Cascading behaviors in Networks

Osamah Al-Ghammari

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Prof: Ahmet Bulut

CONTENTS

Information about Cascades.Diffusion of innovation

Game theoretic model: A networked coordination game.Cascading Behavior:

ExampleViral Marketing.Contagion

Game theoretic model: Local interaction game.

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Information about cascade 2

When people are connected by a network, it becomes possible for them to influence each other’s behavior and decisions.

There are limitless set of situations in which people are influenced by others:Opinions they hold.Products they buy.Political positions they support.Activities they pursue…etc

Diffusion of Innovations

Diffusion of innovations is a theory that seeks to explain how, why, and at what rate new ideas and technology spread.

Analyzing group formation and evolution.– Membership. What are the structural features that influence whether a given individual will join a

particular group?

– Growth. What are the structural features that influence whether a given group will grow significantly (i.e. gain a large net number of new members) over time?

– Change. A given group generally exists for one or more purposes at any point in time; in our datasets, for example, groups are focused on particular “topics of interest.” How do such foci change over time, and how are these changes correlated with changes in the underlying set of group members?

Game theoretic model of diffusion

Based on a two player coordination game:Each node has a choice between two possible

behaviors, A an old behavior, or A a new behavior. If nodes v and w are linked by an edge, then there is

an incentive for them to have their behaviors match.Represented as a game in which v and w are players,

and A and B the possible strategies.

Game theoretic model of diffusion Cont’d

Define the payoffs as follows:– If both and choose behavior, they will receive a payoff of.

– If both and choose behavior, they will receive a payoff of.

– If both choose the opposite behavior, they will receive a payoff of 0.

In the network at large: Each node v plays a copy of the game with each of its neighbors. Payoff of a node = sum of the payoffs played on each edge.

G(v,w) A B

A 0,0

B 0,0

Questions faced by vSuppose some of v’s neighbors adopt A, and some adopt B.What should v do to maximize its payoff?Let v have neighbors, and of its neighbors

adopt A, and have adopted B. Then:

If v chooses A: payoff = (q) If v chooses B: payoff = (1-q)()

Thus, v should adopt behavior B if()> (q , and behavior A if ()< (q Or

Cascading behaviorIn any network, there are two obvious equilibria to the

network-wide coordination game:Everyone adopts A.Everyone adopts B.

We want to understand:How easy is to “tip” the network from one of these equilibria to

the other.What other intermediate equilibria look like (states of

coexistence where A is adopted in some parts of the network and B adopted in others)

Cascading behavior: an exampleSuppose everyone in the network is initially using B.Then a small set of “Initial adopters” all decide to use A.Some of the neighbors of initial adopters may now decide

to swatch to A as well. And then some of their neighbors may switch and so

forth, in a potentially cascading fashion.When does this result in every node eventually switching

to A? when this isn’t the result, what causes the spread of A to stop?

Coordination game setup:

a= 3, b=2 ,

an example

another example

Cascading behavior and viral marketing

Observations from the pervious example: tightly-knit communities can work to hinder the spread of

innovation.As a result, we get coexistence between A and B ( a common real

world phenomenon; eg. Political views, age/life style groups in social networking sites)

Suggests strategy for market competition:Maker of A can increase its reach by raising the quality of its product.Maker of A could try to convince a small set of key people using B to

switch to A.The later issue is considered in research in viral marketing.

What is Viral Marketing?Refers to marketing techniques that use preexisting

social networks to produce increase in brand awareness through self-replicating viral processes, analogous to the spread of an epidemic.

Viral Marketing and Direct marketing

Modes of marketing:Direct marketing: blogs, blogs, E-shopping, E-mail…etc.Viral Marketing (Word-of-mouth marketing): person-to-

person, chat rooms, blogs.The difference between direct marketing and viral marketing is that viral marketing is more profitable. Data mining has been employed with direct marketing in order to predict future purchasing behavior. However, viral marketing uses “the word-of-mouth” strategy which can be much more cost effective

Contagioneach player at each location has a set of available actions

and a payoff function from each of his various interactions, we have a local interaction game.

Local interaction game model:Each player has two different strategies, either 0 or 1. We

write for the payoff of a player from a specific action if he/she chooses and his neighbor chooses. The following payoff matrix which corresponds to the payoff functions:

This game has two Nash equilibria. When and u(1,1) > u(0,1).

0 1

0

1

Contagion Cont’dOn the other side, the other player chooses action 1. The payoff is

parameterized with the critical probability of the payoff matrix:

Ex:– The examples given provide the intuition for the contagion threshold. Note

that is the set of the integers. Interaction on a line. The population is arranged on a line and each player interacts with the next player either on the right or the left.

0 1

0 0,0

1 0,0

– If in the payoff matrix action 1 is the best response whenever at least one neighbor chooses action 1. Therefore, if two neighbors and choose action 1 initially, players and must all choose the same action for the next period.

– Players and must all choose action 1 in the period after that, this process goes on.

– As it can be seen, action 1 spreads to the entire population. But if, no player would switch to action 1 unless both neighbors are already with action1. Therefore, the contagion threshold is.

Contagion Cont’d

Dimensional contagion threshold

QUERIES?

With a note of Thanks.

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