Social and Information - SNAP - Stanford University

CS 322: (Social and Information) Network Analysis
Jure Leskovec
Stanford University

Probabilistic models of network diffusion
How cascades spread in real life:
Viral marketing
Blogs
GGroup membership b hi
Last 20 minutes: mid‐term mid term course evaluation
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis 2

How do viruses/rumors propagate?
Will a flu‐like virus linger, or will it become extinct?
(Virus) birth rate β: probability than an infected
neighbor attacks
(Virus) death rate δ: probability that an infected node
heals
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N 1
Infected
Prob. β
Prob. δ
N
N 2
Healthy
Jure Leskovec, Stanford CS322: Network Analysis 3
N 3

General scheme for epidemic models:
S…susceptible
E…exposed
I…infected
R…recoveredd
Z…immune
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Number
of noodes
time
Susceptible Infected Recovered
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
Assuming g pperfect
mixing, i.e., a
network is a
complete graph
The model dynamics:
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Susceptible‐Infective‐Susceptible Susceptible Infective Susceptible (SIS) model
Cured nodes immediately become susceptible
Virus “strength”: strength : s = β / δ
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Infected by neighbor
with prob. β
Susceptible Infective
Cured internally
with prob. δ
Jure Leskovec, Stanford CS322: Network Analysis 6

Number N of nodes n
time
SSusceptible sceptible Infected
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
Assuming perfect f
mixing (complete
graph):
dS
dt
dI
dt
SI
SI
I
I
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Epidemic threshold of a graph is a value of t, t
such that:
If strength s = β / δ < t epidemic can not happen
(it eventually dies out)
Given a graph compute its epidemic threshold
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Jure Leskovec, Stanford CS322: Network Analysis 8

What should t depend on?
avg. degree? and/or highest degree?
and/or variance of degree?
and/or third moment of degree?
and/or diameter?
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
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We have no epidemic if:
(Virus) ( ) Death
rate
Epidemic threshold
β/δ β < τ = 1/ λ 1,A
(Virus) Birth rate largest eigenvalue
of adj. matrix A
► λ λ1,AA alone captures the property of the graph!
[Wang et al. 2003]
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
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N umber off
Infectedd
Nodes
500
400
300
200
100
0
Oregon
β = 0.001 0001
0 250 500 750 1000
Time
δ: 0.05 0.06 0.07
[Wang et al. 2003]
10,900 nodes and
3, 31,180 edges g
β/δ > τ
(above threshold)
β/δ = τ
(at the threshold)
β/δ β < τ
(below threshold)
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
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Does it matter how many people are
initially infected?
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[Leskovec et al., SDM ’07]
Bloggers write posts and refer (link) to other
posts and the information propagates
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Posts
Blogs
Time
ordered
hyperlinks
Dt Data – Bl Blogs:
We crawled 45,000 blogs for 1 year
10 million posts and 350,000 cascades
Information
cascade
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[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
• 4 million illi people l
• 500,000 products
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[Backstrom et al., KDD ’06]
Use social networks where people belong to
explicitly defined groups
Each group defines a behavior that diffuses
Data – LiveJournal:
On‐line blogging community with friendship links
and user‐defined groups g p
Over a million users update content each month
Over 250,000 groups to join
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
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Prob. Prob of adoption depends on the number of
friends who have adopted [Bass ‘69, Granovetter ’78]
Prob. of o adoptioon
What is the shape?
Distinction has consequences for models and algorithms
k = number of friends adopting k = number of friends adopting
Prob. of o adoptioon
Diminishing returns? Critical mass?
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Probbability
of o purchaasing
0.1
0.09
0.08
0.07
0.06
005 0.05
0.04
0.03
002 0.02
0.01
0
DVD recommendations
(8.2 million observations)
0 10 20 30 40
# recommendations received
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
[Leskovec et al., TWEB ’07]
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LiveJournal community membership
Prrob.
of jooining
k ( (number b of f ffriends i d i in th the community) it )
[Backstrom et al., KDD ’06]
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Sending email:
Email network of large university
Prob. of a link as a function of # of common friends
mail
ob. of em
Pro
k (number of common friends)
[Kossinets‐Watts ‘06]
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For viral marketing:
We see that node v receiving the i‐th
recommendation and then purchased p the pproduct
For communities:
At time t we see the behavior of node v’s friends
Questions:
When did v become aware of recommendations or
fi friends’ d’bh behavior? i ?
When did it translate into a decision by v to act?
How long after this decision did v act?
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Dependence on number of friends
Consider: connectedness of friends
x and y have three friends in the group
x’s friends are independent
’ fi d ll td x y
y’s friends are all connected
Who is more likely to join?
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Competing sociological theories x y
Competing sociological theories
Information argument [Granovetter ‘73]
SSocial i lcapital it l argument t [C [Coleman l ’88]
Information argument:
Unconnected friends give independent support
Social capital argument:
Safety/truest advantage in having friends who
know each other
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LiveJournal: 1 million users, 250,000 groups
Social capital p argument g wins!
Prob. of joining increases with
adjacent members.
[Backstrom et al. KDD ‘06]
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Large anonymous online retailer
(June 2001 to May 2003)
15 15,646,121 646 121 recommendations dti
3,943,084 distinct customers
548 548,523 523 products recommended
Products belonging to 4 product groups:
books
DVDs
music
VHS
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t 1 < t 2 < … < t n
t 1
t 4
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
t 3
t 2
legend
bought but didn’t
receive i a di discountt
bought and
received a discount
received a recommendation
but didn’t buy
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There are relatively few DVD titles, but DVDs account for ~ 50% of recommendations.
Recommendations pper pperson
DVD: 10
books and music: 2
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 p p as DVDs but used only y
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 music the
number of unique edges is smaller than the number of nodes – the networks are highly
disconnected
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What role does the product category play?
products customers
recommenda-
buy + get buy + no
edges
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
FFull ll 542 542,719 719 33,943,084 943 084 15 15,646,121 646 121 33,153,676 153 676 91 91,322 322 79 79,164 164
high
low
people
recommendations
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Some products are easier to recommend than
others
product d t category t
number of buy forward
bits recommendations
Book 65,391 15,769 24.2
DVD 16,459 7,336 44.6
Music 7,843 1,824 23.3
Video 909 250 27.6
Total 90 90,602 602 25 25,179 179 27 27.8 8
percentt
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Does sending more recommendations
influence more purchases?
Number
of Purchaases
7
6
5
4
3
2
1
0
20 40 60 80 100 120 140
Outgoing Recommendations
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What is the effectiveness of subsequent
recommendations?
Probbability
of buying
007 0.07
0.06
0.05
0.04
0.03
002 0.02
5 10 15 20 25 30 35 40
Exchanged recommendations
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consider successful recommendations in terms of
av. 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 p=0 01 level)
fiction
romance (1.78), horror (1.81)
teen (1.94), children’s books (2.06)
comics i (2 (2.30), 30) sci‐fi i fi (2 (2.34), 34) mystery t and d th thrillers ill (2 (2.40) 40)
nonfiction
sports (2.26)
home & garden (2.26)
travel (2.39) (2 39)
higher than average success rate (statistically significant)
professional & technical
medicine (5.68)
professional & technical (4 (4.54) 54)
engineering (4.10), science (3.90), computers & internet (3.61)
law (3.66), business & investing (3.62)
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47,000 47 000 customers responsible for the 2.5 25out out of
16 million recommendations in the system
29% success rate per recommender of an anime
DVD
Giant component covers 19% of the nodes
Overall, recommendations for DVDs are more
likely to result in a purchase (7%), but the anime
community i stands d out
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Variable transformation Coefficient
const -0.940 ***
# recommendations ln(r) 00.426 426 ***
# senders ln(n s) -0.782 ***
# recipients ln(n ln(nr) ) -11.307 307 ***
product price ln(p) 0.128 ***
# reviews ln(v) -0 -0.011 011***
avg. rating ln(t) -0.027 *
R2 R 074
2 0.74
significance at the 0.01 (***), 0.05 (**) and 0.1 (*) levels
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94% of users make first recommendation without having g
received one previously
Size of giant connected component increases from 1% to
2 2.5% % of f the h network k (100,420 (100 20 users) ) – small! ll!
Some sub‐communities are better connected
24% out of f 18,000 18 000 users ffor westerns on DVD V
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
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Products suited for Viral Marketing:
small and tightly knit community
few reviews, senders, and recipients
but sending more recommendations helps
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 small, tightly knit communities
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How big are cascades?
What are the building
blocks of cascades?
973
938
Medical guide book DVD
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Given a (social) network
A process by spreading over the network
creates a graph (a tree)
Social network
Let’s count cascades
Cascade
(propagation graph)
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is the most common cascade subgraph
It accounts for ~75% cascades in books, CD
and VHS, , only y 12% of DVD cascades
is 6 (1.2 for DVD) times more frequent
than
For DVDs is more frequent than
Chains ( ) are more frequent than
iis more ffrequent tth than a collision lli i
( ) (but collision has less edges)
Late split ( ) is more frequent than
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
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Stars ( (“no no propagation propagation”) )
Bipartite cores (“common friends”)
Nodes having same friends
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Coount
10 6
10 4
10 2
10 0 10 0
= 1.8e6 x -4.98
10 1
steep t drop‐off d ff books
very few large cascades
10 2
x = Cascade size (number of nodes)
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DVD cascades can grow large
Possibly as a result of websites where people
sign up to exchange recommendations
Count
10 4
10 2
10 0 10 0
~ x -1.56
10 1
10 2
shallow drop off –fat tail
a number of large cascades
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10 3
x = Cascade size (number of nodes)

The pprobability yof observing g a cascade on x
nodes follows: p(x) ~ x ‐2
Count
x = Cascade size (number of nodes)
10/27/2009 Jure Leskovec, Stanford CS322: Network Analysis
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Cascade sizes follow a heavy‐tailed y distribution
Viral marketing:
Books: steep drop‐off: power‐law exponent ‐5
DVDs: s larger agecascades: cascades exponent epoe ‐1.55
Blogs:
Power‐law exponent ‐2
However, o e e , it is s not o a ssimple peba branching c gpocess process
A simple branching process (a on k‐ary tree):
Every node infects each of k of its neighbors with prob. p
gives exponential cascade size distribution
Questions:
What role does the underlying social network play?
CCan make k a step t towards t d more realistic li ti cascade d generation ti
(propagation) model?
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1) Randomly pick blog to
infect, infect add to cascade. cascade
1
B1 1 B2 1 3
B 3
1
B 4
3) Add infected neighbors
to cascade.
1
B1 1 B2 1 3
B 3
1
B 4
2
2
B 1
2) Infect each in‐linked
neighbor with probability
1
B1 1 B2 1 3
B 3
1
B 4
2
4) Set node infected in (i) to
uninfected.
1
B1 1 B2 B 1 B 1
B4 B4 1 B3 B 3
4
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1
2
B 1

Generative model
produces realistic
cascades
β=0.025
Count
Count
Cascade size
Count
Coount
Cascade node in‐degree
Most frequent cascades Size of star cascade Size of chain cascade
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Blogs – information epidemics
Which are the influential/infectious blogs?
Viral marketing
Who are the trendsetters?
Influential people?
Disease spreading
Where to place p monitoring g stations to detect
epidemics?
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