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Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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CSE 6240: Web Search and Text Mining. Spring 2020

Cascades and Contagion

• Prof. Srijan Kumar

http://cc.gatech.edu/~srijan

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Today’s Lecture

• Introduction
• Decision based models of diffusion

– Single Adoption – Multiple Adoption

• Probabilistic models of diffusion

– SEIR model – Independent cascade model

These slides are borrowed from Prof. Jure Leskovec’s CS224W class.

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Epidemics vs Cascade Spreading

• In decision-based models nodes make

decisions based on pay-off benefits of adopting one strategy or the other.

• In epidemic spreading:

– Lack of decision making – Process of contagion is complex and unobservable

• In some cases it involves (or can be modeled as)

randomness

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Simple model: Branching Process

• First wave: A person carrying a disease

enters the population and transmits to all she meets with probability 𝑟. She meets 𝑒 people, a portion of which will be infected.

• Second wave: Each of the 𝑒 people goes

and meets 𝑒 different people. So we have a second wave of 𝑒 ∗ 𝑒 = 𝑒% people, a portion

• f which will be infected.
• Subsequent waves: same process

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Example with k=3

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Spreading Models of Viruses

Virus Propagation: 2 Parameters:

• (Virus) Birth rate β:

– probability that an infected neighbor attacks

• (Virus) Death rate δ:

– Probability that an infected node heals

Infected Healthy N N1 N3 N2

• Prob. β
• Prob. δ

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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More Generally: S+E+I+R Models

• General scheme for epidemic models:

– Each node can go through phases:

• Transition probs. are governed by the model parameters

S…susceptible E…exposed I…infected R…recovered Z…immune

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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SIR Model

• SIR model: Node goes through phases

– Models chickenpox or plague:

• Once you heal, you can never get infected again
• Assuming perfect mixing: The network is a

complete graph

• The model dynamics are:

Susceptible Infected Recovered time Number of nodes

dI dt = βSI −δI dS dt = −βSI dR dt =δI

I(t) S(t) R(t) 𝛾 𝜀

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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SIS Model

• Susceptible-Infective-Susceptible (SIS)

model

• Cured nodes immediately become susceptible
• Virus “strength”: 𝒕 = 𝜸 / 𝜺
• Node state transition diagram:

Susceptible Infective

Infected by neighbor with prob. β Cured with

• prob. δ

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

SIS Model

• Models flu:

– Susceptible node

becomes infected

– The node then

heals and become susceptible again

• Assuming perfect

mixing (a complete graph):

Susceptible Infected

I SI dt dI d b

• =

I SI dt dS d b +

• =

time Number of nodes

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I(t) S(t)

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Question: Epidemic threshold 𝝊

• SIS Model:

Epidemic threshold of an arbitrary graph G is τ, such that:

– If virus “strength” s = β / δ < τ the epidemic can not happen (it eventually dies out)

• Given a graph what is its epidemic

threshold?

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Epidemic Threshold in SIS Model

• Fact: We have no epidemic if:

β/δ < τ = 1/ λ1,A

► λ1,A alone captures the property of the

graph! (Virus) Birth rate (Virus) Death rate Epidemic threshold largest eigenvalue

• f adj. matrix A of G

[Wang et al. 2003]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

Experiments on an Small Graph

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100 200 300 400 500 250 500 750 1000

Time Number of Infected Nodes

δ: 0.05 0.06 0.07 Oregon β = 0.001

s=β/δ > τ (above threshold) s=β/δ = τ (at the threshold) s=β/δ < τ (below threshold)

10,900 nodes and 31,180 edges

[Wang et al. 2003]

Autonomous Systems Graph

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Experiments

• Does it matter how many people are

initially infected?

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Modeling Ebola with SEIR

[Gomes et al., 2014]

[Gomes et al., Assessing the International Spreading Risk Associated with the 2014 West African Ebola Outbreak, PLOS Current Outbreaks, ‘14]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Example: Ebola

S: susceptible individuals, E: exposed individuals, I: infectious cases in the community, H: hospitalized cases, F: dead but not yet buried, R: individuals no longer transmitting the disease

[Gomes et al., Assessing the International Spreading Risk Associated with the 2014 West African Ebola Outbreak, PLOS Current Outbreaks, ‘14]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Application: Rumor spread modeling using SEIZ model

References: 1. Epidemiological Modeling of News and Rumors on Twitter. Jin et al. SNAKDD 2013 2. False Information on Web and Social Media: A survey. Kumar et al., arXiv :1804.08559

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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SEIZ model: Extension of SIS model

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Recap: SIS model

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Details of the SEIZ model

Notation:

– S = Susceptible – I = Infected – E = Exposed – Z = Skeptics

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Dataset

Tweets collected from eight stories: Four rumors and four real

REAL EVENTS RUMORS

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Method: Fitting SEIZ model to data

• SEIZ model is fit to each cascade to minimize the

difference |𝐽(𝑢) – 𝑢𝑥𝑓𝑓𝑢𝑡(𝑢)|: – 𝑢𝑥𝑓𝑓𝑢𝑡(𝑢) = number of rumor tweets – 𝐽(𝑢) = the estimated number of rumor tweets by the

model

• Use grid-search and find the parameters with

minimum error

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Fitting to “Boston Marathon Bombing”

SEIZ model better models the real data, especially at initial points

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Fitting to "Pope resignation” data

SEIZ model better models the real data, especially at initial points

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Rumor detection with SEIZ model

Notation: S = Susceptible I = Infected E = Exposed Z = Skeptics

New metric:

All parameters learned by model fitting to real data (from previous slides)

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Rumor detection by RSI

Parameters obtained by fitting SEIZ model efficiently identifies rumors vs. news

Rumors

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Today’s Lecture

• Introduction
• Decision based models of diffusion

– Single Adoption – Multiple Adoption

• Probabilistic models of diffusion

– SEIZ model – Independent cascade model

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Linear Threshold Model

, active neighbor of v w v w v

b q ³

å

• A decision-based model
• A node v has random threshold 𝜄𝑤 ~ U[0,1]
• A node v is influenced by each neighbor w

according to a weight 𝑐𝑤,𝑥 such that

• A node v becomes active when >=

(weighted) 𝜾𝒘 fraction of its neighbors are active

, neighbor of

1

v w w v

b £

å

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Linear Threshold Model

Inactive Node Active Node Threshold Active neighbors

v w

0.5 0.3 0.2 0.5 0.1 0.4 0.3 0.2 0.6 0.2

Stop!

U X

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Probabilistic Contagion

• Independent Cascade Model

– Directed finite 𝑯 = (𝑾, 𝑭) – Set 𝑻 starts out with new behavior

• Say nodes with this behavior are “active”

– Each edge (𝒘, 𝒙) has a probability 𝒒𝒘𝒙 – If node 𝒘 is active, it gets one chance to make 𝒙 active, with probability 𝒒𝒘𝒙

• Each edge fires at most once
• Does scheduling matter? No
• If 𝒗, 𝒘 are both active at the same time, it doesn’t

matter which tries to activate 𝒙 first

– But the time moves in discrete steps

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Independent Cascade Model

• Initially some nodes S are active
• Each edge (u,v) has probability (weight) puv
• When node u becomes active/infected:

– It activates each out-neighbor v with prob. puv

• Activations spread through the network!

0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.2

e g f c b a d h i f g e

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Independent Cascade Modal

• Independent cascade model

is simple but requires many parameters!

– Estimating them from data is very hard [Goyal et al. 2010]

• Solution: Make all edges have the same

weight (which brings us back to the SIR model)

– Simple, but too simple

• Can we do something better?

0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.2

e g f c b a d h i f g e

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Exposures and Adoptions

• From exposures to adoptions

– Exposure: Node’s neighbor exposes the node to the contagion – Adoption: The node acts on the contagion

[KDD ‘12]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Exposure Curves

• Exposure curve:

– Probability of adopting new behavior depends on the total number of friends who have already adopted

• What’s the dependence?

k = number of friends adopting

• Prob. of adoption

k = number of friends adopting

• Prob. of adoption

“Probabilistic” spreading: Viruses, Information Critical mass: Decision making … adopters

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Exposure Curves

• From exposures to adoptions

– Exposure: Node’s neighbor exposes the node to information – Adoption: The node acts on the information

• Examples of different adoption curves:

Prob(Infection) # exposures Probability of infection ever increases Nodes build resistance [KDD ‘12]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Diffusion in Viral Marketing

• Senders and followers of

recommendations receive discounts on products

• Data: Incentivized Viral Marketing

program

– 16 million recommendations – 4 million people, 500k products

10% credit 10% off

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

Exposure Curve: Validation

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Probability of purchasing

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10 20 30 40

DVD recommendations (8.2 million observations) # recommendations received

[Leskovec et al., TWEB ’07]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Exposure Curve: LiveJournal

• Group memberships spread over the

network:

– Red circles represent existing group members – Yellow squares may join

• Question:

– How does prob. of joining a group depend on the number of friends already in the group?

[Backstrom et al. KDD ‘06]

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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• LiveJournal group membership

k (number of friends in the group)

• Prob. of joining

[Backstrom et al., KDD ’06]

Exposure Curve: LiveJournal

Srijan Kumar, Georgia Tech, CSE6240 Spring 2020: Web Search and Text Mining

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Today’s Lecture

• Introduction
• Decision based models of diffusion

– Single Adoption – Multiple Adoption

• Probabilistic models of diffusion

– SEIZ model – Independent cascade model