network dynamics i: bayesian learning, information...

Network Dynamics I: Bayesian Learning, InformationCascades

Web Science (VU) (706.716)

Elisabeth Lex

ISDS, TU Graz

April 8, 2019

Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 1 / 71

Repetition

Networks and Random Graphs

Small World Phenomenon

Repetition

Sheep herd

Motivation

Individuals easily influenced by decisions of others, especially in socialand economic situations.

Ex.: opinions, products to buy, political positions

Today: Why does such influence occur and how can we model this?

Motivation

Example: Look Up

In the 60s, some experimenters had groups of people 1-15 look up inthe sky.

They then watched to see if anyone else looked up as well.

With one looking up, most ignored him.

With 5 looking up, many stopped.

With 15, 45% stopped and kept looking up.

Example: Look Up

Example: Restaurant

You go to a unfamiliar city such as San Francisco.

Where do you decide to eat?

Restaurant A and B both look nice on the outside and have a similarmenu.

A has a small crowd.

B is empty.

Which do you go to?

Probably A.

But: What if from the outside B looks slightly better?

How about if a colleague said he heard B was good?

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Example: Restaurant

B is empty.

Which do you go to?

Probably A.

Observations

In all these cases, decisions are made sequentially.

People make decisions based on inferences from what earlier peoplehave done

Individuals may imitate behavior of others but not mindless

Sometimes it is rational for an individual to follow the crowd even ifthe individual’s own information suggests an alternative choice

Observations

Reasons for Imitation

Direct-Benefit Effects: Actions of others affect you directly

E.g. Becoming part of a social networking site - choosing an optionthat has a large user population

Informational Effects: Actions of others affect you indirectly bychanging your information

How to model this?

Decision-based Models: adopt new behaviors if k others do it

Probabilistic Models: adopt a behavior with some probability fromneighbors in the network

How to model this?

Herding & Information Cascades

Herding

A decision needs to be made

People make the decision sequentially

Each person has some private information that helps with thedecision

This private information cannot be directly observed but one cansee what people do

This way, inferences about their private information can be made

Herding

A Simple Herding Example (1/2)

Large group of students - participants

Consider two urns with 3 balls that is either:

Majority-blue: blue, blue, red or

Majority-red: red, red, blue

Student sequentially guess whether urn is majority-blue ormajority-red

Experiment:

One by one, each student:

Draws a ball

Privately looks at its color and puts it back

Publicly announces her guess

If guess correct - receives monetary reward

Experiment:

Draws a ball

Experiment:

Draws a ball

Experiment:

Draws a ball

Experiment:

Draws a ball

Experiment:

Draws a ball

What happens? (1/2)

Example:

1st student: draws blue

Guess: urn is majority-blue

2nd student: draws red

Guess: urn is majority-red

3rd person: draws blue

Guess: urn is majority-blue Why?

Student goes with her own color as students 1 & 2 made differentguesses

What happens? (1/2)

Example:

What happens? (1/2)

Example:

Guess:

urn is majority-blue

What happens? (1/2)

Example:

What happens? (1/2)

Example:

What happens? (1/2)

Example:

Guess:

urn is majority-red

What happens? (1/2)

Example:

What happens? (1/2)

Example:

What happens? (1/2)

Example:

Guess:

urn is majority-blue Why?

What happens? (1/2)

Example:

What happens? (1/2)

Example:

What happens? (1/2)

Example:

What happens? (1/2)

Another example:

1st student: draws red

Guess: urn is majority-red Why?

Go with guesses of 1st & 2nd student as both guessed urn ismajority-red

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

Guess:

urn is majority-red

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

Guess:

urn is majority-red

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

Guess:

urn is majority-red Why?

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

What happens? (1/2)

Another example:

Information Cascades

Definition

An information cascade develops when people abandon their owninformation in favor of inferences based on earlier people’s actions

What happens? (2/2)

4th student and onward:

Say first 2 guesses were both blue, 3rd person will also guess blue,regardless of what she saw

4th heard blue 3 times in a row, but knows that 3rd guess conveys noinformation

4th in the same situation as 3rd - should also guess blue

Continues with all subsequent students since everyone’s best strategyis to rely on limited genuine information available

What happens? (2/2)

Modeling this type of reasoning: Bayes’s Rule (1/2)

Mathematical model for when information cascades occur

Based on computing probabilities of events (e.g. event is “the urn ismajority-blue”)

Whether an event (not) occurs is the result of certain randomoutcomes (e.g. which urn was placed in the room)

We need to estimate the conditional probability of event A given thatevent B has occurred:

The fraction of the area of region B occupied by the joint event A∩B:

P[A|B] =P[A ∩ B]

P[B](1)

We can follow the Bayes’ rule:

P[A|B] =P[A] ∗ P[B|A]

P[B](2)

where P[A] is the prior probability of A and P[A|B] is the posteriorprobability of A given B

P[A|B] =P[A ∩ B]

P[B](1)

P[A|B] =P[A] ∗ P[B|A]

P[B](2)

P[A|B] =P[A ∩ B]

P[B](1)

P[A|B] =P[A] ∗ P[B|A]

P[B](2)

Our Example with Bayes’s Rule (1/3)

Each person tries to estimate conditional probability that urn ismajority-blue or majority-red given what she has seen and heardShe should guess

Pr [majority-blue|what she has seen and heard] > 12 ]

majority-red otherwise

If both conditional probabilities 0.5 - doesn’t matter what she guesses

Our Example with Bayes’ Rule (2/3)

Prior probabilities Pr [majority-blue] and Pr [majority-red] = 12

Probabilities for the 2 urns Pr [blue|majority-blue] =Pr [red|majority-red] = 2

Let’s assume that 1st person draws blue ball

Pr [majority-blue|blue] = Pr [majority-blue]∗Pr [blue|majority-blue]Pr [blue]

Ergo: Conditional probability greater than 12 , 1st person should guess urn

is majority-blue

Calculation for 3rd person:

Pr [majority-blue|blue, blue, red] =Pr [majority-blue]∗Pr [blue, blue, red|majority-blue]

Pr [blue, blue, red]= 2

Ergo: Conditional probability greater than 12 , 3rd person should guess urn

is majority-blue

Calculation for 3rd person:

Pr [majority-blue|blue, blue, red] =Pr [majority-blue]∗Pr [blue, blue, red|majority-blue]

Pr [blue, blue, red]= 2

Ergo: Conditional probability greater than 12 , 3rd person should guess urn

is majority-blue

Discussion

Cascade can occur easily

Can lead to non-optimal outcomes

In our example, with prob 1/3x1/3 = 1/9, the first two would see thewrong color, from then on, the whole population would guess wrong

Cascades fragile despite their potential to produce long runs ofconformity

Suppose, first 2 guesses are majority-blue

People 100 and 101 draw red and cheat, i.e. they show the drawn balls

Person 102 has then 4 pieces of honest information - she should guessbased on her own color

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Discussion

Cascade is broken!

Sequential Decision-Making and Cascades

Each person gets a private signal that remains unknown to others

Each person releases a public action

Information dependence leads to cascade

People align their behavior to what they observe from the crowd

P1 follows own private signal

P2 follows own signal as signal P2 different from P1’s

P3 follows own signal as P1 and P2 different

P4 follows own signal as P3 made genuine guess

P6 follows P5 and P4 as both have same signals → begin of cascade

P6 follows P5 and P4 as both have same signals

→ begin of cascade

Let’s consider the perspective of a person numbered N

If number of accepts among people before N is equal to number ofrejects, N follows own signal

If number of accepts among people before N differs from number ofrejects by 1, N follows own signal

If number of accepts among people before N differs from number ofrejects by 2 or more, N follows majority

Then, N + 1,N + 2, .. know that N ignored own signal. So, theyfollow the majority and cascade has begun

Summary

People initially rely on own private information

Observe what others decide

If number of acceptances and rejections of decision is ≥ 2, peoplefollow majority decision

Over time, population ignores own information and follow the crowd -while still being fully rational

Summary

Problems of cascades

Cascades can be wrong

Cascades can be based on very little information

Cascades are fragile

Careful if you want to follow the crowd :)

Applications and Practical Value

Early adopters in Online Marketing: attempting to initiate a buyingcascade

Collaboration and consensus building: can be wise to makecollaborators reach partial conclusions before entering phase ofcollaboration (e.g. hiring committees)

Models of influence

red star bird, sign of the rebel alliance

Example: Organize a revolt

You live an in oppressive society

You know of a rebel group that plans to fight against the empiretomorrow

If a lot of people show up, the empire will fall

If only a few people show up, the rebels will be arrested and it wouldhave been better had everyone stayed at home

Organize a revolt: A Model

Personal threshold k: “I will show up if am sure at least k people intotal (including myself) will show up”

Each node only knows the thresholds and attitudes of all their directfriends.

Can we predict if a revolt can happen based on the network structure?

Ex.: Which network can have a revolt?

The last one. Why?Because nodes u,v, and w share common knowledge about the thresholds

of their friends

The last one. Why?

Because nodes u,v, and w share common knowledge about the thresholdsof their friends

The last one. Why?Because nodes u,v, and w share common knowledge about the thresholds

of their friends

Models of influence

Decisions often correlated with number/fraction of friends

The more friends, the more influence

Book recommendation: Christakis and Fowler: Connected(http://connectedthebook.com/index.html)

Models of influence

Models to capture influence

Deterministic models

Linear Threshold Model

Independent Cascade Model

Probabilistic Models

SIRSIS

Output of the model fully determined by the parameter values andthe initial conditions

E.g. if k of my friends buy this item, I will buy it as well

Linear Threshold Model (1/2)

Nodes have thresholds

Edges (directed!) have non-negative weights

Nodes that receive weighted influence equal or above their thresholdbecome active

Examples: riots, mobile phone networks

[Kempe, Kleinberg and Tardos, KDD 2003]

Act if number of neighbors N, who also willing to act is ≥ t

Each directed edge (u,w) ∈ E has non negative weight b(u,w)

For each node v ∈ E , total incoming edge weights sum to less than orequal to one

Each node v chooses threshold tv randomly from a uniform distributionin an interval between 0 and 1

In each time step, all nodes remain active that were active in theprevious step

Nodes that satisfy the following condition will be activated:∑w∈Nv ,w is active bw ,v ≥ tv

At each time step, an inactive node influenced by all of its activeneighbors

An active node influences its inactive neighbors according to theweights

At each step, an inactive node becomes active if total weight of itsincoming neighbors is at least tv

Thresholds tv selected randomly due to lack of knowledge of thetendency of node

Express different levels of tendency of nodes to adopt an idea orinnovation

Linear Threshold Model: Example

Figure: Linear Threshold Model Diffusion Process, t = 0.5

Linear Threshold Model - Example applications

Study social influence of users1

Study diffusion of innovation in social networks2

1https://www.rug.nl/research/portal/files/42545633/2016_PNAS_

Threshold_model_networks.pdf2https:

//www.princeton.edu/~naomi/publications/2017/ZhoSriLeoCDC2017.pdfElisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 44 / 71

No thresholds

When node becomes active, it has single chance of activating eachcurrently inactive neighbors

Probability of succeeding given by edge weights

The independent cascade model focuses on the sender’s rather than thereceiver’s view

A node w, once activated at step t, has one chance to activate eachof its neighbors randomly

For a neighboring node v, the activation succeeds with probabilitypw ,v (e.g. p = 0.5)

If the activation succeeds, then v will become active at step t + 1

In the subsequent rounds, w will not attempt to activate v anymore.

The diffusion process starts with an initial activated set of nodes,then continues until no further activation is possible

Probabilistic models

Aka Contagion, Propagation: Randomly occur as a result of socialcontact - no decision making involved

Models of influence or disease spreading

An infected node tries to “push” the disease to an uninfected node

Ex.: Catching a disease with some probability from each activeneighbor in the network

Involves some randomness and given the initial conditions, differentoutcomes possible

SIR model

SIR Model

A model to study propagation of diseases, epidemics (Kermack andMcKendrick, 1927)

Can also be used to study propagation of information, rumors,falsehoods,..

Suitable to model infections that result in immunity (or death) suchas chickenpox

SIR: Susceptible-Infective-Recovered Model (1/2)

Each node of a population N can be in one of the following states:

Susceptible: node healthy but not immune, i.e. can be infected

Infective: has the infection and can actively propagate it

Recovered: had the infection and is now immune

Infection rate β: probability to get infected by neighbor at time step

Recovery rate γ: probability of node to recover at time step

SIR model describes change in the population N of three groups (S, I,R) in terms of two parameters β and γ

SIR: Transition rates

Transition rate between S and I is βI

Transition rate between I and R is γ

Each node stays in the current state for a period of time - typically, arandom variable with exponential distribution

SIR Model: Algorithm

Initially: a fraction of nodes is in state I, all other nodes in the state S

Each node in state I remains infectious for period of time

We iterate over a period of time

We take a node from the network

Say the node is e.g. infected, it can

recover and enter state R if recovery rate γ is greater than R(t)

infect neighbors in state S if infection rate β is greater than S(t)

stay infected for a period of time (e.g. random number) based oninfection rate β - after that period, node enters state R

Nodes with state R (i.e. recovered) cannot be infected again and stayin R (they are immune)

SIR model

Model can also be described by following ordinary differential equations:

dt= −βS(t)I (t)

dt=βS(t)I (t)

N− γI ,

dt= γI (t).

Note that N = S + I + R

SIR Model

Diff equation can be solved numerically - e.g. Euler method (not partof this lecture!)

Gives us the following equations:

St+1 = St − (β ∗ St ∗ It)/NIt+1 = It + ((β ∗ St ∗ It)/N)− Rt

Rt+1 = γ ∗ It

Example: SIR

https://www.public.asu.edu/~hnesse/classes/sir.html

SIR Model: Discussion

Is the model a realistic scenario? What is missing?

Changes in the population: births and deaths

So-called vital dynamics are missing

Is the model a realistic scenario?

What is missing?

SIS model

SIS: Susceptible-Infective-Susceptible Model

Each node may be healthy (susceptible) or infected

Cured nodes immediately become susceptible

Virus strength: s = β/δ (spreading rate)

Epidemic threshold t: if s = β/δ < t, epidemic cannot happen

Epidemic Threshold

What should t be dependent on?

Average degree? Highest degree?

Variance of degree?

Diameter?

Epidemic Threshold

Variance of degree?

Diameter?

Epidemic Threshold

Variance of degree?

Diameter?

Epidemic Threshold

Variance of degree?

Diameter?

Epidemic Threshold

Theorem by Wang et al 2003: No epidemic if

δ< t =

λ1,A(3)

where λ1,A is the largest eigenvalue of the adjacency matrix A

Real-World Application of such models: Viral Marketing

Viral Marketing and Recommender Systems

How do recommendations and purchases propagate? [Leskovec, Adamic,Huberman 2006]

Product recommendation network

Viral Marketing3

Senders and followers of recommendations receive discounts onproducts

Recommendations are made to any number of people at the time ofpurchase

Only the recipient who buys first gets a discount

3The dynamics of viral marketing, J. LeskovecElisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 64 / 71

Viral Marketing3

Dataset

Large anonymous online retailer (June 2001 to May 2003)

15,646,121 recommendations

3,943,084 distinct customers

548,523 products recommended

4 product groups: books, DVDs, music, VHS

Dataset

Identifying the cascades

Role of product category

Figure: Red is high, blue is low. People are nodes, edges are recommendations

Recommendations for DVDs are more likely to result in a purchase

Some observations

Few DVDs, but DVDs make approx. 50% of recommendations

Music recommendations reached approx. same nr of people as DVDsbut used only 1/5 as many recommendations

Book recommendations reached the most people

Networks highly disconnected

Overall, 3.69 recommendations per node on 3.85 different products

Some observations

Viral Marketing

Does sending more recommendations influence more purchases?

Figure: Influence of number of sent recommendations on purchases

Summary

Herding and Information Cascades, Bayes Rule as Model

Decision based models: Linear Threshold Model, IndependentCascade Model

Probability based models: SIR, SIS

Viral Marketing as application scenario

Summary

Thanks for your attention - Questions?

elisabeth.lex@tugraz.at

Slides use figures from Chapter 16 and Chapter 19 of Networks, Crowdsand Markets by Easley and Kleinberg (2010)http://www.cs.cornell.edu/home/kleinberg/networks-book/http://web.stanford.edu/class/cs224w/handouts.htmlhttp://snap.stanford.edu/na09/11-viral-annot.pdf

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