http://cs224w.stanford.edu Probabilistic models of network contagion - - PowerPoint PPT Presentation

▶
http cs224w stanford edu probabilistic models of network
SMART_READER_LITE
LIVE PREVIEW

http://cs224w.stanford.edu Probabilistic models of network contagion - - PowerPoint PPT Presentation

Nov 02, 2022 386 likes •859 views

CS224W: Social and Information Network Analysis Jure Leskovec Stanford University Jure Leskovec, Stanford University http://cs224w.stanford.edu Probabilistic models of network contagion Probabilistic models of network contagion How

slide-1
SLIDE 1

CS224W: Social and Information Network Analysis Jure Leskovec Stanford University Jure Leskovec, Stanford University

http://cs224w.stanford.edu

slide-2
SLIDE 2

 Probabilistic models of network contagion  Probabilistic models of network contagion  How contagions diffuse in real‐life:

g

  • Viral marketing
  • Blogs

Blogs

  • Group membership

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2

slide-3
SLIDE 3

 How do viruses/rumors propagate?

How do viruses/rumors propagate?

 Will a flu‐like virus linger, or will it become extinct?  (Virus) birth rate β:  (Virus) birth rate β:

  • probability than an infected neighbor attacks

 (Virus) death rate δ:

  • probability that an infected node heals Healthy

N2

  • Prob. δ

N N1

2

  • Prob. β

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 3

Infected N3

10/13/2009

slide-4
SLIDE 4

 General scheme for epidemic models:  General scheme for epidemic models:

S…susceptible E…exposed I…infected d

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

R…recovered Z…immune

4

slide-5
SLIDE 5

 Assuming perfect

g p mixing, i.e., a network is a complete graph

  • des

 The model dynamics:

mber of no Nu time

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Susceptible Infected Recovered

5

slide-6
SLIDE 6

 Susceptible Infective Susceptible (SIS) model  Susceptible‐Infective‐Susceptible (SIS) model  Cured nodes immediately become susceptible  Virus “strength”: s = β / δ  Virus strength : s = β / δ

Infected by neighbor with prob. β

Susceptible Infective

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 6

Cured internally with prob. δ

10/13/2009

slide-7
SLIDE 7

f

 Assuming perfect

mixing (complete graph):

nodes

graph):

I SI dS     

Number of n

I SI dI   dt 

N S sceptible Infected

I SI dt    

time

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Susceptible Infected

7

slide-8
SLIDE 8

 Representing SIS epidemic an SIR model  Representing SIS epidemic an SIR model

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 8

slide-9
SLIDE 9

 Epidemic threshold of a graph is a  Epidemic threshold of a graph is a

value of t, such that:

  • If strength s = β / δ < t epidemic can not
  • If strength s = β / δ < t epidemic can not

happen (it eventually dies out)

 Given a graph compute its epidemic threshold

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 9 10/13/2009

slide-10
SLIDE 10

 What should t depend on?  What should t depend on?

  • Avg. degree? And/or highest degree?
  • A d/

i f d ?

  • And/or variance of degree?
  • And/or third moment of degree?

A d/ di ?

  • And/or diameter?

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 10

slide-11
SLIDE 11

[Wang et al. 2003]

 We have no epidemic if:  We have no epidemic if:

(Virus) Death Epidemic threshold

β/δ < τ = 1/ λ1 A

( ) rate

β

1,A

(Virus) Birth rate largest eigenvalue

► λ A alone captures the property of the graph!

  • f adj. matrix A

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

► λ1,A alone captures the property of the graph!

10/13/2009 11

slide-12
SLIDE 12

[Wang et al. 2003]

500

Oregon β 0 001

10,900 nodes and 31,180 edges

400

d Nodes

β = 0.001

β/δ > τ (above threshold)

3 , g

200 300

f Infected

100 200

umber of

β/δ = τ (at the threshold)

250 500 750 1000

N

β/δ < τ

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 12

Time

δ: 0.05 0.06 0.07

β (below threshold)

slide-13
SLIDE 13

 Does it matter how many people are  Does it matter how many people are

initially infected?

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 13

slide-14
SLIDE 14

 Prob of adoption depends on the number of  Prob. of adoption depends on the number of

friends who have adopted [Bass ‘69, Granovetter ’78]

  • What is the shape?

What is the shape?

  • Distinction has consequences for models and algorithms
  • n
  • n
  • f adoptio
  • f adoptio

k = number of friends adopting

  • Prob. o

k = number of friends adopting

  • Prob. o

k = number of friends adopting k = number of friends adopting

Diminishing returns? Critical mass?

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 14

slide-15
SLIDE 15

[Leskovec et al., TWEB ’07]

 Senders and followers of recommendations

receive discounts on products

10% credit 10% off

  • Data – Incentivized Viral Marketing program
  • 16 million recommendations

illi l

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

  • 4 million people
  • 500,000 products

10/13/2009 15

slide-16
SLIDE 16

[Backstrom et al., KDD ’06]

 Use social networks where people belong to  Use social networks where people belong to

explicitly defined groups

 Each group defines a behavior that diffuses  Each group defines a behavior that diffuses  Data – LiveJournal:

  • On‐line blogging community with

friendship links and user‐defined groups p g p

  • Over a million users update content

each month

  • Over 250,000 groups to join

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 16

slide-17
SLIDE 17

[Leskovec et al., TWEB ’07]

DVD recommendations asing

0.09 0.1

DVD recommendations (8.2 million observations)

  • f purcha

0 05 0.06 0.07 0.08

bability o

0 02 0.03 0.04 0.05

Prob

0.01 0.02 10 20 30 40

17

10 20 30 40

# recommendations received

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-18
SLIDE 18

[Backstrom et al., KDD ’06]

 LiveJournal community membership  LiveJournal community membership

  • ining
  • rob. of jo

k ( b f f i d i th it ) Pr

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

k (number of friends in the community)

10/13/2009 18

slide-19
SLIDE 19

 For viral marketing:  For viral marketing:

  • We see that node v receiving the i‐th

recommendation and then purchased the product p p

 For communities:

  • At time t we see the behavior of node v’s friends

 Questions:

  • When did v become aware of recommendations or

f i d ’ b h i ? friends’ behavior?

  • When did it translate into a decision by v to act?
  • How long after this decision did v act?
  • How long after this decision did v act?

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 19

slide-20
SLIDE 20

 Large anonymous online retailer

Large anonymous online retailer (June 2001 to May 2003) 15 646 121 d ti

 15,646,121 recommendations  3,943,084 distinct customers  548 523 products recommended

548,523 products recommended

 Products belonging to 4 product groups:

  • books
  • DVDs
  • music
  • VHS

20 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-21
SLIDE 21

purchase following a recommendation customer recommending a product customer not buying a recommended product

Majority of recommendations do not cause purchases nor cause purchases nor propagation

Notice many star‐like patterns

21 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

patterns

Many disconnected components

slide-22
SLIDE 22

 t < t <

< t

 t1 < t2 < … < tn

t3 legend

bought but didn’t i di t

t1

receive a discount bought and received a discount

t2

received a recommendation but didn’t buy

t5

22

t4

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-23
SLIDE 23

 What role does the product category play?  What role does the product category play?

products customers recommenda- tions edges buy + get discount buy + no discount tions discount discount Book 103,161 2,863,977 5,741,611 2,097,809 65,344 17,769 DVD 19,829 805,285 8,180,393 962,341 17,232 58,189 Music 393,598 794,148 1,443,847 585,738 7,837 2,739 Video 26,131 239,583 280,270 160,683 909 467 F ll 542 719 3 943 084 15 646 121 3 153 676 91 322 79 164 Full 542,719 3,943,084 15,646,121 3,153,676 91,322 79,164

people recommendations

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

high low

10/13/2009 23

slide-24
SLIDE 24

There are relatively few DVD titles, but DVDs account for ~ 50% of recommendations.

Recommendations per person

  • DVD: 10
  • books and music: 2
  • VHS: 1
  • VHS: 1

Recommendations per purchase

  • books: 69
  • DVDs: 108
  • music: 136
  • VHS: 203

Overall there are 3.69 recommendations per node on 3.85 different products.

Music recommendations reached about the same number of people as DVDs but used only 1/5 as many recommendations

Book recommendations reached by far the most people – 2.8 million.

All networks have a very small number of unique edges For books videos and

All networks have a very small number of unique edges. For books, videos and music the number of unique edges is smaller than the number of nodes – the networks are highly disconnected

24 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-25
SLIDE 25

 Does sending more recommendations  Does sending more recommendations

influence more purchases?

BOOKS DVDs

0.5 s 6 7 s 0.3 0.4

  • f Purchase

3 4 5

  • f Purchase

0.1 0.2 Number o 1 2 3 Number o

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 25

10 20 30 40 50 60 Outgoing Recommendations 20 40 60 80 100 120 140 Outgoing Recommendations

slide-26
SLIDE 26

 What is the effectiveness of subsequent  What is the effectiveness of subsequent

recommendations?

10

  • 3

10 12x 10

3

ng 0.06 0.07 ng

BOOKS DVDs

8 bility of buyin 0 04 0.05 bility of buyin 6 Probab 0.03 0.04 Probab

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 26

5 10 15 20 25 30 35 40 4 Exchanged recommendations 5 10 15 20 25 30 35 40 0.02 Exchanged recommendations

slide-27
SLIDE 27

consider successful recommendations in terms of

  • av # senders of recommendations per book category
  • av. # senders of recommendations per book category
  • av. # of recommendations accepted

books overall have a 3% success rate

  • (2% with discount, 1% without)

lower than average success rate (significant at p=0 01 level)

lower than average success rate (significant at p=0.01 level)

  • fiction
  • romance (1.78), horror (1.81)
  • teen (1.94), children’s books (2.06)
  • i

(2 30) i fi (2 34) t d th ill (2 40)

  • comics (2.30), sci‐fi (2.34), mystery and thrillers (2.40)
  • nonfiction
  • sports (2.26)
  • home & garden (2.26)
  • travel (2 39)
  • travel (2.39)

higher than average success rate (statistically significant)

  • professional & technical
  • medicine (5.68)
  • professional & technical (4 54)
  • professional & technical (4.54)
  • engineering (4.10), science (3.90), computers & internet (3.61)
  • law (3.66), business & investing (3.62)

27 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-28
SLIDE 28

 47 000 customers responsible for the 2 5 out  47,000 customers responsible for the 2.5 out

  • f 16 million recommendations in the system

d f

 29% success rate per recommender of an

anime DVD

 Giant component covers 19% of the nodes

O ll d ti f DVD

 Overall, recommendations for DVDs are more

likely to result in a purchase (7%), but the anime community stands out anime community stands out

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 28

slide-29
SLIDE 29

Variable transformation Coefficient const

  • 0.940 ***

# recommendations ln(r) 0 426 *** # recommendations ln(r) 0.426 # senders ln(ns)

  • 0.782 ***

# recipients ln(n )

  • 1 307 ***

# recipients ln(nr) 1.307 product price ln(p) 0.128 *** # reviews ln(v)

  • 0 011 ***

# reviews ln(v)

  • 0.011
  • avg. rating

ln(t)

  • 0.027 *

R2 0 74

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 29

R2 0.74

significance at the 0.01 (***), 0.05 (**) and 0.1 (*) levels

slide-30
SLIDE 30

12x 10

4

10

6

10 nent 4x 10

6

6 8 compon 2 n # nodes 4 6

  • f giant

10 20 m (month) 1.7*106m 2 size by month quadratic fit m (month)

30

1 2 3 4 x 10

6

number of nodes quadratic fit

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-31
SLIDE 31

 94% of users make first recommendation without having

g received one previously

 Size of giant connected component increases from 1% to

2 % f h k (100 20 ) ll! 2.5% of the network (100,420 users) – small!

 Some sub‐communities are better connected

24% f 18 000 f V

  • 24% out of 18,000 users for westerns on DVD
  • 26% of 25,000 for classics on DVD
  • 19% of 47,000 for anime (Japanese animated film) on DVD

 Others are just as disconnected

  • 3% of 180,000 home and gardening
  • 2‐7% for children’s and fitness DVDs

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 31

slide-32
SLIDE 32

Products suited for Viral Marketing:

 small and tightly knit community

  • few reviews, senders, and recipients
  • but sending more recommendations helps

 pricey products  pricey products  rating doesn’t play as much of a role

Observations for future diffusion models:

 purchase decision more complex than threshold or simple infection  influence saturates as the number of contacts expands  links user effectiveness if they are overused

Conditions for successful recommendations:

 professional and organizational contexts  discounts on expensive items  small tightly knit communities  small, tightly knit communities

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 32

slide-33
SLIDE 33

 How big are cascades?  How big are cascades?  What are the building

blocks of cascades? blocks of cascades?

973 938

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Medical guide book DVD

10/13/2009 33

slide-34
SLIDE 34

 Given a (social) network  Given a (social) network  A process by spreading over the network

creates a graph (a tree) creates a graph (a tree)

Cascade (propagation graph) Social network

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Let’s count cascades

10/13/2009 34

slide-35
SLIDE 35

high

 General observations:

  • DVDs have the richest

d ( t

cascades different

g low

cascades (most recommendations, most densely linked)

cascades different Book 122,657 959 DVD 289 055 87 614

  • Books have small

cascades

  • M

i i 3 ti l

DVD 289,055 87,614 Music 13,330 158

  • Music is 3 times larger

than video but does not have much variety in

Video 1,928 109

number of vocabulary

35

cascades

all “words” y size

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-36
SLIDE 36

is the most common cascade subgraph

is the most common cascade subgraph

 It accounts for ~75% cascades in books, CD

and VHS, only 12% of DVD cascades , y

is 6 (1.2 for DVD) times more frequent than

 For DVDs is more frequent than  Chains ( ) are more frequent than

i f t th lli i

is more frequent than a collision ( ) (but collision has less edges)

 Late split (

) is more frequent than

 Late split ( ) is more frequent than

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 36

slide-37
SLIDE 37

 Stars (“no propagation”)  Stars ( no propagation )  Bipartite cores (“common friends”)  Nodes having same friends  Nodes having same friends

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 37

slide-38
SLIDE 38

 Delete late recommendations  Count how many people are in a single cascade  Exclude nodes that did not buy

steep drop‐off books 10

6

= 1.8e6 x-4.98 10

4

very few large cascades 10

2

38

10 10

1

10

2

10

10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-39
SLIDE 39

 DVD cascades can grow large  DVD cascades can grow large  Possibly as a result of websites where people

sign up to exchange recommendations sign up to exchange recommendations

shallow drop off – fat tail

~ x-1.56 10

4

Count

a number of large cascades

10

2

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

10 10

1

10

2

10

3

10

x = Cascade size (number of nodes)

10/13/2009 39

slide-40
SLIDE 40

[Leskovec et al., SDM ’07]

Posts Blogs Time Information cascade

D t Bl

Time

  • rdered

hyperlinks

 Data – Blogs:

  • We crawled 45,000 blogs for 1 year
  • 10 million posts and 350,000 cascades

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 40

slide-41
SLIDE 41

Cascade shapes (ranked by frequency)

The probability of The probability of

  • bserving a cascade
  • n n nodes follows a
  • unt

f Zipf distribution: p(n) ~ n-2

Co x = Cascade size (number of nodes)

10/13/2009 41 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-42
SLIDE 42

 Most of cascades are trees

r of edges diameter Number Effective d Cascade size (number of nodes) Cascade size

Number of

Count Count

Number of cascades per node also follows power‐law

Number of joined cascades C Cascades per node

p distribution.

10/13/2009 42 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

slide-43
SLIDE 43

 Cascade sizes follow a heavy‐tailed distribution  Cascade sizes follow a heavy‐tailed distribution

  • Viral marketing:
  • Books: steep drop‐off: power‐law exponent ‐5

Books: steep drop off: power law exponent 5

  • DVDs: larger cascades: exponent ‐1.5
  • Blogs:
  • Power‐law exponent ‐2

 What’s a good model?

  • What role does the underlying social network play?
  • Can make a step towards more realistic cascade

generation (propagation) model? generation (propagation) model?

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10/13/2009 43

slide-44
SLIDE 44

1) Randomly pick blog to infect add to cascade 2) Infect each in‐linked neighbor with probability  infect, add to cascade.

B1 B2 1 1

neighbor with probability 

B1 B1

B1 B2 1 1 B4 B3 2 1 3 1 B4 B3 2 1 3 1

4

3) Add infected neighbors to cascade. 4) Set node infected in (i) to uninfected.

B1 B2 1 1 2 1 B1 B2 1 1 2 1

B1 B1

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

B4 B3 1 3 B4 B3 1 3

B4 B4

10/13/2009 44

slide-45
SLIDE 45

Generative model

Count Count

produces realistic cascades

Cascade size Cascade node in‐degree

β=0.025

  • unt

nt Co Cou

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Most frequent cascades

Size of star cascade Size of chain cascade

10/13/2009 45

slide-46
SLIDE 46

 Blogs – information epidemics  Blogs – information epidemics

  • Which are the influential/infectious blogs?

 Viral marketing

  • Who are the trendsetters?

Who are the trendsetters?

  • Influential people?

 Disease spreading

  • Where to place monitoring stations to detect

p g epidemics?

46 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu