Cascading behavior in the networks

Social and Economic
Network Analysis
UNIT – IV
CASCADING BEHAVIOUR IN NETWORKS
1

Overview
Probabilistic Models
Decision Based
Models
VANI KANDHASAMY, PSGTECH
2

Summary
3

Probabilistic Model
NETWORKS, CROWDS AND MARKETS - CHAPTER 21
4

Epidemics
▪Epidemics spread is determined by properties of the
pathogen carrying it & also by contact network
▪Modeling the contact network is crucial to understand the
spread of an epidemics
▪Probabilistic models for epidemics – random
▪Diffusion of Ideas and Behaviors (Influence) in social
network
5

Contagion model based on random trees
▪A patient meets d new people
▪With probability q > 0 patient infects each of them
▪How to determine whether the disease will spread
or dies out?
▪Ans: R0 - Basic reproductive number
Expected # of people that get infected = d . q
▪Only R0 matters:
R0 ≥ 1: epidemic never dies and the number of infected people increases exponentially
R0 <1 : Epidemic dies out exponentially quickly
6

Contagion model based on random trees
▪When R0 is close 1, slightly changing d or q can result in epidemics
dying out or happening
▪Quarantining people/nodes [reducing d]
▪Encouraging better sanitary practices reduces germs spreading
[reducing q]
oMeasles has an R0 between 12 and 18
oEbola has an R0 between 1.5 and 2
oCovid-19 has an R0 between 1.4 and 3.9
▪Very simplified model of disease spread
7

Social contagion - Flickr
▪Users can be exposed to a photo via social influence or external links
▪ Did a particular like spread through social links?
▪No, if a user likes a photo and if none of his friends have previously liked the
photo
▪Yes, if a user likes a photo after at least one of her friends liked the photo -
Social cascade
R0 of popular photo is
between 1 and 190
8

SIR Epidemic model
▪Virus Propagation: 2 Parameters:
▪(Virus) Birth rate β:
probability that an infected neighbor attacks
probability of contagion
▪(Virus) Death rate δ:
Probability that an infected node heals
Length of infection
9

Assume nodes y and z are initially
infected & length of infection (t) = 1
10

SIS Epidemic model
Virus “strength”: s = β /δ
11

SIS Epidemic model
Assume node v is initially infected &
length of infection (t) = 1
12

Rumor spread modelling - Twitter
13

Decision Based Model
NETWORKS, CROWDS AND MARKETS - CHAPTER 6 & 19
14

Game Theory
▪Graph Theory – study of network structure
▪Game Theory – study of behaviors
▪Outcome of a persons decision depends not just on how they choose options
but also on choices made by their friends
Product pricing Choosing Route Auction bidding
15

What is a Game?
▪A game is a formal representation of the strategic interaction
between the multiple agents that are called players
▪Prisoner’s Dilemma:
oPrisoners 1 and 2 are caught for a crime and are interrogated in separate
chambers
oInterrogating officer explains the rules
• if both confesses the crime - both get 4 years of jail
• if both denies, some part of the charges still apply - each get 1 years of jail
• if one confesses but the other denies, the crime will be proved - confessor goes free, denier gets 10
years of jail
oAvailable choices for the suspecs: Confess (betray) or Not confess (stays silent)
16

Basic Ingredients of a Game
Not
Confess
Confess
Not
Confess
Each serves
1 year
S2 is free
S1 got 10
years
Confess S2 got 10
S1 years
is free
Each serves 4
years
Suspect 2
Suspect
1
17

Basic Ingredients of a Game
▪Players – set of participants
oBoth the suspects
▪Strategy – set of options
available to the players
oTo confess or not to confess
▪Payoff – the benefits that
each player receives
oReduced sentence/jail term
Not
Confess
Confess
Not
Confess
Each serves
1 year
S2 is free
S1 got 10
years
Confess S2 got 10
S1 years
is free
Each serves 4
years
Suspect 2
Suspect
1
17

Reasoning about Behavior in a Game
▪How the Suspect 1 choose his option?
▪If Suspect 2 were going to confess
oSuspect 1 can also confess and gets payoff (sentence term) of 4 years
oSuspect 1 don’t confess and gets payoff (sentence term) of 10 years
▪If Suspect 2 were not going to confess
▪Suspect 1 can also don’t confess and gets payoff (sentence term) of 1
year
▪Suspect 1 confess and gets free
18

▪How the Suspect 1 choose his option?
▪If Suspect 2 were going to confess
oSuspect 1 can also confess and gets payoff (sentence term) of 4 years
oSuspect 1 don’t confess and gets payoff (sentence term) of 10 years
▪If Suspect 2 were not going to confess
▪Suspect 1 can also don’t confess and gets payoff (sentence term) of 1
year
▪Suspect 1 confess and gets free
Confessing is best strategy in both the cases
18

Arms Races – competitors use
dangerous option to remain
evenly matched
19

Best responses
▪Best response - best choice of one player in response to each possible
choice of his or her opponent
▪Strategy S for Player 1 is a best response to a strategy T for Player 2, if S
produces at least as good a payoff as any other strategies S’ of Player 1
paired with T
▪Strategy S of Player 1 is a strict best response to a strategy T for Player 2,
if S produces a strictly higher payoff than all other strategies S’ of Player 1
paired with T
20

Dominant strategy
▪A dominant strategy for Player 1 is a strategy that is a best response
to every strategy of Player 2
▪A strictly dominant strategy for Player 1 is a strategy that is a strict
best response to every strategy of Player 2
21

Only one player has Strictly Dominant
Strategy
▪60% of people prefer low priced product & 40% prefer upscaled
product
▪Firm 1 is popular and gets 80% of the sales & Firm 2 gets only 20%
22

Strategy
product
Firm 1 has strictly
dominant strategy:
Low priced is a strict
best response to both
strategies of Firm 2
22

Strategy
product
Firm 1 has strictly
dominant strategy:
Low priced is a strict
best response to both
strategies of Firm 2
Firm 2 doesn’t have
dominant strategy:
Low priced is a best
response when Firm 1
plays upscale & vice
versa
22

No player has Strictly Dominant Strategy
Three client game
▪Two firms each hope to do business with one of three large clients A, B, and C
▪A is a larger client, doing business with A is worth 8 and worth of B or C is 2
▪A will only do business with the firms if both approach A
▪Firm 1 is too small to attract business on its own
23

Three client game
For Firm 1
A is strict best response to A by Firm 2
B is strict best response to B by Firm 2
C is strict best response to C by Firm 2
23

Three client game
For Firm 1
For Firm 2
C is strict best response to B by Firm 1
B is strict best response to C by Firm 1
23

Three client game
For Firm 1
For Firm 2
C is strict best response to B by Firm 1
B is strict best response to C by Firm 1
Neither firm has a dominant strategy:
Each strategy by each firm is a strict
best response to some strategy by
other firm
23

Nash Equilibrium
▪The pair of strategies (S, T ) is a Nash equilibrium if S is a best response to T, and
T is a best response to S
▪If the players choose strategies that are best responses to each other, then no
player has an incentive to deviate to an alternative strategy
24

Multiple Equilibria
Coordination game – the two players’ have shared goal to coordinate
on the same strategy
Stag hunt game
25

Multiple Equilibria
Coordination game – the two players’ have shared goal to coordinate
on the same strategy
Stag hunt game
If players mis coordinate
then the one who is trying
for the higher-payoff
outcome gets penalized
more
25

Multiple Equilibria
Unbalanced Coordination game – Battle of the sexes
Romantic
Action
Action
Romantic
26

Multiple Equilibria
Unbalanced Coordination game – Battle of the sexes
Romantic
Action
Action
Romantic
Follow conventions to
resolve disagreements
when players prefer
different ways to
coordinate
26

Game Theoretic model of Cascades
▪Based on 2 player coordination game
o2 players – each chooses technology A or B (Signal/Telegram)
oEach person can only adopt one “behavior”, A or B
oYou gain more payoff if your friend has adopted the same behavior as you
27

Game Theoretic model of Cascades
▪Based on 2 player coordination game
o2 players – each chooses technology A or B (Signal/Telegram)
oEach person can only adopt one “behavior”, A or B
oYou gain more payoff if your friend has adopted the same behavior as you
Direct benefit
effects
27

A Networked Coordination Game
▪Each node v is playing a copy of the game with each of its neighbors
▪Payoff: sum of node payoffs per game
28

▪Each node v is playing a copy of the game with each of its neighbors
▪Payoff: sum of node payoffs per game
Two equilibria
28

▪Let v have d neighbors
▪Assume fraction p of v’s neighbors
adopt A
▪Payoffv = a∙p∙d if v chooses A
= b∙(1-p)∙d if v chooses B
▪A is a better choice if
p -> fraction of v’s friends with A
q -> payoff threshold
29

▪Let v have d neighbors
▪Assume fraction p of v’s neighbors
adopt A
▪Payoffv = a∙p∙d if v chooses A
= b∙(1-p)∙d if v chooses B
▪A is a better choice if
p -> fraction of v’s friends with A
q -> payoff threshold
v chooses A if: p > q
29

Cascading behavior
30

Cascading behavior
▪Graph where everyone starts with all B
▪Small set S of early adopters of A (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 q of my friends take A I’ll also take A
▪When does all nodes will switch to A?
▪What causes the spread of A to stop?
Network structure, Initial adopters, Threshold - q
30

Cascading behavior – Complete cascade
▪v and w as the initial adopters
▪Payoffs a = 3 and b = 2
▪Nodes will switch from B to A, if q =
2/5 fraction of their neighbors are
using A
▪Step 1:
or and t switch to A since 2/3 (>2/5) of their
friends are using A
os and u do not switch as only 1/3 (<2/5) of
their friends are using A
▪Step 2:
os and u switch to A since 2/3 (>2/5) of their
friends are using A
31

Cascading behavior
Step 1: Step 2-4:
32

Cascading behavior
Communities
hinder innovation
Step 1: Step 2-4:
32

How to go viral?
THRESHOLD
▪Increasing quality/payoff of A
from a = 3 to 4
▪Lowers the threshold q from 2/5
to 1/3
▪A would break into other parts
of the network
INITIAL ADOPTERS
▪Choosing initial adopters
carefully
▪12 or 13 can be convinced to
switch to A to get cascade going
▪Choose key nodes based on their
network position
33

Cascading behavior – Lowering
Threshold
Step 1: Step 2-5:
34

Threshold
Step 5: Step 6:
35

Threshold
Step 6: Step 7:
36

Cascading behavior – Convincing key
people
Step 2:
Step 1:
37

people
Step 2: Step 3-4:
38

people
Step 4: Step 5-6:
39

Diffusion models
DECISION BASED MODEL
▪Models of product adoption, decision making
▪Utility based
▪“Node” centric: A node observes decisions of
its neighbors and makes its own decision
▪Example:
Protest recruitment in Twitter, Product adoption
PROBABILISTIC MODEL
▪Models of influence or disease spreading
▪Lack of decision making and involves
randomness
▪An infected node tries to “push” the contagion
to an uninfected node
▪Example:
Rumor detection in Twitter, Power grid failure
40

Cascading behavior in the networks

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