Transcript lecture slides

Using Graphs in Unstructured
and Semistructured Data Mining
Soumen Chakrabarti
IIT Bombay
www.cse.iitb.ac.in/~soumen
ACFOCS 2004
Chakrabarti
1
Acknowledgments






C. Faloutsos, CMU
W. Cohen, CMU
IBM Almaden (many colleagues)
IIT Bombay (many students)
S. Sarawagi, IIT Bombay
S. Sudarshan, IIT Bombay
ACFOCS 2004
Chakrabarti
2
Graphs are everywhere





Phone network, Internet, Web
Databases, XML, email, blogs
Web of trust (epinion)
Text and language artifacts (WordNet)
Commodity distribution networks
Internet Map
[lumeta.com]
ACFOCS 2004
Food Web
[Martinez1991]
Chakrabarti
Protein Interactions
[genomebiology.com]
3
Why analyze graphs?
 What properties do real-life graphs have?
 How important is a node? What is
importance?
 Who is the best customer to target in a
social network?
 Who spread a raging rumor?
 How similar are two nodes?
 How do nodes influence each other?
 Can I predict some property of a node
based on its neighborhood?
ACFOCS 2004
Chakrabarti
4
Outline, some more detail
 Part 1 (Modeling graphs)
 What do real-life graphs look like?
 What laws govern their formation, evolution and
properties?
 What structural analyses are useful?
 Part 2 (Analyzing graphs)




ACFOCS 2004
Modeling data analysis problems using graphs
Proposing parametric models
Estimating parameters
Applications from Web search and text mining
Chakrabarti
5
Modeling and generating
realistic graphs
ACFOCS 2004
Chakrabarti
6
Questions
 What do real graphs look like?
 Edges, communities, clustering effects
 What properties of nodes, edges are
important to model?
 Degree, paths, cycles, …
 What local and global properties are
important to measure?
 How to artificially generate realistic graphs?
ACFOCS 2004
Chakrabarti
7
Modeling: why care?
 Algorithm design
 Can skewed degree distribution make our
algorithm faster?
 Extrapolation
 How well will Pagerank work on the Web 10
years from now?
 Sampling
 Make sure scaled-down algorithm shows same
performance/behavior on large-scale data
 Deviation detection
 Is this page trying to spam the search engine?
ACFOCS 2004
Chakrabarti
8
Laws – degree distributions
 Q: avg degree is ~10 - what is the most
probable degree?
count
??
10
ACFOCS 2004
degree
Chakrabarti
9
Laws – degree distributions
 Q: avg degree is ~10 - what is the most
probable degree?
count
??
10
ACFOCS 2004
count
degree
Chakrabarti
10
degree
10
Power-law: outdegree O
Frequency
Exponent = slope
O = -2.15
-2.15
Nov’97
Outdegree
The plot is linear in log-log scale [FFF’99]
freq = degree (-2.15)
ACFOCS 2004
Chakrabarti
11
Power-law: rank R
outdegree
Exponent = slope
R = -0.74
R
Dec’98
Rank: nodes in decreasing outdegree order
 The plot is a line in log-log scale
ACFOCS 2004
Chakrabarti
12
Eigenvalues
 Let A be the adjacency matrix of graph
 The eigenvalue  satisfies
 A v =  v,
where v is some vector
 Eigenvalues are strongly related to graph
topology
A
B
C
A
B
A
B
C
D
1
1
1
1
C
1
D
1
D
ACFOCS 2004
Chakrabarti
13
Power-law: eigenvalues of E
 Eigenvalues in decreasing order
Eigenvalue
Exponent = slope
E = -0.48
Dec’98
Rank of decreasing eigenvalue
ACFOCS 2004
Chakrabarti
14
The Node Neighborhood
 N(h) = # of pairs of nodes within h hops
 Let average degree = 3
 How many neighbors should I expect within
1,2,… h hops?
 Potential answer:




ACFOCS 2004
1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
Chakrabarti
15
The Node Neighborhood
 N(h) = # of pairs of nodes within h hops
 Let average degree = 3
 How many neighbors should I expect within
1,2,… h hops?
 Potential answer:




ACFOCS 2004
1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
WE HAVE DUPLICATES!
Chakrabarti
16
The Node Neighborhood
 N(h) = # of pairs of nodes within h hops
 Let average degree = 3
 How many neighbors should I expect within
1,2,… h hops?
 Potential answer:




ACFOCS 2004
1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
‘avg’ degree: meaningless!
Chakrabarti
17
Power-law: hop-plot H
# of Pairs
# of Pairs
H = 4.86
Hops
Dec 98
H = 2.83
Hops
Router level ’95
Pairs of nodes as a function of hops N(h)= hH
ACFOCS 2004
Chakrabarti
18
Observation
 Q: Intuition behind ‘hop exponent’?
 A: ‘intrinsic=fractal dimensionality’ of the
network
...
N(h) ~ h1
ACFOCS 2004
N(h) ~ h2
Chakrabarti
19
Any other ‘laws’?
 The Web looks like a “bow-tie”
[Kumar+1999]
 IN, SCC, OUT, ‘tendrils’
 Disconnected components
ACFOCS 2004
Chakrabarti
20
Generators
 How to generate graphs from a realistic
distribution?
 Difficulty: simultaneously preserving many
local and global properties seen in realistic
graphs
 Erdos-Renyi: switch on each edge
independently with some probability
 Problem: degree distribution not power-law
 Degree-based
 Process-based (“preferential attachment”)
ACFOCS 2004
Chakrabarti
21
Degree-based generator
 Fix the degree distribution (e.g., ‘Zipf’)
 Assign degrees to nodes
 Add matching edges to satisfy degrees
 No direct control over
other properties
 “ACL model”
[AielloCL2000]
ACFOCS 2004
Chakrabarti
22
Process-based: Preferential attachment






Start with a clique with m nodes
Add one node v at every time step
v makes m links to old nodes
Suppose old node u has degree d(u)
Let pu = d(u)/ wd(w)
v invokes a multinomial distribution defined
by the set of p’s
 And links to whichever u’s show up
 At time t, there are m+t nodes, mt links
 What is the degree distribution?
ACFOCS 2004
Chakrabarti
23
Preferential attachment: analysis




ki(t) = degree of node i at time t
Discrete random variable
Approximate as continuous random variable
Let i(t) = E(ki(t)), expectation over random
linking choices
 At time t, the infinitesimal expected growth
rate of i(t) is, by linearity of expectation,
ti
i (t )  m
 i (t )
 i (t )  i (t )

 m

t
t
2mt
2t
m degrees to add Total degree at t
ACFOCS 2004
Time at which node i was born
Chakrabarti
24
Preferential attachment, continued
 Expected degree of each node grows as
square-root of age:  i (t )  m ti
t
 Let the current time be t
 A node must be old enough for its
degree to
i
t  m 2t k 2
be large; for i(t) > k, we need
 Therefore, the fraction
(m  t )of
k 2 nodes with
k 2 degree
t


larger than k is


mt
m
2
2
 Pr(degree = k)  const/k3 (data closer to 2)
ACFOCS 2004
Chakrabarti
25
Bipartite cores
 Basic preferential attachment does not
 Explain dense/complete bipartite cores (100,000’s in a
O(20 million)-page crawl)
 Account for influence of search engines
2:3 core
(n:m core)
Number of cores
 The story isn’t over yet
n=2
n=7
log m
ACFOCS 2004
Chakrabarti
26
Other process-based generators
 “Copying model” [KumarRRTU2000]




New node v picks old reference node r u.a.r.
v adds k new links to old nodes; for ith link:
W.p. a add a link to an old node picked u.a.r.
W.p. 1–a copy the ith link from r
 More difficult to analyze
 Reference node  compression techniques!
 H.O.T.: connect to closest, high-connectivity
neighbor [Fabrikant+2002]
 Winner does not take all [Pennock+2002]
ACFOCS 2004
Chakrabarti
27
Reference-based graph compression
 Well-motivated: pack graph into limited fast
memory for, e.g., query-time Web analysis
 Standard approach
 Assign integer IDs to URLs, lexicographic order
 Delta or Gamma encoding of outlink IDs
 If link-copying is rampant, should be able to
compress Outlinks(u) by recording
 A reference node r
 Outlinks(r)  Outlinks(u) …the “correction”
 Finding r : what’s optimal? practical? [Adler,
Mitzenmacher, Boldi, Vigna 2002—2004]
ACFOCS 2004
Chakrabarti
28
Reference-based compression, cont’d
 r is a candidate reference for u if
Outlinks(r)Outlinks(u) is “large enough”
 Given G, construct G’ in which…
 Directed edge from r to u with edge cost =
number of bits needed to write down
Outlinks(u)  Outlinks(r)
 Dummy node z, z has no outlinks in G
 z connected to each u in G’
 cost(z,u) = #bits to write Outlinks(u) w/o ref
 Shortest path tree rooted at z
 In practice, pick “recent” r… 2.58 bits/link
ACFOCS 2004
Chakrabarti
29
log(count)
Summary: Power laws are everywhere
J. Ullman
log(#citations)





log(freq)
“a”
“the
”
log(rank)
Bible: rank vs. word frequency
Length of file transfers [Bestavros+]
Web hit counts [Huberman]
Click-stream data [Montgomery+01]
Lotka’s law of publication count (CiteSeer data)
ACFOCS 2004
Chakrabarti
30
Resources
 Generators
 R-MAT [email protected]
 BRITE www.cs.bu.edu/brite/
 INET topology.eecs.umich.edu/inet
 Visualization tools
 Graphviz www.graphviz.org
 Pajek vlado.fmf.uni-lj.si/pub/networks/pajek
 Kevin Bacon web site
www.cs.virginia.edu/oracle
 Erdös numbers etc.
ACFOCS 2004
Chakrabarti
31
R-MAT: Recursive MATrix generator
 Goals
 Power-law in- and outdegrees
 Power-law eigenvalues
 Small diameter (“six
degrees of separation”)
 Simple, few parameters
From
 Approach
a
(0.5)
b
(0.1)
c
(0.15)
d
(0.25)
2n
 Subdivide the adjacency
matrix
 Choose a quadrant with
probability (a,b,c,d)
To
2n
ACFOCS 2004
Chakrabarti
32
R-MAT algorithm, cont’d
b
c
d
a
 By construction
 Rich gets richer for inand out-degree
 Self-similar
(communities within
communities)
 Small diameter
ACFOCS 2004
a
2n
 Subdivide the adjacency
matrix
 Choose a quadrant with
probability (a,b,c,d)
 Recurse till we reach a
11 cell
d
c
2n
Chakrabarti
33
Evaluation on clickstream data
Count vs Indegree
Count vs Outdegree
Hop-plot
Singular value vs
Rank
Left “Network value” Right “Network value”
R-MAT matches it well
ACFOCS 2004
Chakrabarti
34
Topic structure of the Web
 Measure correlations between link proximity
and content similarity
 How to characterize “topics”?
Test doc
 Started with http://dmoz.org
 Keep pruning until all leaf topics
Classifier
have enough (>300) samples
Topic
Prob
 Approx 120k sample URLs
Arts
0.1
Computers
0.3
 Flatten to approx 482 topics
Science
0.6
 Train a text classifier
 Characterize new document d as a vector of
probabilities pd = (Pr(c|d) c)
ACFOCS 2004
Chakrabarti
35
Sampling the background topic distrib.
 What fraction of Web pages are about
/Health/Diabetes?
 How to sample the Web?
 Invoke the “random surfer” model
(Pagerank)
 Walk from node to node
 Sample trail adjusting for Pagerank
 Modify Web graph to do better sampling
 Self loops
 Bidirectional edges
ACFOCS 2004
Chakrabarti
36
Convergence
Stride=30k
Stride=75k
Distribution
difference
0.4
Background distribution
0.3
0.2
0.1
Sports
Society
Shopping
Science
Reference
Recreation
Home
Health
1000
Games
500
Computers
Arts
0
Business
1
0.8
0.6
0.4
0.2
0
#hops
0
 Start from pairs of diverse topics
 Two random walks, sample from each walk
 Measure distance between topic distributions
 L1 distance |p1 – p2| = c|p1(c) – p2(c)| in [0,2]
 Below .05 —.2 within 300—400 physical pages
ACFOCS 2004
Chakrabarti
37
Biases in topic directories
 Use Dmoz to train a
classifier
 Sample the Web
 Classify samples
 Diff Dmoz topic distribution
from Web sample topic
distribution
 Report maximum deviation
in fractions
 NOTE: Not exactly Dmoz
ACFOCS 2004
Chakrabarti
Dmoz over-represents
Games.Video_Games
Society.People
Arts.Celebrities
...Education.Colleges
...Travel.Reservations
Dmoz under-represents
…WWW…Directories!
Sports.Hockey
Society.Philosophy
Education…K12…
Recreation…Camping
38
Topic-specific degree distribution
 Preferential attachment:
connect u to v w.p.
proportional to the
degree of v, regardless
of topic
 More realistic: u has a
topic, and links to v with
related topics
 Unclear if power-law
should be upheld
Intra-topic
linkage
Inter-topic
linkage
ACFOCS 2004
Chakrabarti
39
Random forward walk without jumps
/Arts/Music
/Sports/Soccer
1.4
1
1.2
L_1 Distance
L_1 Distance
1.2
0.8
0.6
1
0.8
From background
From hop0
0.4
0.6
0.2
From background
From hop0
0.4
0
5
10
15
Wander hops
20
0
5
10
15
Wander hops
20
 Sampling walk is designed to mix topics well
 How about walking forward without jumping?
 Start from a page u0 on a specific topic
 Forward random walk (u0, u1, …, ui, …)
 Compare (Pr(c|ui) c) with (Pr(c|u0) c) and with the
background distribution
ACFOCS 2004
Chakrabarti
40
Observations and implications
 Forward walks wander away from
starting topic slowly
 But do not converge to the
background distribution
 Global PageRank ok also
for topic-specific queries
W.p. d jump to
a random node
 Jump parameter d=.1—.2
 Topic drift not too bad within
path length of 5—10
 Prestige conferred mostly by
same-topic neighbors
 Also explains why focused crawling works
ACFOCS 2004
Chakrabarti
W.p. (1-d)
jump to an
out-neighbor
u.a.r.
Jump
Highprestige
node
41
Citation matrix
 Given a page is about topic i, how likely is it
to link to topic j?
 Matrix C[i,j] = probability that page about topic i
links to page about topic j
 Soft counting: C[i,j] += Pr(i|u)Pr(j|v)
u
 Applications
v
 Classifying Web pages into topics
 Focused crawling for topic-specific pages
 Finding relations between topics in a directory
ACFOCS 2004
Chakrabarti
42
Citation, confusion, correction
From topic
True topic
From topic
To topic 
Guessed topic 
ACFOCS 2004
Arts
Business
Computers
Games
Health
Home
Recreation
Reference
Science
Shopping
Society
Sports
To topic 
Classifier’s confusion
on held-out documents
can be used to correct
confusion matrix
Chakrabarti
43
Fine-grained views of citation
Prominent off-diagonal
entries raise design
issues for taxonomy
editors and maintainers
Clear block-structure derived
from coarse-grain topics
Strong diagonals reflect
tightly-knit topic communities
ACFOCS 2004
Chakrabarti
44
Outline, some more detail
 Part 1 (Modeling graphs)
 What do real-life graphs look like?
 What laws govern their formation, evolution and
properties?
 What structural analyses are useful?
 Part 2 (Analyzing graphs)




ACFOCS 2004
Modeling data analysis problems using graphs
Proposing parametric models
Estimating parameters
Applications from Web search and text mining
Chakrabarti
45
Centrality and prestige
ACFOCS 2004
Chakrabarti
46
How important is a node?








Degree, min-max radius, …
Pagerank
Maximum entropy network flows
HITS and stochastic variants
Stability and susceptibility to spamming
Hypergraphs and nonlinear systems
Using other hypertext properties
Applications: Ranking, crawling, clustering,
detecting obsolete pages
ACFOCS 2004
Chakrabarti
47
Importance/prestige as Pagerank
 A node is important if it is connected to
important nodes
 “Random surfer” walks along links for ever
 If current node has 3 outlinks, take each
with probability 1/3
 Importance = steady-state
u
probability (ssp) of visit
v
OutDegree(u)=3
 “Maxwell’s equation
for the Web”
PR (u )
PR (v )  
(u ,v )E OutDegree(u )
ACFOCS 2004
Chakrabarti
48
(Simplified) Pagerank algorithm




Let A be the node adjacency matrix
Column normalize AT
Want vector p such that AT p = p
I.e. p is the eigenvector corresponding to the
largest eigenvalue, which is 1
From
To
2
1
4
ACFOCS 2004
5
3
AT
1
1
1
1/2
1/2
Chakrabarti
p1
p1
p2
p2
=
1/2
p3
1/2
p4
p4
p5
p5
p3
49
Intuition
 A as vector transformation
x’
A
2
1
2
1 =
x
1
3
x’
1
0
x
1
3
2
1
ACFOCS 2004
Chakrabarti
50
Intuition
 By defn., eigenvectors remain parallel to
themselves (‘fixed points’)
1
v1
0.52
3.62 * 0.85 =
ACFOCS 2004
A
2
1
v1
1
3
0.52
0.85
Chakrabarti
51
Convergence
 Usually, fast:
 depends on ratio
2
1 : 2
ACFOCS 2004
1
Chakrabarti
52
Eigenvalues and epidemic networks
 Will a virus spread across an arbitrary
network create an epidemic?
 Susceptible-Infected-Susceptible (SIS)




Was healthy, did not got infected
Was infected, got cured without further attack
Was infected, got cured immediately after attack
Cured nodes immediately become susceptible
Infected by neighbor
Susceptible/
healthy
ACFOCS 2004
Cured
internally
Chakrabarti
Infected
&
infectious
53
The infection model
 (virus) Birth rate b: probability than an
infected neighbor attacks
 (virus) Death rate d: probability that an
infected node heals
Healthy
Prob. d
N2
Prob. β
N1
N
Infected
ACFOCS 2004
N3
Chakrabarti
54
Working parameters
 pi,t = prob. node i is infected at time t
 i ,t   p j ,t 1 (1  b )  (1  p j ,t 1 )   1  p j ,t 1b


( i , j )E


( i , j )E
Prob. that node i does NOT receive an infection from neighbors


 Was healthy, did not got infected 1  pi ,t 1  i ,t
 Was infected, got cured without further
attack pi ,t 1d i ,t
 Was infected, got attacks from neighbors
and yet cured itself in this time step
1
…
somewhat
arbitrary
p
1

d
i
,
t

1
i
,
t
2

ACFOCS 2004

Chakrabarti
55
Time recurrence for pi,t
 Assuming various probabilities are “suitably
small”  
1 p b  1 b
p
i ,t

j ,t 1


( i , j )E
( i , j )E
j ,t 1
 Recurrence of the probability of infection
can be approximated by this linear form:
pi ,t  1  d  pi ,t 1  b
p
( i , j )E
j ,t 1
 In other words, the (symmetric) transition
matrix is
S  1  d I  bA
ACFOCS 2004
Chakrabarti
56
Epidemic threshold
 Virus ‘strength’ s = b/d
Prob. δ
Healthy
N2
Prob. β
N1
N
Infected
N3
 Epidemic threshold of a graph is defined as
the value of t, such that if strength s = b / d
< t, then an epidemic can not happen
 Problem: compute epidemic threshold
ACFOCS 2004
Chakrabarti
57
Epidemic threshold t
What should t depend on?
 avg. degree? and/or highest degree?
 and/or variance of degree?
 and/or third moment of degree?
ACFOCS 2004
Chakrabarti
58
Analysis
 Eigenvectors of S and A are the same
 Eigenvalues are shifted and scaled
i ,S  1  d  bi , A
 From spectral decomposition
p
( n)
S p
n
( 0)
 i  u u
n
T
i ,S i ,S i ,S
 A sufficient condition for infection dying
down is that   1  d  b  1
1,S
1, A
or b1,A  d or b / d  1, A
ACFOCS 2004
Chakrabarti
59
Epidemic threshold
 An epidemic must die down if
epidemic threshold
recovery prob.
β/δ <τ = 1/ λ1,A
attack prob.
largest eigenvalue
of adj. matrix A
Proof: [Wang+03]
ACFOCS 2004
Chakrabarti
60
Experiments
Number of Infected Nodes
500
Oregon
β = 0.001
b/d > τ
(above threshold)
400
300
200
b/d = τ
(at the threshold)
100
0
0
250
500
750
Time
δ:
ACFOCS 2004
0.05
0.06
1000
b/d < τ
(below threshold)
0.07
Chakrabarti
61
Remarks
 “Primal” problem: topology design
 Design a network resilient to infection
 “Dual” problem: viral marketing
 Selectively convert important customers who
can influence many others
 Will come back to this later
ACFOCS 2004
Chakrabarti
62
Back to the random surfer
 Practical issues with Pagerank [BrinP1997]
 PR converges only if E is aperiodic and
irreducible; make it so:
d
PR (u )
PR (v )   (1  d ) 
N
(u ,v )E OutDegree(u )
 d is the (tuned) probability of “teleporting” to
one of N pages uniformly at random (0.1—
0.2 in practice)
 (Possibly) unintended consequences: topic
sensitivity, stability
ACFOCS 2004
Chakrabarti
63
Prestige as network flow





yij =#surfers clicking from i to j per unit time
Hits per unit time on page j is H j  
y ij
( i , j )E
Flow is conserved at j :
(i , j )E y ij  ( j ,k )E y jk
The total traffic is Y 
 j H j  i , j y ij
Normalize: p ij  y ij /Y
Can interpret pij as a probability
 Standard Pagerank corresponds to one
solution: p ij  H i (Y OutDegree(i ))
 Many other solutions possible
ACFOCS 2004
Chakrabarti
64
Maximum entropy flow [Tomlin2003]
 Flow conservation modeled using feature
 1 j  r , (i ,r )  E

r  1,,N : f r ( x ij )   1 i  r , (r , j )  E
0
otherwise

 And the constraints 0  E (f r ( x ij ))  i , j p ij f r ( x ij )
 Goal is to maximize  i , j p ij log p ij
subject to i , j p ij  1
 Solution has form p ij  exp( 0  i   j )
 i is the “hotness” of page i
ACFOCS 2004
Chakrabarti
65
Maxent flow results
i ranking is better
than Pagerank; Hi
ranking is worse
Two IBM
intranet
data sets
with known
top URLs
Average
rank (106)
of known
top URLs
when
sorted by
Pagerank
Depth up to which
dmoz.org URLs are
used as ground truth
Hi
i
(Smaller rank is better)
Average rank (108)
ACFOCS 2004
Chakrabarti
66
HITS [Kleinberg1997]
 Two kinds of prestige
 Good hubs link to good authorities
 Good authorities are linked to by good hubs
a (v )  (u ,v )E h (u ); h (u )  (u ,v )E a (v )
 In matrix notation, iterations amount to
a  ET h
h  Ea
i.e., a  ET Ea
with interleaved normalization of h and a
 Note that scores are copied not divided
ACFOCS 2004
Chakrabarti
67
HITS graph acquisition
 Steps
 Get root set via keyword query
 Expand by one move forward and backward
 Drop same-site links (why?)
 Each node assigned both a
hub and an auth score
 Graph tends to be “topicspecific”
IN
Root set
 Whereas Pagerank is usually run on “the
entire Web” (but need not be)
ACFOCS 2004
Chakrabarti
OUT
68
HITS vs. Pagerank, more comments
 Dominant eigenvectors of different matrices
 HITS: Eigensystems of EET (h) and ETE (a)
 Pagerank: eigensystem of dJ  (1  d )LT
where
1/ N
J   
1/ N
 1/ N 
1/OutDegree(i ) (i , j )  E


  and L(i , j )  
0
otherwise

 1/ N 
 HITS copies scores, Pagerank distributes
 HITS drops same-site links, Pagerank does
(can?) not
 Implications?
ACFOCS 2004
Chakrabarti
69
HITS: Dyadic interpretation [CohnC2000]
 Graph includes many communities z
 Query=“Jaguar” gets auto, game, animal links
 Each URL is represented as two things
 A document d
 A citation c
 Max 
Pr(d ,c )
(d ,c )E


d c
d c
( , )E
( , )E
Pr(d ) Pr(c | d )
Pr(d )z Pr( z | d ) Pr(c | z )
 Guess number of aspects zs and use
[Hofmann 1999] to estimate Pr(c|z)
 These are the most authoritative URLs
ACFOCS 2004
Chakrabarti
70
Dyadic results for “Machine learning”
Clustering based on citations + ranking within clusters
ACFOCS 2004
Chakrabarti
71
Spamming link-based ranking
 Recipe for spamming HITS
 Create a hub linking to genuine authorities
 Then mix in links to your customers’ sites
 Highly susceptible to adversarial behavior
 Recipe for spamming Pagerank
 Buy a bunch of domains, cloak IP addresses
 Host a site at each domain
 Sprinkle a few links at random per page to other
sites you own
 Takes more work than spamming HITS
ACFOCS 2004
Chakrabarti
72
 Why?
 How to design more stable
algorithms?
ACFOCS 2004
Chakrabarti
HITS Authority
 Compute HITS authority
scores and Pagerank
 Delete 30% of nodes/links at
random
 Recompute and compare
ranks; repeat
 Pagerank ranks more stable
than HITS authority ranks
1
2
3
4
5
6
10
8
Pagerank
Stability of link analysis [NgZJ2001]
1
2
3
4
5
6
7
8
3
1
1
5
3
3
12
6
6
52 20 23
171 119 99
135 56 40
179 159 100
316 141 170
1
2
5
3
6
4
7
8
1
2
6
5
3
4
7
8
1
2
4
5
6
3
7
8
1
2
3
4
5
8
7
6
1
2
5
4
3
6
7
9
73
Stability depends on graph and params
 Auth score is eigenvector for ETE = S, say
 Let 1 > 2 be the first two eigenvalues
 There exists an S’ such that
 S and S’ are close ||S–S’||F = O(1 –2)
 But ||u1 – u’1||2 = (1)
 Pagerank p is eigenvector of (U  (1   ) E )T
 U is a matrix full of 1/N and  is the jump prob
 If set C of nodes are changed in any way, the
new Pagerank vector p’ satisfies


p' p 2  2uC pu / 
ACFOCS 2004
Chakrabarti
74

T
(t )
a
  1  (1   ) Erow h

( t 1)
(t )
h
  1  (1   ) Ecol a
( t 1)
 Much more stable than HITS
 Results more meaningful
  near 1 will always stabilize
 Here  was 0.2
ACFOCS 2004
Chakrabarti
Randomized HITS
 Each half-step, with
probability , teleport to a
node chosen uniformly at
random
1
4
2
3
5
6
7
8
3
1
2
4
6
5
7
8
3
1
2
4
6
5
7
8
2
1
3
4
6
5
7
8
1
2
4
3
5
6
7
8
Pagerank
Randomized HITS
1
3
2
4
5
6
7
8
1
2
3
4
6
7
5
9
1
2
3
4
7
6
5
9
1
2
3
4
5
6
7
9
2
1
3
4
5
6
7
11
75
Another random walk variation of HITS
 SALSA: Stochastic HITS [Lempel+2000]
 Two separate random walks
1/3
a1
 From authority to authority via hub 1/3
 From hub to hub via authority
1/3
 Transition probability Pr(aiaj) =
1
1
h :(h ,ai ),(h ,a j )E InDegree(a ) OutDegree(h )
i
1/2
a2
1/2
 If transition graph is irreducible,  a  InDegree(a )
 For disconnected components, depends on
relative size of bipartite cores
 Avoids dominance of larger cores
ACFOCS 2004
Chakrabarti
76
SALSA sample result (“movies”)
HITS: The Tightly-Knit Community (TKC) effect
SALSA: Less TKC influence (but no reinforcement!)
ACFOCS 2004
Chakrabarti
77
Links in relational data [GibsonKR1998]
 (Attribute, value) pair is a node
 Each node v has weight wv
 Each tuple is a hyperedge
 Tuple r has weight xr
 HITS-like iterations to update
weight wv
 For each tuple
r  (v, u1 ,, uk )
xr  ( wu1 ,  , wuk )
 Update weight
wv  r xr
 Combining operator  can be
sum, max, product, Lp avg, etc.
ACFOCS 2004
Chakrabarti
78
Database
Theory
Distilling links in relational data
ACFOCS 2004
Author
Author
Chakrabarti
Forum
Year
79
Searching and annotating graph data
ACFOCS 2004
Chakrabarti
80
Searching graph data
 Nodes in graph contain text
 RandomIntelligent surfer [RichardsonD2001]
 Topic-sensitive Pagerank [Haveliwala2002]
 Assigning image captions using random walks
[PanYFD2004]
 Rotting pages and links [BarYossefBKT2004]
 Query is a set of keywords
 All keywords may not match a single node
 Implicit joins [Hulgeri+2001, Agrawal+2002]
 Or rank aggregation [Balmin+2004] required
ACFOCS 2004
Chakrabarti
81
Intelligent Web surfer
Prq ( j )  (1  b ) Pr' q ( j )  b (i , j )E Prq (i ) Prq (i  j )
Keyword
Probability
of teleporting
to node j
Relevance
of node k wrt q
Pr'q ( j ) 

Rq ( j )
kV
Rq (k )
Pr'q (i  j ) 
ACFOCS 2004

Rq ( j )
( i , k )E
PR Q ( j )  qQ Pr( q ) Prq ( j )
Query=set
of words
Probability of walking
from i to j wrt q
Rq (k )
Pick out-link to walk on in
proportion to relevance of
target out-neighbor
Pick a query word per
some distribution, e.g. IDF
Chakrabarti
82
Implementing the intelligent surfer
 PRQ(j) approximates a walk that picks a query
keyword using Pr(q) at every step
 Precompute and store Prq(j) for each keyword q in
lexicon: space blowup = avg doc length
 Query-dependent PR rated better by volunteers
ACFOCS 2004
Chakrabarti
83
Topic-sensitive Pagerank
 High overhead for per-word Pagerank
 Instead, compute Pageranks for some
collection of broad topics PRc(j)
 Topic c has sample page set Sc
 Walk as in Pagerank
 Jump to a node in Sc uniformly at random
 “Project” query onto set of topics
Pr(c | Q)  Pr( c)qQ Pr( q | c)
 Rank responses by projection-weighted
Pageranks Score(Q, j ) 
Pr(c | Q) PR ( j )

c
ACFOCS 2004
Chakrabarti
c
84
Topic-sensitive Pagerank results
 Users prefer topic-sensitive Pagerank on most
queries to global Pagerank + keyword filter
ACFOCS 2004
Chakrabarti
85
Image captioning
 Segment images
into regions
 Image has
caption words
 Three-layer
graph: image,
regions, caption
words
 Threshold on
region similarity
to connect
regions (dotted)
ACFOCS 2004
Chakrabarti
86
Random walks with restarts
Regions
Images
Test
image
Words
 Find regions in test image
 Connect regions to other nodes in the region layer
using region similarity
 Random walk, restarting at test image node
 Pick words with largest visit probability
ACFOCS 2004
Chakrabarti
87
More random walks: “Link rot”
 How stale is a page?
 Last-mod unreliable
 Automatic dead-link cleaners mask disuse
 A page is “completely stale” if it is “dead”
 Let D be the set of pages which cannot be
accessed (404 and other problems)
 How stale is a page u? Start with p  u
 If pD declare decay value of u to be 1, else
 With probability  declare decay value of u = 0
 W.p. 1– choose outlink v, set pv, loop
ACFOCS 2004
Chakrabarti
88
Page staleness results
Decay
404s
 Decay score is correlated with, but generally larger
than the fraction of dead outlinks on a page
 Removing direct dead links automatically does not
eliminate live but “rotting” pages
ACFOCS 2004
Chakrabarti
89
Graph proximity search: two paradigms
 A single node as query response
 Find node that matches query terms…
 …or is “near” nodes matching query terms
[Goldman+ 1998]
 A connected subgraph as query response
 Single node may not match all keywords
 No natural “page boundary” [Bhalotia+2002]
[Agrawal+2002]
ACFOCS 2004
Chakrabarti
90
Single-node response examples
 Travolta, Cage
Movie
 Actor, Face/Off
 Travolta, Cage,
Movie
“is-a”
Gathering
Grease
“acted-in”
 Gathering, Grease
 Kleiser, Woo, Actor
“directed”
 Face/Off
 Kleiser, Movie
Face/Off
A3
Travolta
Cage
“is-a”
Actor
 Travolta
Kleiser
Woo
“is-a”
Director
ACFOCS 2004
Chakrabarti
91
Basic search strategy
 Node subset A activated because they
match query keyword(s)
 Look for node near nodes that are activated
 Goodness of response node depends
 Directly on degree of activation
 Inversely on distance from activated node(s)
ACFOCS 2004
Chakrabarti
92
Proximity query: screenshot
http://www.cse.iitb.ac.in/banks/
ACFOCS 2004
Chakrabarti
93
Ranking a single node response
 Activated node set A
 Rank node r in “response set” R based on
proximity to nodes a in A
 Nodes have relevance R and A in [0,1]
 Edge costs are “specified by the system”
 d(a,r) = cost of shortest path from a to r
 Bond between a and r
 A (a )  R ( r )
b(a, r ) 
t
d (a, r )
 Parameter t tunes relative emphasis on
distance and relevance score
 Several ad-hoc choices
ACFOCS 2004
Chakrabarti
94
Scoring single response nodes
 Additive
 Belief
score(r )  aA b(a, r )
score(r )  1  aA 1  b(a, r )
 Goal: list a limited number of find nodes with
the largest scores
 Performance issues
 Assume the graph is in memory?
 Precompute all-pairs shortest path (|V |3)?
 Prune unpromising candidates?
ACFOCS 2004
Chakrabarti
95
Hub indexing
 Decompose APSP problem using sparse
vertex cuts
 |A|+|B | shortest paths to p
 |A|+|B | shortest paths to q
 d(p,q)
 To find d(a,b) compare




d(apb) not through q
d(aqb) not through p
d(apqb)
d(aqpb)
A
B
p
a
b
q
 Greatest savings when |A||B|
 Heuristics to find cuts, e.g. large-degree nodes
ACFOCS 2004
Chakrabarti
96
ObjectRank [Balmin+2004]
 Given a data graph with nodes having text
 For each keyword precompute a keywordsensitive Pagerank [RichardsonD2001]
 Score of a node for multiple keyword search
based on fuzzy AND/OR
 Approximation to Pagerank of node with restarts
to nodes matching keywords
 Use Fagin-merge [Fagin2002] to get best
nodes in data graph
ACFOCS 2004
Chakrabarti
97
Connected subgraph as response
 Single node may not match all keywords
 No natural “page boundary”
 On-the-fly joins make up a “response page”
 Two scenarios
 Keyword search on relational data
• Keywords spread among normalized relations
 Keyword search on XML-like or Web data
• Keywords spread among DOM nodes and subtrees
ACFOCS 2004
Chakrabarti
98
Keyword search on relational data
 Tuple = node
 Some columns have text
 Foreign key constraints =
edges in schema graph
 Query = set of terms
 No natural notion
AuthorID
A1
of a document
A2
Cites
Citing
Cited

Author
AuthorID
AuthorName

PaperID
P1
P2
P2
Paper
PaperID
PaperName

Writes
AuthorID
PaperID

AuthorID AuthorName
A1
Chaudhuri
A2
Sudarshan
A3
Hulgeri
 Normalization
A3
 Join may be needed
Citing Cited PaperID PaperName
to generate results
P2
P1
P1
DBXplorer
P2
BANKS
 Cycles may exist in schema
graph: ‘Cites’
ACFOCS 2004
Chakrabarti
99
DBXplorer and DISCOVER
 Enumerate subsets of relations in schema graph
which, when joined, may contain rows which have
all keywords in the query
 “Join trees” derived from schema graph
 Output SQL query for each join tree
 Generate joins, checking rows for matches
[Agrawal+2001], [Hristidis+2002]
K1,K2,K3
T1
T2
T4
T3
ACFOCS 2004
K3
T3
T5
K2
T5
T2
T2
T4
T4
T2
T3
Chakrabarti
T2
T3
T5
100
Discussion
 Exploits relational
schema information to
contain search
 Pushes final extraction
of joined tuples into
RDBMS
 Faster than dealing
with full data graph
directly
ACFOCS 2004
 Coarse-grained
ranking based on
schema tree
 Does not model
proximity or (dis)
similarity of individual
tuples
 No recipe for data with
less regular (e.g. XML)
or ill-defined schema
Chakrabarti
101
Motivation from Web search
 “Linux modem driver
for a Thinkpad A22p”
IBM Thinkpads
•A20m
Thinkpad
•A22p
Drivers
•Windows XP
Download
•Linux
Installation tips
•Modem
•Ethernet
 Hyperlink path matches
query collectively
 Conjunction query
would fail
 Projects where X and
P work together
 Conjunction may
retrieve wrong page
 General notion of
graph proximity
ACFOCS 2004
The B System
Home Page of
Professor X
Papers
•VLDB…
Students
•P
•Q
Chakrabarti
Group members
•P
•S
•X
P’s home page
I work on the B
project.
102
Data structures for search
 Answer = tree with at least one leaf
containing each keyword in query
 Group Steiner tree problem, NP-hard
 Query term t found in source nodes St
 Single-source-shortest-path SSSP iterator
 Initialize with a source (near-) node
 Consider edges backwards
 getNext() returns next nearest node
 For each iterator, each visited node v
maintains for each t a set v.Rt of nodes in St
which have reached v
ACFOCS 2004
Chakrabarti
103
Generic expanding search
 Near node sets St with S = t St
 For all source nodes   S
 create a SSSP iterator with source 
 While more results required




Get next iterator and its next-nearest node v
Let t be the term for the iterator’s source s
crossProduct = {s}  t ’tv.Rt’
For each tuple of nodes in crossProduct
• Create an answer tree rooted at v with paths to each
source node in the tuple
 Add s to v.Rt
ACFOCS 2004
Chakrabarti
104
Search example (“Vu Kleinberg”)
Quoc Vu
Jon Kleinberg
writes
writes
Organizing Web pages
by “Information Unit”
cites
writes
Authoritative sources in a
hyperlinked environment
A metric
labeling problem
cites
cites
Divyakant Agrawal
author
ACFOCS 2004
paper
writes
writes
cites
Chakrabarti
writes
Eva Tardos
105
First response
Quoc Vu
Jon Kleinberg
writes
writes
Organizing Web pages
by “Information Unit”
cites
writes
Authoritative sources in a
hyperlinked environment
A metric
labeling problem
cites
cites
Divyakant Agrawal
author
ACFOCS 2004
paper
writes
writes
cites
Chakrabarti
writes
Eva Tardos
106
Subgraph search: screenshot
http://www.cse.iitb.ac.in/banks/
ACFOCS 2004
Chakrabarti
107
Similarity, neighborhood, influence
ACFOCS 2004
Chakrabarti
108
Why are two nodes similar?
 What is/are the best paths connecting two nodes
explaining why/how they are related?
 Graph of co-starring, citation, telephone call, …
 Graph with nodes s and t; budget of b nodes
 Find “best” b nodes capturing relationship between
s and t [FaloutsosMT2004]:
 Proposing a definition of goodness
 How to efficiently select best connections
Negroponte
Palmisano
Esther Dyson
ACFOCS 2004
Gerstner
Chakrabarti
109
Simple proposals that do not work
 Shortest path
 Pizza boy p gets same
attention as g
 Network flow
a
s
 sabt is as good as sgt
 Voltage
b
g
t
p
 Connect +1V at s, ground t
 Both g and p will be at +0.5V
 Observations
 Must reward parallel paths
 Must reward short paths
 Must penalize/tax pizza boys
ACFOCS 2004
Chakrabarti
110
Resistive network with universal sink
 Connect +1V to s
 Ground t
 Introduce universal sink
 Grounded
 Connected to every node
a
s
b
g
t
p
 Universal sink is a “tax
collector”
 Penalizes pizza boys
 Penalizes long paths
 Goodness of a path is the
electric current it carries
ACFOCS 2004
Chakrabarti
Connected
to every node
111
Resistive network algorithm








Ohm’s law: I (u, v)  C (u, v)[V (u )  V (v)] u, v
Kirchhoff’s current law: v  s, t : u I (u, v)  0
Boundary conditions (without sink):
V ( s)  1, V (t )  0
Solution:
V
(
v
)
C
(
u
,
v
)

V (u )  v
, for u  s, t
w C (u, w)
Here C(u,v) is the conductance from u to v
Add grounded universal sink z with V(z)=0
Set u : C (u, z )  a w z C (u, w)
Display subgraph carrying high current
ACFOCS 2004
Chakrabarti
112
Distributions influenced via graphs
 Directed or undirected graph; nodes have
 Observable properties
 Some unobservable (random) state
 Edges indicate that distributions over
unobservable states are coupled
 Many applications




ACFOCS 2004
Hypertext classification (topics are clustered)
Social network of customers buying products
Hierarchical classification
Labeling categorical sequences: pos/ne
tagging, sense disambiguation, linkage analysis
Chakrabarti
113
Basic techniques
 Directed (acyclic) graphs: Bayesian network
 Markov networks
 (Loopy) belief propagation
 Conditional Markov networks
 Avoid modeling joint distribution over
observable and hidden properties of nodes
 Some computationally simple special cases
ACFOCS 2004
Chakrabarti
114
Hypertext classification
 Want to assign labels to Web pages
 Text on a single page may be too little or
misleading
 Page is not an isolated instance by itself
 Problem setup





Web graph G=(V,E)
Node uV is a page having text uT
Edges in E are hyperlinks
Some nodes are labeled
Make collective guess at missing labels
 Probabilistic model? Benefits?
ACFOCS 2004
Chakrabarti
115
Graph labeling model
 Seek a labeling f of all unlabeled nodes so
as to maximize
Pr( f (V )) Pr( E , {u : u  V } | f (V ))
Pr( f (V ) | E , {uT : u  V }) 
T
 Pr( ) Pr( E,{u
T
: u V } |  )
Luckily we don’t need to worry about this
 Let VK be the nodes with known labels and
f(VK) their known label assignments
 Let N(v) be the neighbors of v and
NK(v)N(v) be neighbors with known labels
 Markov assumption: f(v) is conditionally
independent of rest of G given f(N(v))
ACFOCS 2004
Chakrabarti
116
Markov graph labeling
Probability of labeling
specific node v…
…given edges, text,
parital labeling
Pr( f (v) | E ,V T , f (V K )) 


Sum over all possible labelings
of unknown neighbors of v

U
T
K
Pr
f
(
v
),
f
(
N
(
v
))
|
E
,
V
,
f
(
V
)

f ( N U ( v )) v

 
U
T
K
U
T
K
Pr
f
(
N
(
v
))
|
E
,
V
,
f
(
V
)
Pr
f
(
v
)
|
f
(
N
(
v
)),
E
,
V
,
f
(
V
)


f ( N U ( v )) v
Markov assumption: label of v does not depend on
unknown labels outside the set NU(v)
 Circularity between f(v) and f(NU(v))
 Some form of iterative Gibb’s sampling or MCMC
ACFOCS 2004
Chakrabarti
117
Iterative labeling
Label estimates of v in the next iteration


Pr( r 1) f (v) | E ,V T , f (V K ) 

f ( N U ( v )) v

wN
U

 
T
K
U
T
K
Pr
f
(
w
)
|
E
,
V
,
f
(
V
)
Pr
f
(
v
)
|
f
(
N
(
v
)),
E
,
V
,
f
(
V
)
(r )
(v)
Joint distribution over neighbors
approximated as product of marginals

Take the expectation of this
term over NU(v) labelings
 Sum over all possible NU(v) labelings still too
expensive to compute
 In practice, prune to most likely configurations
 Let us look at the last term more carefully
ACFOCS 2004
Chakrabarti
118
A generative node model



Pr f (v) | f ( N U (v)), V T , E, f (V K )  Pr f (v) | f ( N (v)), vT

By the Markov assumption, finally we need a distribution
coupling f(v) and vT (the text on v) and f(N(v))
 Can use Bayes classifier as with ordinary text:
estimate a parametric model for

Pr f ( N (v)), vT | f (v)

the class-conditional joint distribution between the
text on the page v and the labels of neighbors of v
 Must make naïve Bayes assumptions to keep
practical
ACFOCS 2004
Chakrabarti
119
Pictorially…
 c=class, t=text,
N=neighbors
 Text-only model: Pr[t|c]
 Using neighbors’ text
to judge my topic:
Pr[t, t(N) | c] (hurts)
 Better model:
Pr[t, c(N) | c]
 Estimate histograms and
update based on
neighbors’ histograms
ACFOCS 2004
Chakrabarti
?
120
Generative model: results
ACFOCS 2004
Chakrabarti
40
%Error
 9600 patents from 12
classes marked by
USPTO
 Patents have text and
cite other patents
 Expand test patent to
include neighborhood
 ‘Forget’ fraction of
neighbors’ classes
30
20
10
0
0
50
100
%Neighborhood known
Text
Link
Text+Link
121
Detour: generative vs. discriminative
 x = feature vector, y = label {0,1} say
 Generative method models Pr(x|y) or Pr(x,y)
“the generation of data x given label y”
 Use Bayes rule to get Pr(y|x)
 Inaccurate; x may have many dimensions
 Discriminative: directly estimates Pr(y|x)
 Simple linear case: want w s.t.
w.x > 0 if y=1 and w.x  0 if y=0
 Cannot differentiate for w; instead pick some
1
smooth “loss function”
Pr( y | x) 
1  exp  w  x 
 Works very well in practice
ACFOCS 2004
Chakrabarti
122
Discriminative node model
 OA(X) = direct “own” attributes of node X
 LD(X) = link-derived attributes of node X
 Mode-link: most frequent label of neighbors(X)
 Count-link: histogram of neighbor labels
 Binary-link: 0/1 histogram of neighbor labels
Pr(c | wo , OA( X ))  1 exp( cw OA( X )  1)
T
o
Local model params
Neighborhood model params
Pr(c | wl , LD( X ))  1 exp( cw LD( X )  1)
Cˆ ( X )  arg max Pr(c | OA( X )) Pr(c | LD( X ))
T
l
c
 Iterate as in generative case
ACFOCS 2004
Chakrabarti
123
Discriminative model: results [Li+2003]
 Binary-link and count-link outperform content-only
at 95% confidence
 Better to separately estimate wl and wo
 In+Out+Cocitation better than any subset for LD
ACFOCS 2004
Chakrabarti
124
Undirected Markov networks
 Clique cC(G) a set of
completely connected
nodes
 Clique potential c(Vc) a
function over all possible
configurations of nodes
in Vc
 Decompose Pr(v) as
(1/Z)cC(G)c(Vc)
 Parametric form
Label coupling
c ( v c )  exp w c  f c ( v c )
Params of model
ACFOCS 2004
Instance
Local feature variable
Label variable
Feature functions
Chakrabarti
125
Conditional and relational networks
 x = vector of observed features at all nodes
 y = vector of labels
1
Pr( y | x) 
c (xc , y c )

Z (x) cC (G )
where Z (x)  
'

(
x
,
y
 c c c)
y ' cC ( G )
c (xc , y c )  exp w c  Fc (xc , y c )
 A set of “clique templates”
specifying links to use
 Other features: “in the same
HTML section”
ACFOCS 2004
Chakrabarti
126
“Toy problem”: Hierarchical classification
 Obvious approaches
 Flatten to leaf topics, losing hierarchy info
 Level-by-level, compounding error probability
 Cascaded generative model
Pr(c | d )  Pr( r | d ) Pr(c | d , r )
r
c
 Pr(c|d,r) estimated as Pr(c|r)Pr(d|c)/Z(r)
 Estimate of Pr(d|c) makes naïve independence
assumptions if d has high dimensionality
 Pr(c|d,r) tends to 0/1 for large dimensions and
 Mistake made at shallow levels become
irrevocable
ACFOCS 2004
Chakrabarti
127
Global discriminative model
 Each node has an associated bit X
 Propose a parametric form
exp( wc  F (d , xr ))
Pr( X c  1 | d , xr ) 
1  exp( wc  F (d , xr ))
 Each training instance sets one path to 1, all other
nodes have X=0
T
d
xr=0
xr
xr=1
2T+1
wc
ACFOCS 2004
F(d,xr)
%Accuracy
SVM 1v1 Ctree
Reuters 1%
41.9
47.3
Reuters 5%
68
71
News20 1%
17.2
21.8
Chakrabarti
128
Network value of customers
 Customer = node X in graph, neighbors N
 M is a marketing action (promotion, coupon)
 Want to predict Pr( X i | X K , Y, M )
Aggregate
marketing action
Response of customer i
Known response of
other customers
Product attributes
 Broader objective is to design action M
 Again, we approximate as
U
K
 Pr( X i | Ni , Y, M) Pr( Ni | X , Y, M)
C ( N iU )
Sum over unknown neighbor configurations
ACFOCS 2004
Chakrabarti
129
Network value, continued
 Let the action be boolean
 c is the cost of marketing
 r0 is the revenue without marketing, r1 with
 Expected lift in profit by marketing to
customer i in isolation
K
K
1
ELPi X , Y, M   r1 Pr( X i  1 | X , Y, f i (M))
 r0 Pr( X i  1 | X , Y, f i (M))  c
 Global effect
K
ACFOCS 2004
0
Chakrabarti
130
Special case: sequential networks
 Text modeled as sequence of tokens drawn
from a large but finite vocabulary
 Each token has attributes
 Visible: allCaps, noCaps, hasXx, allDigits,
hasDigit, isAbbrev, (part-of-speech, wnSense)
 Not visible: part-of-speech, (isPersonName,
isOrgName, isLocation, isDateTime),
{starts|continues|ends}-noun-phrase
 Visible (symbols) and invisible (states)
attributes of nearby tokens are dependent
 Application decides what is (not) visible
 Goal: Estimate invisible attributes
ACFOCS 2004
Chakrabarti
131
Hidden Markov model
 A generative sequential model for the joint
distribution of states (s) and symbols (o)


S
S
S
s  s0 , s1 ,...sn
o  o1 , o2 ,...on
|o|
 
Pr( s , o )   Pr( st 1  st ) Pr( st  ot ) O
O
O
t-1
t-1
t
t
t+1
t+1
...
...
t 1
ACFOCS 2004
Chakrabarti
132
Using redundant token features
 Each o is usually a vector of features
extracted from a token
 Might have high dependence/redundancy:
hasCap, hasDigit, isNoun, isPreposition
 Parametric model for Pr(stot) needs to
make naïve assumptions to be practical
 Overall joint model Pr(s,o) can be very
inaccurate
 (Same argument as in naïve Bayes vs. SVM
or maximum entropy text classifiers)
ACFOCS 2004
Chakrabarti
133
Discriminative graphical model
 Assume one-stage Markov dependence
 Propose direct parametric form for
conditional probability of state sequence
given symbol sequence
St-1

|o|
 
Pr( s | o ) 
1
  Pr( st | st 1 ) Pr(ot | st )
Pr(o) t 1
O
Model
|o|

1
  s ( st , st 1 )o ( st , st 1 , o, t )
Z (o ) t 1

o (t )  exp k k f k (st 1 , st , o, t )
t-1


Ot
St+1
...
Ot+1
...
Log-linear form
Feature function; might
depend on whole o
Parameters to fit
ACFOCS 2004
St
Chakrabarti
134
Feature functions and parameters

|o |
 1

2k
 
L   log    exp   k f k ( st , st 1 , o, t )     2
s , o D
 k
  k 2 Penalize
 Z (o ) t 1
large params
Maximize total
conditional
likelihood over
all instances
 Find L/k for each k and
perform a gradient-based
numerical optimization
 Efficient for linear state
dependence structure
ACFOCS 2004
Chakrabarti
135
Conditional vs. joint: results
Penn Treebank: 45 tags, 1M words training data
DT
NN
NN ,
NN
, VBZ RB
JJ
IN
The asbestos fiber , crocidolite, is unusually resilient once
PRP VBZ DT NNS , IN RB JJ
NNS TO PRP VBG
it enters the lungs , with even brief exposures to it causing
Algorithm/Features
HMM with words
CRF with words
CRF with words and orthography
%Error %OOVE*
5.69
45.99
5.55
48.05
4.27
23.76
Orthography: Use words, plus overlapping features:
isCap, startsWithDigit, hasHyphen, endsWith… -ing, ogy, -ed, -s, -ly, -ion, -tion, -ity, -ies
ACFOCS 2004
Chakrabarti
Out-of-vocabulary error
NNS WDT VBP RP NNS JJ ,
NNS
VBD .
symptoms that show up decades later , researchers said .
136
Summary
 Graphs provide a powerful way to model
many kinds of data, at multiple levels
 Web pages, XML, relational data, images…
 Words, senses, phrases, parse trees…
 A few broad paradigms for analysis
 Factors affecting graph evolution over time
 Eigen analysis, conductance, random walks
 Coupled distributions between node attributes
and graph neighborhood
 Several new classes of model estimation
and inferencing algorithms
ACFOCS 2004
Chakrabarti
137
References
 [BrinP1998] The Anatomy of a Large-Scale
Hypertextual Web Search Engine, WWW.
 [GoldmanSVG1998] Proximity search in
databases. VLDB, 26—37.
 [ChakrabartiDI1998] Enhanced hypertext
categorization using hyperlinks. SIGMOD.
 [BikelSW1999] An Algorithm that Learns What’s in
a Name. Machine Learning Journal.
 [GibsonKR1999] Clustering categorical data: An
approach based on dynamical systems. VLDB.
 [Kleinberg1999] Authoritative sources in a
hyperlinked environment. JACM 46.
ACFOCS 2004
Chakrabarti
138
References
 [CohnC2000] Probabilistically Identifying
Authoritative Documents, ICML.
 [LempelM2000] The stochastic approach for linkstructure analysis (SALSA) and the TKC effect.
Computer Networks 33 (1-6): 387-401
 [RichardsonD2001] The Intelligent Surfer:
Probabilistic Combination of Link and Content
Information in PageRank. NIPS 14 (1441-1448).
 [LaffertyMP2001] Conditional Random Fields:
Probabilistic Models for Segmenting and Labeling
Sequence Data. ICML.
 [BorkarDS2001] Automatic text segmentation for
extracting structured records. SIGMOD.
ACFOCS 2004
Chakrabarti
139
References
 [NgZJ2001] Stable algorithms for link analysis.
SIGIR.
 [Hulgeri+2001] Keyword Search in Databases.
IEEE Data Engineering Bulletin 24(3): 22-32.
 [Hristidis+2002] DISCOVER: Keyword Search in
Relational Databases. VLDB.
 [Agrawal+2002] DBXplorer: A system for keywordbased search over relational databases. ICDE.
 [TaskarAK2002] Discriminative probabilistic
models for relational data.
 [Fagin2002] Combining fuzzy information: an
overview. SIGMOD Record 31(2), 109–118.
ACFOCS 2004
Chakrabarti
140
References
 [Chakrabarti2002] Mining the Web: Discovering
Knowledge from Hypertext Data
 [Tomlin2003] A New Paradigm for Ranking Pages
on the World Wide Web. WWW.
 [Haveliwala2003] Topic-Sensitive Pagerank: A
Context-Sensitive Ranking Algorithm for Web
Search. IEEE TKDE.
 [LuG2003] Link-based Classification. ICML.
 [FaloutsosMT2004] Connection Subgraphs in
Social Networks. SIAM-DM workshop.
 [PanYFD2004] GCap: Graph-based Automatic
Image Captioning. MDDE/CVPR.
ACFOCS 2004
Chakrabarti
141
References
 [Balmin+2004] Authority-Based Keyword Queries
in Databases using ObjectRank. VLDB.
 [BarYossefBKT2004] Sic transit gloria telae:
Towards an understanding of the Web’s decay.
WWW2004.
ACFOCS 2004
Chakrabarti
142