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Organizing Principles for Social Networks and Dynamics

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PPT of talk given at the Center for Vision Research at U Central Florida March 25 - 26 2013

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15. SUBJECT TERMS

Tipping points, scalar stochastic diff eqn, expected times to consensus

Chjan C. Lim

Rensselaer Polytechnic Institute

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Organizing Principles for Social Networks and Dynamics

Report Title

ABSTRACT

PPT of talk given at the Center for Vision Research at U Central Florida March 25 - 26 2013

Invited tour of facility by Dr. Mubarak Shah

Organizing Principles in NetworkOrganizing Principles in Network Science- Scalar SDE,Tipping

Points of opinion dynamics

Chjan Lim, Mathematical Sciences, jRPI

FundingFunding

Main: ARO grant 2009 – 2013 we thank CMain: ARO grant 2009 2013 we thank C. Arney, R. Zachary; ARO grant 2012 –2015 we thank P Iyer2015 we thank P. IyerSecondary: ARL grants 2009 – 2012

MiscMisc

Collaborators: Dr W Zhang, Y Treitman, B g,Szymanski, G Korniss

Papers: Weituo Zhang, Chjan C. Lim, B. Szymanksi, “Analytic Treatment ofWeituo Zhang, Chjan C. Lim, B. Szymanksi, Analytic Treatment of

Tipping Points for Social Consensus in Large Random Networks”, Phys Rev E 86 (6), 061134, 2012Weituo Zhang, Chjan Lim, and Boleslaw Szymanski, Tipping

Points of Diehards in Social Consensus on Large Random Networks, Complex Networks, Proc. 3rd Workshop on Complex Networks, CompleNet, Melbourne, FL, March 7-9, 2012, Studies in Computational Intelligence, vol. 424, Springer, Berlin, Germany, g g y2013, pp. 161-168.

Yosef Treitman, Chjan Lim, W. Zhang and A. Thompson, “Naming Game with Greater Stubbornness and Unilateral Zealots”, IEEE NSW Conf April 29 May 1 2013 West Point NYNSW Conf., April 29 – May 1 2013, West Point, NY

Tipping Point of NGTipping Point of NG A minority of committed agents can persuade

the whole network to a global consensusthe 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 bifurcationSaddle node bifurcation

Node Saddle NodeBelow Critical

unstable NodeAbove Critical

Meanfield Assumption and Complete N t kNetwork

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

f ti ( b ) f ll t f dfractions(or numbers) of all types of nodes.Describe the dynamics by an equation of

the meanfield (macrostate).

Scale of consensus time on complet hgraph

382 word Naming Game on complete graph

34

36

38

numerical simulationanalytical result by linear solver

30

32

26

28T/N

20

22

24

3.5 4 4.5 5 5.5 6 6.5 718

20

log(N)

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

40

50

60

)

10

20

30

T(n A, n

B

60

80

100

60

80

1000

0

20

40

0

20

40

60

nBnA

NG with Committed Agents0 12>q=0.06<qc

q=0.12>qc

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

2 Word Naming Game as a 2D d lk2D random walk

n_B

Transient State

P(B+)

P(A+)P(A-) Transient State

Absorbing StateP(B-)

n_A

Linear Solver for 2-Name NGLinear Solver for 2 Name NG

Have equations:Have equations:

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

SDE models for NG, NG and NGSDE models for NG, NG and NG

Diffusion vs DriftDiffusion vs Drift

Diffusion scales are clear from broadeningDiffusion scales are clear from broadening of trajectories bundles

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

Assume a very natural social – political condition, generalizing NG

Where the network is divided into k+1 sub-populations

Each with a different fixed propensity to signal/vote/utter the opinion A

When hearing the word A a node from subgroup j < k will move to subgroup j+1When hearing the word A a node from subgroup j < k will move to subgroup j+1Likewise hearing B a node from j > 0 will move to subgroup j-1

The probability that a node from subgroup j will signal A is j/k

Same node has probability 1 – j/k of signalling word B

Additional rules of k-NGAdditional rules of k NG

A node s is drawn at random from theA node s is drawn at random from the network and sends out a signal A with por s signals B with prob = 1 por s signals B with prob = 1 – pNext a node L is drawn at random to

i th i l Areceive the signal A or node L is drawn to receive the signal B

k-NGk NG

The subgroups j = 0 k correspond toThe subgroups j = 0, k correspond to those nodes that are completely convinced of the B and A opinion resp ORof the B and A opinion resp. OR

E i l tl th d th t ith b 1Equivalently those nodes that with prob 1Signals B, A resp.

Voting or PollingVoting or Polling

The key network quantity is an averageThe key network quantity is an average network opinion obtained by polling:

p = sum over subgrps j = 0 to k of n(j) j / kN It gives the probability of a speaker chosen g p y p

at random signalling the word A

Stochastic DynamicsStochastic Dynamics

Derivation of k-dim coupled Random Walk:Derivation of k dim coupled Random Walk:

F j 1 t k 1For j = 1 to k-1,n(j,t+1) = n(j,t) + 1, -1 with resp. prob.P(+1) = p(t) n(j-1,t)/N + (1-p(t)) n(j+1,t)/NP(-1) = (1-p(t)) n(j t)/N + p(t) n(j t)/NP( 1) (1 p(t)) n(j,t)/N + p(t) n(j,t)/N = n(j,t)/N

Random walkRandom walk

For the distinguished subgrps at both endsFor the distinguished subgrps at both ends of the opinion spectrum, j = 0 , k

n(0,t+1) = n(0,t) + 1, 0, -1 with prob.P(+1) = (1-p(t)) n(1,t)/N P(0) = (1-p(t)) n(0,t)/N( ) ( p( )) ( , )P(-1) = p(t) n(0,t)/N

Subgroup kSubgroup k

n(k t+1) = n(k t) + 1 0 -1 with probn(k,t+1) = n(k,t) + 1, 0, 1 with prob.

P( 1) (t) (k 1 t)/NP(+1) = p(t) n(k-1,t)/NP(0) = p(t) n(k,t)/NP(-1) = (1-p(t)) n(k,t)/N

Shadow WalkShadow Walk

Define a scalar nonlazy random walk onDefine a scalar nonlazy random walk on the quantity p whose consensus times are lower bounds for the actual expected ptimes.

p(t+1) = p(t) + 1/kN, -1/kN with prob

P(+1) = p(t) P(-1) = 1-p(t)P(-1) = 1-p(t)

SDE or Diffusion ModelSDE or Diffusion Model

Solution of SDESolution of SDE

Expected Times to ConsensusExpected Times to Consensus

Expected times - exit times – stop timesExpected times exit times stop times

continuedcontinued

Higher stubbornness – same qualitative, b t ltrobust result

Higher stubbornness – same qualitative, b t ltrobust result

Other NG variants – same 1D manifoldOther NG variants same 1D manifold

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

Consensus time distributionConsensus time distribution

Recursive relationship of P(X, T), the probability p ( ) p yfor 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.

Take each row for the same X as a vector:

Consensus time distributionConsensus time distribution

x 10-4 p=0.1 x 10-7 p=0.06

1.4

1.6

1.8x 10 p

numericalanalytical

3

3.5

4x 10 p

numericalanalytical

0.8

1

1.2

P(T

c)

1 5

2

2.5

P(T

c)

0.2

0.4

0.6

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

Tc

0 0.5 1 1.5 2 2.5

x 107

0

Tc

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

Consensus Time distributionConsensus Time distribution

1p=0.08

p=0.12

0.7

0.8

0.9 N=50N=100N=150y=exp(x+1) 0.5

0.6

st. normalN=50N=100N=200N=400N=800

0.3

0.4

0.5

0.6

P(T

) * s

td(T

)

0.3

0.4

P(T

c) * s

td(T

c) N 800

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

0.1

0.2

4 2 0 2 4 60

0.1

0.2

T-E[T] / std(T) -4 -2 0 2 4 6( Tc-E[Tc] ) / std(Tc)

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.

NG on RGGNG on RGG

NG on RGG past Tipping pointNG on RGG past Tipping point

Homogeneous Pairwis AssumptionHomogeneous Pairwis AssumptionThe mean field is not uniform but varies for the nodes with different opinionwith different opinion.

AP(·|A)

Mean field BP(·|B)

ABP(·|A)

Numerical comparisonNumerical comparison

0.9

1

0.7

0.8

p

0.5

0.6

pApB

pAB

Theoretical

0.3

0.4Mean field

0 1

0.2

0 3

0 2 4 6 8 10 12 14 16 18 200

0.1

t

Trajectories mapped to 2D macrostate space

0.9

1

0.7

0.8mean field<k>=3<k>=4<k>=5

0.5

0.6

p B

<k>=10<k>=50

0.3

0.4

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

pA

Concentration of the consensus timeConcentration of the consensus time

10 5

11

10

10.5

9

9.5

T 0.95

)

8

8.5ln(T

<k>=5 simulation

7

7.5

k 5 simulation<k>=5 pair approx<k>=10 simulation<k>=10 pair appromean field

4.5 5 5.5 6 6.5 7 7.5 86.5

ln(N)

Change of the tipping point w.r.t. the daverage degree

T he local mean field for the node with opinion C : )

P(· IC) = [P (AIC) ,P (B IC),P(ABIC)Jr,c = A,B,AB

The number of different type of links:

P(· IA)(i) = P(AIA) P(BIA)

P(ABIA)

P(·IB)(i) = P(AIB) P(BIB)

P(ABIB)

P(· IAB)(i) =

P(AIAB) P(BIAB)

P(ABIAB)

1 ---------

2LA-A + LA-B + LA-AB

1 --------

L A-B+ 2LB-B + LB-AB

1

LA-B

2LB- B

LB- AB

- ---------L A-AB

LB- AB 2LAB-AB LA - AB + LB- AB + 2LAB- AB

Analyze the dynamicsAnalyze the dynamics1.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 g p yis the local mean field P(·|A).

C

Direct

Related×(<k>-1)

A BCDirect

C

Local mean field equationLocal mean field equation

Normalized equation:

SDE model of NGSDE model of NG

Merits of SDE modelMerits of SDE model

Include all types of NG and other Include all types of NG and other communication models in one framework and distinguish them by two parametersand distinguish them by two parameters.Present the effect of system size explicitly.

C ll li t d d i i t 1 dCollapse complicated dynamics into 1-d SDE equation on the center manifold.

ThanksThanks