Transcript snap.stanford.edu
CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University
http://cs224w.stanford.edu
Spreading through networks:
Cascading behavior Diffusion of innovations Network effects Epidemics
Behaviors that cascade from node to node like an epidemic Examples:
Biological:
Diseases via contagion
Technological:
Cascading failures Spread of information
Social:
Rumors, news, new technology Viral marketing 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Product adoption:
Senders and followers of recommendations
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Behavior/contagion spreads over the edges of the network
It creates a propagation tree, i.e., cascade 4/27/2020 Network
Terminology:
• Stuff that spreads: Contagion • “Infection” event: Adoption, infection, activation • We have: Infected/active nodes, adoptors Cascade (propagation graph) Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Probabilistic models:
Models of influence or disease spreading An infected node tries to “push” the contagion to an uninfected node
Example:
You “catch” a disease with some prob. from each active neighbor in the network
Decision based models:
Models of product adoption, decision making A node observes decisions of its neighbors and makes its own decision
Example:
You join demonstrations if k of your friends do so too 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Two ingredients:
Payoffs:
Utility of making a particular choice
Signals:
Public information: What your network neighbors have done (Sometimes also) Private information: Something you know Your belief
Now you want to make the optimal decision
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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[Granovetter ‘78]
Collective Action
[Granovetter, ‘78] Model where everyone sees everyone else’s behavior
Examples:
Clapping or getting up and leaving in a theater Keeping your money or not in a stock market Neighborhoods in cities changing ethnic composition Riots, protests, strikes
How the number of people participating in a given activity with network effects would grow or shrink over time?
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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n people – everyone observes all actions
Each person i has a threshold t
i
Node i will adopt the behavior iff at least t
i
other people are adopters:
Small t
i
: early adopter
Large t
i
: late adopter 1 0 t i
The population is described by {t
1
,…,t
n
}
F(x) … fraction of people with threshold t
i
x
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Think of the step-by-step change in number of people adopting the behavior:
F(x) … fraction of people with threshold x s(t) … number of participants at time t
Easy to simulate:
s(0) = 0 s(1) = F(0) s(2) = F(s(1)) = F(F(0)) s(t+1) = F(s(t)) = F t+1 (0) Here we really need to split into 2 slides and explain the axes and show in text and in animation why the amount of people grows y=x F(x)
Fixed point: F(x)=x
Updates to s(t) to converge to a stable fixed point There could be other fixed points but starting from 0 we never reach them F(0) Iterating to y=F(x).
Fixed point.
Threshold, x 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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What if we start the process somewhere else?
We move up/down to the next fixed point How is market going to change?
Show lines how change if we red dot.
y=x F(x) 4/27/2020 Threshold, x Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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4/27/2020 Fragile fixed point y=x Show lines how the behavior will change if we start at blue vs red dot.
Robust fixed point Threshold, x Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Each threshold t
i
is drawn independently from some distribution F(x) = Pr[thresh
x]
Suppose: Normal with =n/2, variance
Small
: Large
:
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Small
Medium
F(x) F(x) No cascades!
Fixed point is low
Small cascades Bigger variance let’s you build a bridge from early adopters to mainstream
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Big
Huge
Fixed point is high!
Big cascades!
Fixed point gets lower!
But if we increase the variance even more we move the higher fixed point lower
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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It does not take into account:
No notion of social network:
Some people are more influential It matters who the early adopters are, not just how many
Models people’s awareness just actual number of people participating
Modeling perceptions of who is adopting the behavior vs. who you believe is adopting of size of participation not Non-monotone behavior – dropping out if too many people adopt People get “locked in” to certain choice over a period of time
Modeling thresholds
Richer distributions Deriving thresholds from more basic assumptions game theoretic models 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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You live in an oppressive society Work on these slides to show how the network You know of a demonstration against the government planned for tomorrow If a lot of people show up, the government will fall If only a few people show up, the demonstrators will be arrested and it would have been better had everyone stayed at home 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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You should do something if you believe you are in the majority!
Dictator tip: Pluralistic ignorance estimates about the prevalence of certain opinions in the population – erroneous Survey conducted in the U.S. in 1970 showed that while a clear minority of white Americans at that point favored racial segregation, significantly more than 50% believed that it was favored by a majority of white Americans in their region of the country 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Personal threshold k: “I will show up to the protest if I am sure at least k people in total (including myself) will show up” Each node in the network knows the thresholds of all their friends 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Will uprising occur?
4/27/2020 No!
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Will uprising occur?
4/27/2020 No!
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Will uprising occur?
4/27/2020 Yes!
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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[Morris 2000]
Based on 2 player coordination game
2 players – each chooses technology A or B Each person can only adopt one “behavior”, A or B You gain more payoff if your friend has adopted the same behavior as you 4/27/2020 Local view of the network of node v Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Payoff matrix:
If both v and w adopt behavior A, they each get payoff a>0 If v and w adopt behavior B, they reach get payoff b>0 If v and w adopt the opposite behaviors, they each get 0
In some large network:
Each node v is playing a copy of the game with each of its neighbors Payoff: sum of node payoffs per game 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Explain the threshold better – relative reward matters, not the absolute
Threshold:
v
choses
q
a A b
if
p>q b
4/27/2020 Let v have d neighbors Assume fraction p of v’s neighbors adopt A
Payoff v
Thus:
=
a∙p∙d
=
b∙(1-p)∙d
v chooses A if:
if
v
chooses A if
v
chooses B
a∙p∙d > b∙(1-p)∙d
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Scenario:
Graph where everyone starts with B. Small set S of early adopters of A Hard wire S – they keep using A no matter what payoffs tell them to do Assume payoffs are set in such a way that nodes say:
If
more than
50% of my friends are red I’ll be red
(this means: a = b-ε and q>1/2) 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S
{
u
,
v
} If more than 50% of my friends are red I’ll be red 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S
{
u
,
v
} u v If more than 50% of my friends are red I’ll be red 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S
{
u
,
v
} u v If more than 50% of my friends are red I’ll be red 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S
{
u
,
v
} u v If more than 50% of my friends are red I’ll be red 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S
{
u
,
v
} u v If more than 50% of my friends are red I’ll be red 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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S
{
u
,
v
} u v If more than 50% of my friends are red I’ll be red 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Observation:
The use of A spreads monotonically
(Nodes only switch from B to A, but never back to B)
Why?
Proof sketch:
Nodes keep switching from B to A:
B
A
Now, suppose some node switched back from A B, consider the first node v to do so (say at time t) Earlier at time t’ (t’
A
So at time t’ v was above threshold for A But up to time t no node switched back to B, so node v could only had more neighbors who used A at time t compared to t’. There was no reason for v to switch.
!! Contradiction !!
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
4/27/2020 37
Consider infinite graph G
v
choses
A q
if
p>q
a b
b
(but each node has finite number of neighbors) We say that a finite set S causes a cascade G with threshold q if, when S adopts A, in eventually every node adopts A Example:
Path
If q<1/2 then cascade occurs
S
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
4/27/2020 38
Infinite Tree:
If q<1/3 then cascade occurs S
Infinite Grid:
S If q<1/4 then cascade occurs
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Def: The cascade capacity of a graph G is the
largest q
for which some
finite set S
can cause a cascade Fact: There is no G where cascade capacity > ½
Proof idea:
Suppose such G exists: q>½, finite S causes cascade
Show contradiction:
nodes stop switching after a finite # of steps Argue that
X
40 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
Fact: There is no G where cascade capacity > ½
Proof sketch:
Suppose such G exists: q>½, finite S causes cascade
Contradiction:
Switching stops after a finite # of steps Define “potential energy” Argue that it starts finite (non-negative) and strictly decreases at every step “Energy”: = |d out (X)| |d out (X)| := # of outgoing edges of active set X The only nodes that switch have a strict majority of its neighbors in S |d out (X)| strictly decreases It can do so only a finite number of steps
X
41 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
What prevents cascades from spreading?
Def:
Cluster of density
ρ
is a
set of nodes C
where each node in the set has at least
ρ
fraction of edges in C.
4/27/2020
ρ=3/5 ρ=2/3
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Let S be an initial set of adopters of A
All nodes apply threshold q to decide whether to switch to A
Two facts:
ρ=3/5
1) If G\S contains a cluster of density >(1-q) then S can not cause a cascade
S No cascade if q>2/5
2) If S fails to create a cascade, then there is a cluster of density >(1-q) in G\S 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Next year:
Move the herding slides back into lecture 1 Then we will have 2 lectures: Herding and the collective action model And the 2 game-theoretic models Points to ask students: What are Collective action model deficiencies?
What are other examples of cascading behavior?
Ask them how to complete the proof, what is the argument 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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So far:
Behaviors A and B compete Can only get utility from neighbors of same behavior: A-A get a, B-B get b, A-B get 0
Let’s add extra strategy “A-B”
AB-A: gets
a
AB-B: gets
b
AB-AB: gets max(a, b) Also: Some cost c for the effort of maintaining both strategies (summed over all interactions) 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Every node in an infinite network starts with B Then a finite set S initially adopts A Run the model for t=1,2,3,… Each node selects behavior that will optimize payoff (given what its neighbors did in at time t-1) a a a+b-c
AB
b
A
A
AB
Edge payoff
How will nodes switch from B to A or AB?
B
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Path:
Start with all Bs, a>b (A is better)
One node switches to A – what happens?
With just A, B: A spreads if b
a
With A, B, AB: Does A spread?
Assume a=2, b=3, c=1
A
a=2
A
0
B
b=3
B
b=3
B
4/27/2020
A
a=2 a=2 b=3 b=3
A B
-1
Cascade stops
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
B
48
Let a=5, b=3, c=1
4/27/2020
A A A A A
0
B
b=3
B
b=3 a=5 a=5 a=5 a=5 b=3 b=3
A B
-1 a=5 a=5 b=3
A
-1 -1 a=5 a=5 b=3
A A
-1 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
B B B
-1 49
Infinite path, start with all Bs
Payoffs:
A:a, B:1, AB:a+1-c
A
What does node w in A-w-B do?
B vs A AB vs B a+1-c=1 c
A B A
4/27/2020
w
1
B
AB vs A a+1-c=a
AB
1
AB
a Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
B
50
Payoffs:
A:a, B:2, AB:a+2-c
Notice: Now also AB spreads
What does node w in AB-w-B do?
B vs A
AB
c
A B A w
AB vs B 4/27/2020 1 AB vs A
B AB AB
1 2 a Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
B
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Joining the two pictures:
c
A B
1
AB
1 2
B →AB → A
a 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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You manufacture default B and new/better A comes along:
Infiltration:
If B is too compatible then people will take on both and then drop the worse one (B) c
B stays
Direct conquest:
If A makes itself not compatible – people on the border must choose. They pick the better one (A)
B →AB
Buffer zone:
If you choose an optimal level then you keep a static “buffer” between A and B 4/27/2020
A spreads B → A
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
B →AB→ A
a 53
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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[Banerjee ‘92] 4/27/2020
Influence of actions of others
Model where everyone sees everyone else’s behavior
Sequential decision making
Example: Picking a restaurant Consider you are choosing a restaurant in an unfamiliar town Based on Yelp reviews you intend to go to restaurant A But then you arrive there is no one eating at A but the next door restaurant B is nearly full
What will you do?
Information that you can infer from other’s choices may be more powerful than your own Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Herding:
There is a decision to be made People make the decision sequentially Each person has some private information that helps guide the decision You can’t directly observe private information of the others but can see what they do
You can make inferences about the private information of others
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Consider an urn with 3 marbles. It can be either:
Majority-blue:
2 blue, 1 red, or
Majority-red:
1 blue, 2 red Each person wants to best guess whether the urn is majority-blue or majority-red Guess
red
if P(
majority-red
| what she has seen or heard) > ½
Experiment:
One by one each person: Draws a marble Privately looks are the color and puts the marble back Publicly guesses or majority-blue whether the urn is majority-red
You see all the guesses beforehand. How should you make your guess?
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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[Banerjee ‘92]
Informally, What happens?
See ch. 16 of Easley-Kleinberg for formal analysis #1 person: Guess the color you draw from the urn.
#2 person: Guess the color you draw from the urn. Why? If same color as 1 st , then go with it If different, break the tie by doing with your own color
#3 person:
If the two before made different guesses, go with your color Else, go with their guess (
regardless
your color) – cascade starts!
#4 person:
Suppose the first two guesses were
R
, you go with
R
Since 3 rd person always guesses
R
Everyone else guesses
R
(regardless of their draw) 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Three ingredients:
State of the world:
Whether the urn is
MR
or
MB
Payoffs:
Utility of making a correct guess
Signals:
Models private information: The color of the marble that you just draw Models public information: The
MR
vs
MB
guesses of people before you 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Decision: Guess
MR
if 𝑃 𝑴𝑹 𝑝𝑎𝑠𝑡 𝑎𝑐𝑡𝑖𝑜𝑛𝑠 > 1 2
Analysis (Bayes rule):
#1 follows her own color (private signal)!
Why?
P
(
MR
| r
ed
]
P
(
MR
)
P
(
r P
(
r
) |
MR
) 1 / 2 1 / 2 2 / 3 2 / 3
P
(
r
)
P
(
r
|
MB
)
P
(
MB
)
P
(
r
|
MR
)
P
(
MR
) 1 1 2 3
#2 guesses her own color (private signal)!
1 2 2 3 1 / 2 #2 knows #1 revealed her color. So, #2 gets 2 colors.
If they are the same, decision is easy.
If not, break the tie in favor of her own color 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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4/27/2020
#3 follows majority signal!
Knows #1, #2 acted on their colors. So, #3 gets 3 signals.
If #1 and #2 made opposite decisions, #3 goes with her own color. Future people will know #3 revealed its signal
P
(
MR
|
r
,
r
,
b
] 2 / 3 If #1 and #2 made same choice, #3’s decision conveyed no info. Cascade has started!
How does this unfold?
You are N-th person
#MB
=
#MR
: you guess your color |
#MB
-
#MR
|=1 : your color makes you indifferent, or reinforces you guess |
#MB
-
#MR
| ≥ 2 : Ignore your signal. Go with majority.
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
61
Cascade begins when the difference between the number of blue and red guesses reaches 2
Guess B Guess R Guess R Guess B Guess B Guess B
Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
4/27/2020 62
Easy to occur given the right structural conditions
Can lead to bizarre patterns of decisions
Non-optimal outcomes
With prob. ⅓ ⅓=⅟ 9 first two see the wrong color, from then on the whole population guesses wrong
Can be very fragile
Suppose first two guess blue People 100 and 101 draw red showing their marbles and cheat by Person 102 now has 4 pieces of information, she guesses based on her own color
Cascade is broken
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
64
Spread of new agricultural practices
[Ryan-Gross ‘43] Adoption of a new hybrid-corn between the 259 farmers in Iowa Interpersonal network plays important role in adoption Diffusion is a social process
Spread of new medical practices
[Coleman et al. ‘66] Adoption of a new drug between doctors in Illinois Clinical studies and scientific evaluations were not sufficient to convince the doctors It was the social power of peers that led to adoption 4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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Basis for models:
Probability of adopting new behavior depends on the number of friends who have adopted What’s the dependence?
k = number of friends adopting 4/27/2020 Diminishing returns: Viruses, Information k = number of friends adopting Critical mass: Decision making Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
66
Strongly connected directed graph has a path from each node to every other node and vice versa (e.g., A-B path and B-A path) Weakly connected directed graph is connected if we disregard the edge directions
E F B A
Graph on the left is not strongly connected.
D C G
4/27/2020 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu
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