1. And Competitive Influence
Non-cooperative Game
And Competitive Influence
Ding-Zhu Du
University of Texas at Dallas
First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies. 
It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on..
The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups.
Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

2. Outline Non-cooperative Game Competitive Influence
Approximate Nash Equilibrium
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

3. “Decisions are made by a set of non-cooperative agents whose action spaces are subsets of an underlying groundset. The actions of the agents induce some social utility, measured by a set function. The goal of the agents, though, is not to maximize the overall social utility; rather, they seek to maximize their own private utility functions.” The only assumptions we make are
The social utility and private utility functions are measured in the same standard unit

4. Mathematical Formulation

5. groundset
actions
acts

6. Notations

7. Notations

8. Utility Functions

9. Nash Equilibrium
Theorem (Nash, 1951)

10. A Beautiful Mind- John Nash

11. Utility System

12. Valid Utility System

13. Basic Utility System
Theorem

14. Proof of basic => valid
Submodularity
basic

15. Remark

16. Lemma

17. Union and subtraction

18. Proof
Submodular

19. Theorem

20. Proof
utility system
Valid utility system
utility system

21. Theorem

22. Proof

23. Outline Non-cooperative Game Competitive Influence
Approximate Nash Equilibrium
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

24. Independent Cascade (IC) Model
When node v becomes active, it has a single chance of activating each currently inactive neighbor w.
The activation attempt succeeds with probability pvw .
The deterministic model is a special case of IC model. In this case, pvw =1 for all (v,w).
We again start with an initial set of active nodes A0, and the process unfolds in discrete steps according to
the following randomized rule. When node v first becomes active
in step t, it is given a single chance to activate each currently inactive
neighbor w; it succeeds with a probability pv;w —a parameter
of the system — independently of the history thus far. (If w has
multiple newly activated neighbors, their attempts are sequenced in
an arbitrary order.) If v succeeds, then w will become active in step
t+1; but whether or not v succeeds, it cannot make any further attempts
to activate w in subsequent rounds. Again, the process runs
until no more activations are possible.

25. Example Y
0.6
Inactive Node
0.2
0.2
0.3
Active Node
Newly active
node X U
0.1
0.4
Successful
attempt
0.5
0.3
0.2
Unsuccessful
attempt
0.5 w v
Stop!

26. IC Model (Competitive Version)
each of b players selects a color and a set Si of at most ki nodes.
A node activated by players with one color will take the color.
A node activated by players with multi-color will take the color of one of the players uniformly at random.
Process ends until no new activations occur.

27. Example Two players red and green.6 2 1 5 3 4
Two players red and green.
Step 1:1--2, becomes red or green.
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28. Example Two kinds of influence cascades: rumors and protectors.6 2 1 5 3 4
Two kinds of influence cascades: rumors and protectors.
Each individual has three status: inactive, rumored, protected.
The active individual only activates one of its neighbors successfully.
When rumors and protectors influence an individual at the same time, then the individual is protected.
Each individual has unlimited opportunities to influence their neighbors.
Each node will never change its status if it has been activated.
Step 2:1--3, becomes red.
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29. Example Step 3:1--2, 3--4, 6--4. 4 becomes red or green. 6 2 1 5 3 4
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30. Example Step 4:1--3, 3--2, 6--4, 4--5. 5 is protected. 6 2 1 5 3 4
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31. Example 6 2 1 5 3 4
end:no more node can be activated.
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32. Lemma

33. Lemma
This lemma seems useless?
It can be used later!
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34. Game
Each node is an act.
Each action for player i is a subset of ki nodes.
Private utility function
Social utility function

35. Valid Utility system
Theorem
Proof
Nondecreasing

36. Outline Non-cooperative Game Competitive Influence
Approximate Nash Equilibrium
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

37. Theorem

38. Corollary

39. Thank you!
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