tipping points, statistics of opinion dynamics
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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