Empirical Investigations of WWW Surfing Paths

Jim PitkowUser Interface Research

Xerox Palo Alto Research Center

August 1999

Agenda

• A few characteristics of the Web and clicks• Aggregate click models

– user surfing behaviors– post-hoc hit prediction

• Individual click models– entropy– path prediction

August 1999

1

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1000000

10000000

Aug-92

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Serv

ers

Source: World Wide Web Consortium, Mark Gray, Netcraft Server Survey

Web is big, Web is good

1 new server every 2 seconds7.5 new pages per second

August 1999

Users, sessions, and clicksCurrent Internet Universe Estimate 97.1 millionTime spent/month 7:28:16Number of unique sites visited/month 15Page views/month 313Number of sessions/month 16Page views/session 19Time spent/session 0:28:01Time spent/site 0:31:27Duration of a page viewed 0:01:28

Source Nielsen//NetRatings February 1999

August 1999

Popularity of pages—Zipf• Zipf Distribution:

– frequency is inversely proportionate to rank

• Zipf Law:– slope equals

minus one

August 1999

Enter Exit

Users enter a website at variouspages and begin surfing

Continuing surfers distribute themselves down various paths

Surfers arrive at pages having traveled different paths

After some number of page visits surfers leave the web site

(a)

(b)

(c)

(d)

p1

p3

p2

August 1999

Model of surfing

VL = VL-1 + L

• Where L is the number of clicks and is L varies as independent and identically distributed Gaussian random variables

• Surfing proceeds until the perceived cost is larger

than the discounted expected future value

August 1999

Random walk with a stopping threshold• Two parameter inverse Gaussian distribution

• mean (L) = and variance (L) = 3/

LL

LLP 2

3

3 2exp

2)(

August 1999

Experimental design• Client data

– Georgia Tech (3 weeks August, 1994)– Boston University (1995)

• tens of thousands of requests

• Proxy data– AOL (5 days in December, 1997)

• tens of millions of requests

• Server data– Xerox WWW Site (week during May 1997)

August 1999

0

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0 5 10 15 20 25 30 35 40 45 50Clicks

Freq

uenc

yProbability distribution function

Clicks 1 click/site mode3-4 clicks/site median8-10 clicks/site mean

August 1999

0.00

0.25

0.50

0.75

1.00

0 25 50 75 100Clicks

Prob

abili

ty

Experimentalinverse Gaussian

Cumulative distribution function

*** Experimental

— inverse Gaussian

75% of the distribution accounted for in three clicks

August 1999

Two observations

• Inverse Gaussian distribution has very long tail– expect to see large deviations from average

• Due to asymmetric nature of the Inverse Gaussian distribution, typical behavior does not equal average behavior

August 1999

An interesting derivation

• Up to a constant given by the third term, the probability of finding a group surfing at a given level scales inversely in proportion to its depth

2og

2)(og

23)(og 2

2

lL

LLlLPl

2/3)( LLP

August 1999

Number of surfers at each level

1

10

100

1000

1 10 100 1000

log(Clicks)

log(

Freq

uenc

y)

2/3)( LLP

August 1999

Implications of the Law of Surfing

• Implications on techniques designed to enhance performance– HTTP Keep-Alive, Pre-fetching of content

• Can adapt content based location on curve in a cost sensitive manner– different user modalities (browser, searcher, etc.)– expend different CPU resources for different users

• Web site visitation modeling

August 1999

Spreading Activation

Pump activation into source

Activation spreads through

the network

Activation settles into asymptotic pattern

August 1999

Matrix Formulation

0010000

123456

0 0 0 0 1 00 0 1 0 0 01 1 0 0 1 00 1 0 0 1 00 0 1 1 0 00 1 1 0 0 0

1 2 3 4 5 6

C R

..5

.74311

ASources Network Activation

Content + Usage + Topology Technique

A(t) = C + M A(t - 1)

M = (1 - ) I + RNetworksUser Paths

Text SimilarityTopology

August 1999

Application to hit prediction• Let fL be the fraction of users who, having surfed along

L-1 links, continue to surf to depth L.

• Define the activation value Ni,L as the number of users who are at node i after surfing through L clicks

Lkk

kiLLi NSfN ,,1,

),,1(1),,(1

LFLFfL

August 1999

Predicted versus observed

0

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100

1,000

10,000

100,000

1 10 100 1,000 10,000 100,000

log(Observed Hits)

log(

Pred

icte

d Hi

ts)

August 1999

Proportion of surfers traversing link

Freq

uenc

y

Surfing probabilities

by outlinkdensity

No. Links, Li 2 (df = 6)0 < Li 3 25.213 < Li 6 26.446 < Li 9 172.99

9 < Li 12 59.7012 < Li 3666.30

August 1999

Investigating user paths

• Each user path can be thought of as an ngram and represented as tuples of the form <X1, X2, … Xn> to indicate sequences of page clicks– Distribution of ngrams is the Law of Surfing

• Determine the conditional probability of seeing the next page given a matching prior ngram

),...,|Pr( 1 knnnn XXxX

August 1999

Uncertainty and entropy

• Conditional probabilities are also know as as kth-order Markov approximations/models

• Entropy is the expected (average) uncertainty of the random variable measured in bits– minimal number of bits to encode information– uncertainty in the sequence of letters in languages

August 1999

Mathematics of entropy

Xx

Xx

xpxpxp

xp

XpEXH

)(log)()(

1log)(

)(1log)(

2

2

2

August 1999

Conditional entropy

nn Xx

knknnnnnknnn xXxXXHxpXXXH ),...,|()(),...|( 111

Xx Yy

yxpyxpYXH ),(log),(),( 2

),...,|()|()(),...,( 111211 nnn XXXHXXHXHXXH

Conditional probability

Chaining rule

Joint probabilities

August 1999

Entropy versus ngram length

01

234

567

89

1 2 3 4 5 6 7 8 9 10Order of transition

Bits

2345678910

NgramTotal Information Gain

0

1

2

3

4

5

6

7

8

9

2 3 4 5 6 7 8 9 10Path Length

Tota

l Bits

Gai

ned

August 1999

Path predictions

Pr(PPM) the probability that a penultimate path, <xn-1,…xn-k>, observed in the test data was matched by the same penultimate path in the training data

Pr(Hit|PPM) the probability that page xn is visited, given that <xn-1,…xn-k>, is the penultimate path and the highest probability conditional on that path is p(xn|xn-1,…xn-k )

Pr(Hit) = Pr(Hit|PPM)*Pr(PPM), the probability that the page visited in the test set is the one estimated from the training as the most likely to occur

August 1999

Pr(Path Match)

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0.80

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0 2 4 6 8 10Length of Path Used in Prediction

Prob

2345678910

August 1999

Pr(Hit|Path Match)

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10

Length of Path Used in Prediction

Prob

2345678910

August 1999

Pr(Hit)

0.00

0.05

0.10

0.15

0.20

0.25

0 2 4 6 8 10Length of Path Used in Prediction

Prob

2345678910

August 1999

Principles of path prediction

• Paths are not completely modeled by 1st order Markov approximations

• Path Specificity Principle: longer paths contain more predictive power than lower order paths

• Complexity Reduction: keep models as simple and as small as possible

August 1999

Agenda revisited

• A few characteristics of the Web and clicks• Aggregate click models

– user surfing behaviors– post-hoc hit prediction

• Individual click models– entropy– path prediction

August 1999

Areas of further investigation

• Advance the state of predictive modeling– Move from post-hoc to a-priori prediction of user

interactions on the Web– Test hypothetical models of Web site usage

• Validate existing and new models on more representative data sets

• Understand new and emerging applications– streaming video and audio, mobile, etc.

August 1999

More information

pitkow@parc.xerox.com

http://www.parc.xerox.com/istl/projects/uir/projects/Webology.html