Cascading behaviors can occur in networks when people are connected and influence each other's decisions. This document discusses several models of cascading behaviors: 1) A diffusion of innovations model looks at how new ideas and technologies spread through social networks based on factors like group membership, growth, and change. 2) A game theoretic model of diffusion represents the spread of behaviors as a coordination game, where players receive higher payoffs for matching their neighbors' behaviors. This can lead to cascading shifts from one equilibrium behavior to another. 3) Viral marketing techniques aim to leverage social networks to spread brand awareness through self-replicating viral processes, similar to epidemic contagion models from game theory. The

1. Cascading behaviors
in Networks
Osamah Al-Ghammari
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Prof: Ahmet Bulut

2. CONTENTS
Information about Cascades.
Diffusion of innovation
 Game theoretic model: A networked coordination game.
Cascading Behavior:
 Example
Viral Marketing.
Contagion
 Game theoretic model: Local interaction game.
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3. Information about cascade
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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

4. 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?

6. 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.

7. Game theoretic model of diffusion Cont’d
Define the payoffs as follows:
– If both 𝑣 and 𝑤 choose behavior 𝐴, they will receive a payoff of 𝑞.
– If both 𝑣 and 𝑤 choose behavior 𝐵, they will receive a payoff of 1 − 𝑞.
– 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 𝑞, 𝑞 0,0
B 0,0 1 − 𝑞, 1 − 𝑞

8. Questions faced by v
Suppose some of v’s neighbors adopt A, and some adopt B.
What should v do to maximize its payoff?
Let v have 𝑑 𝑣 neighbors, and 𝑑 𝑣
𝐴
of its neighbors
adopt A, and 𝑑 𝑣
𝐵
have adopted B.
Then:
 If v chooses A: payoff = (q𝑑 𝑣
𝐴
)
 If v chooses B: payoff = (1-q)(𝑑 𝑣
𝐵)
Thus, v should adopt behavior B if
(𝑑 𝑣
𝐵
)> (q𝑑 𝑣) , and behavior A if (𝑑 𝑣
𝐵
)< (q𝑑 𝑣)
Or 𝑝 ≥
𝑏
𝑎+𝑏

9. Cascading behavior
In 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

10. Cascading behavior: an example
Suppose 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?

11. Coordination game
setup:
a= 3, b=2 ,
𝑝 =
2
3+2
=
2
5
an example

12. another example

13. 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.

14. 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.

15. 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

16. Contagion
each 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 𝑢(𝑎, 𝑎’) for
the payoff of a player from a specific action if he/she chooses 𝑎 and his
neighbor chooses 𝑎’. The following payoff matrix which corresponds to the
payoff functions:
This game has two Nash equilibria.
When 𝑢(0,0) > 𝑢(1,0)
and u(1,1) > u(0,1).
0 1
0 𝑢(0,0), 𝑢(0,0) 𝑢(0,1), 𝑢(1,0)
1 𝑢(1,0), 𝑢(0,1) 𝑢(1,1), 𝑢(1,1)

17. Contagion Cont’d
On the other side, the other player chooses action 1. The payoff is
parameterized with the critical probability of 𝑞 𝜖 (0,1). the payoff matrix:
Ex:
– The examples given provide the intuition for the contagion threshold. Note
that 𝑍 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.
– 𝑋 = 𝑍, 𝑥’~𝑥 𝑖𝑓 𝑥’ = 𝑥 − 1 𝑜𝑟 𝑥’ = 𝑥 + 1
0 1
0 𝑞, 𝑞 0,0
1 0,0 1 − 𝑞, 1 − 𝑞

18. – If 𝑞 < ½ in the payoff matrix (2,1), action 1 is the best response whenever
at least one neighbor chooses action 1. Therefore, if two neighbors 𝑥 and
𝑥 + 1 choose action 1 initially, players 𝑥 − 1, 𝑥, 𝑥 + 1 and 𝑥 + 2 must all
choose the same action for the next period.
– Players 𝑥 − 2, 𝑥 − 1, 𝑥, 𝑥 + 1, 𝑥 + 2 and 𝑥 + 3 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 𝑞 > ½ ,
no player would switch to action 1 unless both neighbors are already with
action1. Therefore, the contagion threshold is ½.
Contagion Cont’d

19. Dimensional contagion threshold

20. QUERIES?
With a note of Thanks.
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