Tipping Points, Statistics of opinion dynamics

Chjan Lim, Mathematical Sciences, RPI

Collaboration with B. Szymanski, W Zhang, Y. Treitman, G. Korniss

Funding

Main: ARO grant 2009 – 2013 Prog Officer C. Arney, R. Zachary; ARO grant 2012 – 2015 Prog Officer P. Iyer

Secondary: ARL grants 2009 – 2012, ONR

Applications

Predict Average Outcomes, Properties in Networks of Semi-Autonomous sensor-bots / drones.

Less direct and more qualitative

predictions of social-political-cultural-economic networks

NG NG NG and more

Background of Naming Games (NG)Other variants of signaling gamesNG on Different networksOn Complete graphs – simple mean fieldNG on ER graphSDE model of NG

NG in detection community structure

Q. Lu, G. Korniss, and B.K. Szymanski, J. Economic Interaction and Coordination,4, 221-235 (2009).

ABC

AB

AC BC

A B

A

C

B

C

Two Names NG are End-Games for 3 Names case

Tipping Point of NGA minority of committed agents can persuade

the whole network to a global consensus. The critical value for phase transition is called

the “tipping point”.

J. Xie, S. Sreenivasan, G. Korniss, W. Zhang, C. Lim and B. K. Szymanski PHYSICAL REVIEW E (2011)

Saddle node bifurcation

Node Saddle Node

unstable Node

Below Critical

Above Critical

Meanfield Assumption and Complete Network

The network structure is ignored. Every node is only affected by the meanfield.

The meanfield depends only on the fractions(or numbers) of all types of nodes.

Describe the dynamics by an equation of the meanfield (macrostate).

Scale of consensus time on complet graph

3.5 4 4.5 5 5.5 6 6.5 718

20

22

24

26

28

30

32

34

36

38

log(N)

T/N

2 word Naming Game on complete graph

numerical simulation

analytical result by linear solver

Expected Time Spend on Each Macrostate before Consensus (without committed agents)

0

20

40

60

80

100

0

20

40

60

80

1000

10

20

30

40

50

60

nBn

A

T(n

A,

n B )

NG with Committed Agentsq=0.06<qc

q=0.12>qc

q is the fraction of agents committed in A.When q is below a critical value qc, the process may stuck in a meta-stable state for a very long time.

Higher stubbornness – same qualitative, robust result

ABC

AB

AC BC

A B

A

C

B

C

Two Names NG are End-Games for 3 Names case

2 Word Naming Game as a 2D random walk

Transient State

Absorbing State

n_A

n_B

P(B+)

P(A+)

P(B-)

P(A-)

Linear Solver for 2-Name NG

Have equations:

Then we assign an order to the coordinates, make , into vectors, and finally write equations in the linear system form:

SDE models for NG, NG and NG

Higher stubbornness – same qualitative, robust result

Diffusion vs Drift

Diffusion scales are clear from broadening of trajectories bundles

Drift governed by mean field nonlinear ODEs can be seen from the average / midlines of bundles

Other NG variants – same 1D manifold

3D plot of trajectory bundles – stubbornness K = 10 as example of variant (Y. Treitman and C. Lim 2012)

Consensus time distribution

Recursive relationship of P(X, T), the probability for consensus at T starting from X, Q is the transition matrix.

Take each column for the same T as a vector:

Take each row for the same X as a vector:

Calculate the whole table P(X,T) iteratively.

Consensus time distribution

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

-4

Tc

P(T

c)

p=0.1

numerical

analytical

0 0.5 1 1.5 2 2.5

x 107

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-7

Tc

P(T

c)

p=0.06

numerical

analytical

Red lines are calculated through the recursive equation.Blue lines are statistics of consensus times from numerical simulation(very expensive),(done by Jerry Xie)

Consensus Time distribution

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T-E[T] / std(T)

P(T

) *

std(

T)

p=0.08

N=50

N=100

N=150

y=exp(x+1)

-4 -2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

( Tc-E[T

c] ) / std(T

c)

P(T

c) *

std

(Tc)

p=0.12

st. normalN=50

N=100

N=200

N=400N=800

Below critical, consensus time distribution tends to exponential. Above critical, consensus time distribution tends to Gaussian.

For large enough system, only the mean and the variance of the consensus time is needed.

Variance of Consensus time

Theorem: the variance of total consensus time equals to the sum of variances introduced by every macrostate:

is the expected total number of steps spend on the given macrostate before consensus.

is the variance introduced by one step stay in the given macrostate.

Naming Game on other networks

Mean field assumption

Local meanfield assumptionHomogeneous Pairwise assumptionHeterogeneous Pairwise assumption

Homogeneous Pairwis Assumption

A

Mean field

P(·|A)

BP(·|B)

ABP(·|A)

The mean field is not uniform but varies for the nodes with different opinion.

Numerical comparison

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

pA

pB

pAB

Theoretical

Mean field

Trajectories mapped to 2D macrostate space

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

pA

p B

mean field

<k>=3

<k>=4<k>=5

<k>=10

<k>=50

Concentration of the consensus time

4.5 5 5.5 6 6.5 7 7.5 86.5

7

7.5

8

8.5

9

9.5

10

10.5

11

ln(N)

ln(T

0.95

)

<k>=5 simulation

<k>=5 pair approx

<k>=10 simulation<k>=10 pair appro

mean field

Analyze the dynamics

A B

C

C

CDirect

Related×(<k>-1)

1.Choosing one type of links, say A-B, and A is the listener.2.Direct change: A-B changes into AB-B.3.Related changes: since A changes into AB, <k>-1 related links C-A change into C-AB. The probability distribution of C is the local mean field P(·|A).

Merits of SDE model

Include all types of NG and other communication models in one framework and distinguish them by two parameters.

Present the effect of system size explicitly.Collapse complicated dynamics into 1-d

SDE equation on the center manifold.

Thanks