Information Diffusion Mary McGlohon CMU 10-802 3/23/10

Information DiffusionMary McGlohonCMU 10-8023/23/10

Outline•Intro: Models for diffusion

▫Epidemiological: SIS/SIR/SIRS▫Threshold models

•Case studies▫SIR: Info diffusion in blogs▫SIS: Cascades in blogs▫Timing: Cascades in chain letters▫A closer look: Network-based Marketing

Epidemiological: SIS•Susceptible, Infected, Susceptible

▫Infected for tI timesteps▫While infected, transmits with probability b▫After tI steps, returns to susceptible

Epidemiological: SIR•Susceptible, Infected, Removed

▫Infected for tI timesteps▫While infected, transmits with probability b▫After tI steps, goes to removed/recovered

Epidemiological: SIRS•Susceptible, Infected, Removed,

Susceptible▫Combination of SIS+SIR▫After tI steps, goes to removed/recovered

▫After tR steps, returns to susceptible

Epidemiological: Networks•Historically, SIS/SIR assumed a person

could infect anybody else, full clique. There is an epidemic threshold in SIS.

•For random power-law networks, threshold=0 [Pastor-Satorras+Vespignani]▫(But not for PL networks with high

clustering coefficients [Egu´ıluz and Klemm])

Threshold Models•Each node in network has weighted

threshold•If adopted neighbors reaches threshold,

the node adopts.




Info Diffusion in Blogs•D. Gruhl, R. Guha, Liben D. Nowell, A.

Tomkins. Information Diffusion Through Blogspace. In WWW '04 (2004).

•Goal: How do topics trend in blogs, and how can we model diffusion of topics?

Info Diffusion in Blogs•Data: Crawled 11K blogs, 400K posts.•Found 34o topics:

▫apple arianna ashcroft astronaut blair boykin bustamante chibi china davis diana farfarello guantanamo harvard kazaa longhorn schwarzenegger udell siegfried wildfires zidane gizmodo microsoft saddam

Info Diffusion in Blogs•Topics =

Chatter + Spikes▫Chatter:

Alzheimer▫Spike: Chibi▫Spiky Chatter:

Microsoft

Info Diffusion in Blogs•Modeled as SIR

▫Some set of authors is infected to write about a topic

▫Then propagate, as others write new posts on that topic

▫Measure the topic over time and other properties

•Fit using EM▫Compute probability of propagation along

each edge

Info Diffusion in Blogs•Validation:

▫Synthetic Used modified Erdos-Renyi graph, created

propagation Found that EM was able to identify

transmission of most edges▫Real

Found “internet-only” topics Looked at most highly ranked expected

transmission links, identified a real link in 90% of cases

Info Diffusion in Blogs•Limitations of SIR

▫No multiple postings▫No “stickiness”, which topics resonate with

whom ▫No time limiting factor in topics▫“Closed world assumption”

No outside influences after initial infection




Cascades in Blogs•Jure Leskovec, Mary Mcglohon,

Christos Faloutsos, Natalie Glance, Matthew Hurst. Cascading Behavior in Large Blog Graphs: Patterns and a Model. In Society of Applied and Industrial Mathematics: Data Mining (SDM07) (2007)

•Goal: What do cascades (conversation trees) in blogs look like, and how can we model them?

17

Cascades in Blogs•Data:

▫Gathered from August-September 2005▫Used set of 44,362 blogs, 2.4 million posts▫245,404 blog-to-blog links

Time [1 day]

Nu

mb

er

of

post

s

Jul 4

Aug 1Sep 29

18

Cascades in Blogs

Blogosphere

B1 B2

B4B3

Cascades

d

e

b c

e

a

a

b c

de

“Star” “Chain”

•What is the timing of links?•What are cascade sizes?•What are cascade shapes?

19

Cascades in Blogs

• What is the timing of links?• Does popularity decay at a constant rate?

• With an exponential (“half life”)?

Linear-linear scale Log-linear scale Log-log scale

20

Cascades in Blogs•Observation: The probability that a post written at time tp acquires a link at time tp + Δ is:

p(tp+Δ) ∝ Δ-1.5

log(days after post)

log

( #

in

-lin

ks)

slope=-1.5

(Linear-linear scale)

21

Cascades in Blogs

• How are cascade sizes distributed?

• Geometric distribution?

Linear-linear scale Log-linear scale Log-log scale

d

e

b c

e

a

22

Cascades in Blogs•Q: What size distribution do cascades follow? Are large cascades frequent?

•Observation: The probability of observing a cascade of n blog posts follows a Zipf distribution:

p(n) ∝ n-2

log(Cascade size) (# of nodes)

log

(Cou

nt)

slope=-2d

e

b c

e

a

23

Cascades in Blogs

• How are cascade shapes distributed?

• More stars? More chains?

d

e

b c

e

a

log(Size) of chain (# nodes)

log

(Cou

nt)

a=-8.5

log(Size) of star (# nodes)

log

(Cou

nt) a=-3.1

Cascades in Blogs• Q: What is the distribution of particular cascade

shapes?• Observation: Stars and chains in blog cascades

also follow a power law, with different exponents (star -3.1, chain -8.5).

24

25

Cascades in Blogs• Based on SIS model in epidemiology

▫Randomly pick blog to infect, add post to cascade

▫Infect each in-linked neighbor with probability ▫Add infected neighbors’ posts to cascade.▫Set old infected node to uninfected.

B1 B2

B4B3

26




B1 B2

B4B3

p1,1

27




B1 B2

B4B3

p1,1

28




B1 B2

B4B3

p1,1

p4,1 p2,1

29




B1 B2

B4B3

p1,1

p4,1 p2,1

30




B1 B2

B4B3

p1,1

p4,1 p2,1

31




B1 B2

B4B3

p1,1

p4,1 p2,1

p4,1

32

Cascades in BlogsMost frequent cascades

model

data

log(Cascade size) (# nodes)

log

(Cou

nt

)

log

(Cou

nt

)

log(Star size)

log

(Cou

nt)

log(Chain size)

DataModel

Cascades in Blogs•Limitations of SIS

▫Closed world assumption▫Forced to set infection probability low to

avoid large epidemics– possibly limits stars.▫No time limit, possibly overestimates

chains.




Chain Letter Cascades•David Liben-Nowell, Jon Kleinberg.

Tracing the Flow of Information on a Global Scale Using Internet Chain-Letter Data. Proceedings of the National Academy of Sciences, Vol. 105, No. 12.(March 2008), pp. 4633-4638.

•Goal: How can we trace the path of a meme, and explain these paths?

Chain Letter Cascades•Data: NPR chain letter records.

▫People directed to sign and send back to admin

▫Had several copies of lists, overlaps▫Reconstructed the trees using edit distance

Chain Letter Cascades•A reconstruction:

Chain Letter Cascades•The tree:

Chain Letter Cascades•How to model?

▫These trees have much longer paths▫2 considerations

Spatial distance (geographic) Timing

Chain Letter Cascades•Model: based on a delay distribution•Nodes reply-to-all, so latecomers just

append.

Chain Letter Cascades•Validation: Simulated on a real social

network (Livejournal), produced similar trees.

•Limitations:▫The chain letter mechanism is somewhat

nontraditional diffusion▫Closed-world assumption is perhaps OK



•Case studies▫SIR: Info diffusion in blogs▫SIS: Cascades in blogs▫Timing: Cascades in chain letters▫A closer look: Network-Based Marketing

Network-Based Marketing•Shawndra Hill, Foster Provost, Chris

Volinsky. Network-based marketing: Identifying likely adopters via consumer networks. Statistical Science, Vol. 22, No. 2. (2006), pp. 256-275.

•Question: Is there statistical evidence that network linkage directly affects product adoption?

Network-Based Marketing•Data: Direct-mail marketing campaign for

adopting a new communications service.▫21 target segments, millions of customers▫Divided based on:

Loyalty Previous adoptions Predictive scores based on other

demographics Different marketing campaigns (postcards,

calls)

Network-Based Marketing

Network-Based Marketing•Hypothesis: A customer who has had

direct communication with a subscriber is more likely to adopt.▫Data: (incomplete) network information

ID of users, Timestamp, Duration•To test, added a “NN” (network neighbor)

flag to features if a customer had communicated with a subscriber. (0.3% overall)

Network-Based Marketing•Created baseline statistical model based

on node attributes.▫“Loyalty”- how consumer used services in

past▫Geographic - city, state, etc.▫Demographic- census-type data, credit

score

•Added a variable for NN, performed logistic regression on each segment, with response variable being “take rate”.

Network-Based Marketing•Log-odds ratio for NN variable

Network-Based Marketing•Take rates Lift ratios

Network-Based Marketing•Added a “segment 22” consisting of only

NN, but made up of less promising customers.

Network-Based Marketing•What about causality? What if the

adoption is due to homophily?

•To address this, sample from non-NN to make a similar data set to the NN group.

•Performed logistic regression, showed that network impact is highest for the least loyal group.

Network-Based Marketing•Lift curve for NN

Network-Based Marketing•What about other network features?

▫Degree, transactions, connectedness, etc.•Added network features to existing

regression model, tested lift.

Network-Based Marketing•Lift in sales for both models

Conclusion•Several ways of approaching the study of

diffusion•No model is perfect. Considerations:

▫Closed world assumption vs. external effects

▫Homophily and node attributes▫Network structure

•Network information is valuable, but (usually) does not account for everything.

Documents

Information Diffusion Mary McGlohon CMU 10-802 3/23/10