú–{‚ÌŠwZ � a‰„¬ŠwZƒz[ƒƒy[ƒW � � � � � � 100 Schools Home Pages (English) K-12 from Japan 10/...rnet and Education ) http://www...iglobe.ne.jp/~IKESAN ‚l‚f‚j¬ŠwZ‚U”N‚P‘g•¨Œê � � ÒŠ—’¬—§ÒŠ—“Œ¬ŠwZ � � � � Koulutus ja oppilaitokset TOYODA HOMEPAGE Education Cay's Homepage(Japanese) –y“ì¬ŠwZ‚Ìƒz[ƒƒy[ƒW � � � � � UNIVERSITY ‰J—³¬ŠwZ DRAGON97-TOP � � Â‰ª¬ŠwZ‚T”N‚P‘gƒz[ƒƒy[ƒW � � � � � � ¶µ°é¼ÂÁ© ¥á¥Ë¥å¡¼ ¥á¥Ë¥å¡¼


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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?


A word on evolution


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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]


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Example

Time

I’ve been thinking about your idea with

the asparagus…

Uh huh I think I see…

Uh huh Yeah, that’s what I’m saying…

So then I said “Hey, let’s give

it a try”

And anyway she said

maybe, okay?

0.0051 2

0.01State 1:Output rate: very low

State 2:Output rate: very high

Pr[1] ~ 1 Pr[1] ~ 10 Pr[1] ~ 5 Pr[1] ~ 10 Pr[1] ~ 2 Pr[1] ~ 1 Pr[1] ~ 2

Pr[2] ~ 5 Pr[2] ~ 2 Pr[2] ~ 5 Pr[2] ~ 2 Pr[2] ~ 7 Pr[2] ~ 10 Pr[2] ~ 10

1 1 1 1 2 2 2

Most likely “hidden” sequence:


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More bursts

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


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Integrating bursts and graph analysis

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

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]


IMA Tutorial (part III):Generative and probabilistic models of data

May 5, 2003


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Probabilistic generative models

Observation: These distributions have the same form:1. Fraction of laptops that fail catastrophically during

tutorials, by city

2. 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


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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?


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An Introduction to the Power Law

Definition: a distribution is said to have a power law if Pr[X >= x] cx

Normally: 0<<=2 (Var(X) infinite)

Sometimes: 0<<=1 (Mean(X) infinite)

Exponentially-decaying distribution

Power law distribution


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Early Observations: Pareto on Income

[Pareto1897] observed that the random variable I denoting the income of an individual has a power law distribution

More strongly, he observed that Pr[X>x] = (x/k)

For density function f, note that ln f(x) = (--1)ln(x) + c for constant c

Thus, in a plot of log(value) versus log(probability), power laws display a linear tail, while Pareto distributions are linear always.


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Early Observations: Yule/Zipf

[Yule26] observed (and explained) power laws in the context of number of species within a genus

[Zipf32] and [Estoup16] studied the relative frequency of words in natural language, beginning a cottage industry that continues to this day.

A “Yule-Zipf” distribution is typically characterized by rank rather than value:• The ith most frequent word in English occurs with

probability proportional to 1/i. This characterization relies on finite vocabulary


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Early Observations: Lotka on Citations

[Lotka25] presented the first occurrence of power laws in the context of graph theory, showing a power law for the indegree of the citation graph


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Ranks versus Values

Commonly encountered phrasings of the power law in the context of word counts:1. Pr[word has count >= W] has some form

2. Number of words with count >= W has some form

3. The frequency of the word with rank r has some form

• The first two forms are clearly identical.• What about the third?


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Equivalence of rank versus value formulation

Given: number of words occurring t times ~ t Approach:

• Consider single most frequent word, with count T

• Characterize word occurring t times in terms of T

• Approximate rank of words occurring t times by counting number of words occurring at each more frequent count.

Conclusion: Rank-j word occurs (cj + d)times (power law)

But... high ranks correspond to low values – must keep straight the “head” and the “tail”

[Bookstein90, Adamic99]


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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


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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 time t, the book contains t words. The process of Simon generates the t+1st word based on the current book.


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Constructing a book: snapshot at time t

When in the course of human events, it becomes necessary…

Current word frequencies:

Rank Word Count

1 “the” 1000

2 “of” 600

3 “from” 300

… “...” …

4,791 “necessary” 5

“...” “...”

11,325 “neccesary” 1

Let f(i,t) be the number of words of count i at time t


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The Generative Model

Assumptions:1. Constant probability that a neologism will be introduced

at any timestep

2. Probability of re-using a word of count i is proportional to if(i,t), that is, number of occurrences of count i words.

Algorithm:• With probability a new word is introduced into the text

• With remaining probability, a word with count i is introduced with probability proportional to if(i,t)


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Constructing a book: snapshot at time t

Current word frequencies:

Rank Word Count

1 “the” 1000

2 “of” 600

3 “from” 300

… “...” …

4,791 “necessary” 5

“...” “...”

11,325 “neccesary” 1

Let f(i,t) be the number of words of count i at time t

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)


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What’s going on?

One unique word (which occurs 1 or more times)

1 2 3 4 5 6

Each word in bucket i occurs i times in the current document

….


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What’s going on?

1 2 3 4 5 6

With probability a new word is introduced into the text


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What’s going on?

1 2 3 4 5 6

How many times do words in this bucket occur?

With probability 1- an existing word is reused


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What’s going on?

2 3 4

Size of bucket 3 at time t+1 depends only on sizes of buckets 2 and 3 at time t

??

Must show: fraction of balls in 3rd bucket approaches some limiting value


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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]


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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.


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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?


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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.


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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


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Generally, would require…

Separate processes for:• Node creation

• Node deletion

• Edge creation

• Edge deletion


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A specific model

Nodes are created in a sequence of discrete time steps• e.g. at each time step, a new node is created with

d1) out-links

Probabilistic copying– links go to random nodes with probability – copy d links from a random node with probability 1-


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Example

New node arrivesWith probability , it linksto a uniformly-chosen page


ITo copy, it first choosesa page uniformlyThen chooses a uniformout-edge from that pageThen links to the destinationof that edge ("copies" the edge)

Under copying, your rate of getting new inlinks is proportional to your in-degree.

Example

With probability (1-), itdecides to copy a link.


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Degree sequences in this model

Pr[page has k inlinks] =~ kk

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-)


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Model extensions

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


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A model of Mandelbrot

Key idea: Generate frequencies of English words to maximize information transferred per unit cost

Approach:• Say word i occurs with probability p(i)

• Set the transmission cost of word i to be log(i)

• Average information per word: –p(i) log(p(i))

• Cost of a word with probability p(j): log (j)

• Average cost per word: p(j) log(j)

• Choose probabilities p(i) to maximize information/cost Result: p(j) = c j


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Discussion of Mandelbrot’s model

Trade-offs between communication cost (log(p(j)) and information.

Are there other tradeoff-based models that drive similar properties?


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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 points j– d(j): distance from new node to existing node j (“nearness”)– h(j): distance from node j to 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]


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Monkeys on Typewriters

Consider a creation model divorced form concerns of information and cost

Model:• Monkey types randomly, hits space bar with probability

q, character chosen uniformly with remaining probability

Result:• Rank j word occurs with probability qjlog(1-q)-1 = c j


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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


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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.


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One final direction…

The Central Limit Theorem tells us how sums of independent random variables behave in the limit

Example: ln Xj = ln X0 + ln Fj Xj 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?

Measurement and modeling of the web and related data sets

Technology

web domain

web sql

web pervasive

entire web

web extraction of information

scc nodes

graph faloutsos

large scc