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.

Outline Non-cooperative Game Competitive Influence Approximate Nash Equilibrium 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

“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

Mathematical Formulation

groundset actions acts

Notations

Notations

Utility Functions

Nash Equilibrium Theorem (Nash, 1951)

A Beautiful Mind- John Nash

Utility System

Valid Utility System

Basic Utility System Theorem

Proof of basic => valid Submodularity basic

Remark

Lemma

Union and subtraction

Proof Submodular

Theorem

Proof utility system Valid utility system utility system

Theorem

Proof

Outline Non-cooperative Game Competitive Influence Approximate Nash Equilibrium 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

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.

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!

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.

Example Two players red and green. 6 2 1 5 3 4 Two players red and green. Step 1:1--2, 6--2. 2 becomes red or green. 8/25/2019

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, 6--2. 3 becomes red. 8/25/2019 28

Example Step 3:1--2, 3--4, 6--4. 4 becomes red or green. 6 2 1 5 3 4 8/25/2019

Example Step 4:1--3, 3--2, 6--4, 4--5. 5 is protected. 6 2 1 5 3 4 8/25/2019

Example 6 2 1 5 3 4 end:no more node can be activated. 8/25/2019

Lemma

Lemma This lemma seems useless? It can be used later! 8/25/2019

Game Each node is an act. Each action for player i is a subset of ki nodes. Private utility function Social utility function

Valid Utility system Theorem Proof Nondecreasing

Outline Non-cooperative Game Competitive Influence Approximate Nash Equilibrium 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Theorem

Corollary

Thank you! 8/25/2019