A Game Theoretical Approach to Modeling Information

Dissemination in Social Networks

Dmitry Zinoviev, Vy Duong, Honggang ZhangMathematics and Computer Science Department

Suffolk UniversityBoston

Actors and Assertions⏏ Our first paper considers two people engaged in a one-way

communication.⏏ One person (“S[ender]”) has an assertion Φ that she wants to

share with another person (“R[eceiver]”). Both S and R are actors.⏏ An assertion is an atomic piece of knowledge.⏏ R may already have the assertion Φ, but S does not know about it.⏏ R may have other assertions as well, but is not allowed to share

them.

Assertions and Feedback⏏ Sender S must decide whether to speak (post, publish, etc.) or not⏏ Publishing can hurt—and so can not publishing⏏ Receiver R must decide whether to trust S or not, and also

whether to comment on S's post or not to comment⏏ Commenting can hurt—and so can not commenting

Two Actors—Two Policies⏏ As a result, each actor has two

strategies: to post or not to post and to comment (if posted) or not to comment.

⏏ Each actor has to make a choice that maximizes his/her utility.

⏏ This forms a mathematical game—a square game with two players and two strategies per player

⏏ Solve the game—get the strategies!

⏏ What is the utility?

Actor's Utility

⏏ Actor's utility is a convex linear combination of three factors: credibility, popularity, and knowledge:

Ui=τiTi+πiPi+κiKi

τi+π

i+κi=1

0≤T,P,K≤N⏏ T is credibility—the extent to which S trusts R and R trusts S

τ is the importance of trust to the actor⏏ P is popularity—a measure of “social visibility” of the actor

π is the importance of popularity to the actor⏏ K is the measure of knowledge

κ is the importance of knowledge to the actor⏏ N is simply a reasonably large number.

If in the course of simulation T, P or K are re-normalized as needed

Personality Types⏏ Depending on the values of τ, π, and κ, one can define

several personality types; for example:⏏ “Internet Trolls” have high π and low τ and κ⏏ “Experts” have medium κ, high τ, and low π⏏ “Mad Professors” have high κ, medium or low τ, and low π

Knowledge⏏ Actor S's knowledge is a collection of S's assertions; S

knows KS assertions

⏏ The number of assertions in the system, N, is finite and fixed⏏ Each assertion can be of three types:

⏏ Privately believed to be true—a positive fact (+); S knows F+

S true assertions

⏏ Privately believed to be false—a negative fact (-); S knows F-

S false assertions

⏏ Privately not known to be true or false—a rumor (○); S knows F○

S rumors

Rumor Discount⏏ λ is a rumor discount coefficient:

⏏ λ=0 means that rumors are not included in the total knowledge

⏏ λ=1 means that rumors are fully included⏏ The measure of S's knowledge is K (0≤K≤N):

K=F+S+F-

S+λF○

S

Knowledge Types⏏ Depending on the value of k=K/N, one can define several

knowledge types; for example:⏏ “Ignoramuses” have low k⏏ “Mediocres” have medium k⏏ “Gurus” have high k

What Is Global Truth?⏏ The probability of an assertion to be globally true is φ⏏ Only an external oracle (a “God”) knows which particular

assertions are globally true

What Is Perceived Truth?⏏ Upon receiving an assertion, R must assess (or fail to assess) it—

that is, calculate the probabilities of Φ being a true assertion (g+), a false assertion (g-), or a rumor (g○); g++g-+g○=1

⏏ This process is based on:⏏ R's own knowledge⏏ R's trust in S⏏ The probability of the assertion being true by nature⏏ The sender's opinion of the assertion

⏏ Situations:⏏ k

R=0: R is an Ignoramus, he must trust S to the extent of S's

credibility⏏ k

R=1: R is a Guru, he makes a decision himself using φ as a

guidance⏏ 0<k

R<1: R is a Mediocre, he blends the two extreme strategies

Equations for g

g+=kRφ+(1-kR)rSf

+S

g-=kR(1-φ)+(1-kR)rSf-S

g○=(1-kR)(rSf○

S+(1-rS))

g+ g-

g○

Learning⏏ After receiving an assertion from S, R will:

⏏ Learn the assertion if he didn't know it before (with probability of p

1=1-k

R), and give S a popularity credit of 1

⏏ Reassess the assertion if he already knows it, but has a different opinion (with probability of p

2=f

R(1-(g-f-

R+g+f+

R+g○f○

R)),

and give S a popularity credit of 1; reassessment does not change F

R!

⏏ Ignore the assertion, otherwise, and ignore S altogether; we assume that in the absence of popularity credits, S's popularity slowly deteriorates by -δP per communication cycle

⏏ The overall change of the receiver's knowledge is:ΔK

R=λ(1-f

R)+(1-λ)(g-+g+-f

R(f+

R+f-

R))

⏏ The overall change of the sender's popularity is:ΔP

S=1-f

R(g+f+

R+g-f-

R+g○f○

R)

Learning Outcomes

Gain popularity ΔP

Gain knowledge ΔK

Feedback: Receiver's Side⏏ R can influence his own trust level by providing feedback in the

form of a comment⏏ If R is a Guru (k

R=1, he knows all assertions), he can always

assess a passed assertion correctly and earn a trust credit of +1⏏ If R is an Ignoramus (k

R=0), then we compare R's assessment of

an assertion with the oracle's assessment of the assertion; if they match, R get a trust credit of +1; otherwise, he gets a trust penalty of -1

⏏ Overall:ΔR

R=k

R+(1-k

R)r

S(2φ-1)(f+

S-f-

S)

Feedback: Sender's Side⏏ If receiver R sends a feedback, he can influence sender's S trust,

too⏏ If R's assessment of an assertion matches the S's assessment of

the same assertion, S earns a trust credit of +1 (discounter by the receiver's trust level!)

⏏ Otherwise, she gets a trust penalty of -1:ΔR

S=R

R((1-2g--2g+)(1-2f-

S-2f+

S)-2(f+

Sg++f-

Sg-))

⏏ The feedback can also change the knowledge distribution of S by forcing her to reassess her assertions, based on her trust in R:ΔK

S=r

R(1-λ)(g-+g+-(f+

S+f-

S))

⏏ The number of assertions at S will not change

Feedback Outcomes

Gain/lose trust ΔR and knowledge ΔK

Gain popularity ΔP and gain/lose trust ΔR

The Game⏏ S and R form a two-player square game⏏ We assume that in general the game is non-cooperative S and R

do not coordinate their strategies to maximize their joint utility)⏏ The game has a pure-strategy Nash equilibrium

Case Study: a MOSN

Let's experiment with a simulated massive online social network (MOSN)

Network Design⏏ Massive online social network (MOSN) represented as a

connected bidirectional graph where nodes represent actors and edges represent “friendship” connections or other information dissemination channels

⏏ 1,000 actors, fully connected (anyone can talk to anyone)⏏ At each simulation step, exactly two actors talk (still a 2⨯2 game,

not n⨯n)⏏ The probability of a fact being true is φ=0.8. The actor popularity

decay factor is δP=-0.1. The rumor discount coefficient is λ= 0.5. The maximum number of facts in the network is N = 2000.

Network Population⏏ Three experiments:

⏏ All “trolls”⏏ All “experts”⏏ 50% “trolls,” 50% “experts”

⏏ In each experiment, 1/3 of actors are “ignoramuses,” 1/3—”mediocres,” and 1/3—”gurus”

⏏ ri, p

i, f+

i, f-

i are drawn uniformly at random between 0 and 1

Goal⏏ Execute 10,000,000 random communications (10,000

communications per actor)⏏ Monitor the distribution of knowledge k and its quality f+ and f-

“Trolls”⏏ The “troll” community converges to the state of “total knowledge”

after a finite number of iterations (“Ignore credibility, talk!”)

“Experts”⏏ The distribution of information in the “expert” community changes

marginally over time (“Think before you say!”)⏏ Dispersion of the learning speed: some “Ignoramuses” and

“mediocres” learn faster

Difference in Learning Speed⏏ Actors with lower credibility and lower initial knowledge learn

faster to increase their utility⏏ Actors with higher credibility or higher initial knowledge learn

slower, because they have less incentive to learn

Future Directions⏏ Study the variability of κ, π, and ρ for different actors⏏ Analyze a full-duplex (two-way) communication scenario where

the actors are both senders and receivers—just finished, submitted to the Summer Simulation Conference-2010

⏏ Analyze a groupcast (one-to-many) communication scenario that is more common in massive online social networks

⏏ Collect experimental data that supports the model—so far, done only for the popularity component

Acknowledgment

This research has been supported in part by the College of Arts and Sciences, Suffolk University,

through an undergraduate research assistantship grant.