network dynamics i: bayesian learning, information...
TRANSCRIPT
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 2 / 71
Repetition
Networks and Random Graphs
Small World Phenomenon
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 2 / 71
Repetition
Networks and Random Graphs
Small World Phenomenon
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 2 / 71
Sheep herd
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 3 / 71
Sheep herd
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 3 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 4 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 4 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 4 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 4 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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.
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 5 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 6 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 7 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 7 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 7 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 7 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 7 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 7 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 8 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 8 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 8 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 8 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 9 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 9 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 9 / 71
Herding & Information Cascades
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 10 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 11 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 11 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 11 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 11 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 11 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 11 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 12 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 12 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 12 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 12 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 12 / 71
A Simple Herding Example (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 13 / 71
A Simple Herding Example (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 13 / 71
A Simple Herding Example (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 13 / 71
A Simple Herding Example (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 13 / 71
A Simple Herding Example (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 13 / 71
A Simple Herding Example (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 13 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 14 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess:
urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess:
urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess:
urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red
Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
What happens? (1/2)
Another example:
1st student: draws red
Guess: urn is majority-red
2nd student: draws red
Guess: urn is majority-red
3rd person: draws blue
Guess: urn is majority-red Why?
Go with guesses of 1st & 2nd student as both guessed urn ismajority-red
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 15 / 71
Information Cascades
Definition
An information cascade develops when people abandon their owninformation in favor of inferences based on earlier people’s actions
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 16 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 17 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 17 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 17 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 17 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 17 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 18 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 18 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 18 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 18 / 71
Modeling this type of reasoning: Bayes’s Rule (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 19 / 71
Modeling this type of reasoning: Bayes’s Rule (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 19 / 71
Modeling this type of reasoning: Bayes’s Rule (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 19 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 20 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 20 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 20 / 71
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
3
Let’s assume that 1st person draws blue ball
Pr [majority-blue|blue] = Pr [majority-blue]∗Pr [blue|majority-blue]Pr [blue]
= 23
Ergo: Conditional probability greater than 12 , 1st person should guess urn
is majority-blue
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 21 / 71
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
3
Let’s assume that 1st person draws blue ball
Pr [majority-blue|blue] = Pr [majority-blue]∗Pr [blue|majority-blue]Pr [blue]
= 23
Ergo: Conditional probability greater than 12 , 1st person should guess urn
is majority-blue
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 21 / 71
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
3
Let’s assume that 1st person draws blue ball
Pr [majority-blue|blue] = Pr [majority-blue]∗Pr [blue|majority-blue]Pr [blue]
= 23
Ergo: Conditional probability greater than 12 , 1st person should guess urn
is majority-blue
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 21 / 71
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
3
Let’s assume that 1st person draws blue ball
Pr [majority-blue|blue] = Pr [majority-blue]∗Pr [blue|majority-blue]Pr [blue]
= 23
Ergo: Conditional probability greater than 12 , 1st person should guess urn
is majority-blue
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 21 / 71
Our Example with Bayes’ Rule (3/3)
Calculation for 3rd person:
Pr [majority-blue|blue, blue, red] =Pr [majority-blue]∗Pr [blue, blue, red|majority-blue]
Pr [blue, blue, red]= 2
3
Ergo: Conditional probability greater than 12 , 3rd person should guess urn
is majority-blue
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 22 / 71
Our Example with Bayes’ Rule (3/3)
Calculation for 3rd person:
Pr [majority-blue|blue, blue, red] =Pr [majority-blue]∗Pr [blue, blue, red|majority-blue]
Pr [blue, blue, red]= 2
3
Ergo: Conditional probability greater than 12 , 3rd person should guess urn
is majority-blue
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 22 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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!
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 23 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 24 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals
→ begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
P5 follows own signal as P4 made genuine guess
P6 follows P5 and P4 as both have same signals → begin of cascade
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 25 / 71
Sequential Decision-Making and Cascades
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 26 / 71
Sequential Decision-Making and Cascades
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 26 / 71
Sequential Decision-Making and Cascades
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 26 / 71
Sequential Decision-Making and Cascades
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 26 / 71
Sequential Decision-Making and Cascades
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 26 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 27 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 27 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 27 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 27 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 27 / 71
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 :)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 28 / 71
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 :)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 28 / 71
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 :)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 28 / 71
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 :)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 28 / 71
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 :)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 28 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 29 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 29 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 29 / 71
Models of influence
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 30 / 71
red star bird, sign of the rebel alliance
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 31 / 71
red star bird, sign of the rebel alliance
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 31 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 32 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 32 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 32 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 32 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 32 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 33 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 33 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 33 / 71
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?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 33 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 34 / 71
Ex.: Which network can have a revolt?
The last one. Why?
Because nodes u,v, and w share common knowledge about the thresholdsof their friends
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 34 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 34 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 35 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 35 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 35 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 35 / 71
Models to capture influence
Deterministic models
Linear Threshold Model
Independent Cascade Model
Probabilistic Models
SIRSIS
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 36 / 71
Models to capture influence
Deterministic models
Linear Threshold Model
Independent Cascade Model
Probabilistic Models
SIRSIS
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 36 / 71
Models to capture influence
Deterministic models
Linear Threshold Model
Independent Cascade Model
Probabilistic Models
SIRSIS
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 36 / 71
Models to capture influence
Deterministic models
Linear Threshold Model
Independent Cascade Model
Probabilistic Models
SIRSIS
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 36 / 71
Models to capture influence
Deterministic models
Linear Threshold Model
Independent Cascade Model
Probabilistic Models
SIRSIS
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 36 / 71
Deterministic models
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 37 / 71
Deterministic models
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 38 / 71
Deterministic models
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 38 / 71
Deterministic models
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 38 / 71
Linear Threshold Model
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 39 / 71
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]
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 40 / 71
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]
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 40 / 71
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]
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 40 / 71
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]
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 40 / 71
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]
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 40 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (1/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 41 / 71
Linear Threshold Model (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 42 / 71
Linear Threshold Model (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 42 / 71
Linear Threshold Model (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 42 / 71
Linear Threshold Model (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 42 / 71
Linear Threshold Model (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 42 / 71
Linear Threshold Model (2/2)
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 42 / 71
Linear Threshold Model: Example
Figure: Linear Threshold Model Diffusion Process, t = 0.5
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 43 / 71
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
Independent Cascade Model
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 45 / 71
Independent Cascade Model
No thresholds
When node becomes active, it has single chance of activating eachcurrently inactive neighbors
Probability of succeeding given by edge weights
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 46 / 71
Independent Cascade Model
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 47 / 71
Independent Cascade Model
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 47 / 71
Independent Cascade Model
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 47 / 71
Independent Cascade Model
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 47 / 71
Independent Cascade Model
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 47 / 71
Independent Cascade Model
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 47 / 71
Probabilistic models
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 48 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 49 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 49 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 49 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 49 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 49 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 49 / 71
SIR model
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 50 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 51 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 52 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 53 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 53 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 53 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 53 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
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)
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 54 / 71
SIR model
Model can also be described by following ordinary differential equations:
dS
dt= −βS(t)I (t)
N,
dI
dt=βS(t)I (t)
N− γI ,
dR
dt= γI (t).
Note that N = S + I + R
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 55 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 56 / 71
Example: SIR
https://www.public.asu.edu/~hnesse/classes/sir.html
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 57 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 58 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 58 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 58 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 58 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 58 / 71
SIS model
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 59 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 60 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 60 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 60 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 60 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 60 / 71
Epidemic Threshold
What should t be dependent on?
Average degree? Highest degree?
Variance of degree?
Diameter?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 61 / 71
Epidemic Threshold
What should t be dependent on?
Average degree? Highest degree?
Variance of degree?
Diameter?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 61 / 71
Epidemic Threshold
What should t be dependent on?
Average degree? Highest degree?
Variance of degree?
Diameter?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 61 / 71
Epidemic Threshold
What should t be dependent on?
Average degree? Highest degree?
Variance of degree?
Diameter?
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 61 / 71
Epidemic Threshold
Theorem by Wang et al 2003: No epidemic if
β
δ< t =
1
λ1,A(3)
where λ1,A is the largest eigenvalue of the adjacency matrix A
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 62 / 71
Real-World Application of such models: Viral Marketing
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 63 / 71
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 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 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 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 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 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
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 65 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 65 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 65 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 65 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 65 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 65 / 71
Identifying the cascades
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 66 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 67 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 68 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 68 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 68 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 68 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 68 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 68 / 71
Viral Marketing
Does sending more recommendations influence more purchases?
Figure: Influence of number of sent recommendations on purchases
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 69 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 70 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 70 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 70 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 70 / 71
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 70 / 71
Thanks for your attention - Questions?
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
Elisabeth Lex (ISDS, TU Graz) Web Science VU April 8, 2019 71 / 71