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Influencing with committed minorities

S. Sreenivasan, J. XieW. Zhang, C. LimG. Korniss, B.K. Szymanski

Supported by  ARL NS‐CTA, ONR

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“Never doubt that a small group of thoughtful, committed, citizens can change the world. Indeed, it is the only thing that ever has.“

‐Margaret Mead

“The role of inflexible minorities in the breaking of democratic opinion dynamics”, Galam and Jacobs , Physica A 381, 366 (2007).  (homogeneous mixing/mean‐field)

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Q. Can a committed set of minority opinion holderson a network, reverse the majority opinion?

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Q. Can a committed set of minority opinion holderson a network, reverse the majority opinion?

B (vaccinations cause autism)

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Q. Can a committed set of minority opinion holderson a network, reverse the majority opinion?

A (vaccinations do not cause autism)B (vaccinations cause autism)

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Q. Can a committed set of minority opinion holderson a network, reverse the majority opinion?

Applications: Influencing public opinion on preventative healthcare,Eradicating hostile opinions in terrorist states.

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Model of social influence: Binary agreement model(2-word Naming Game)

Influencing is symmetric in both opinions.(for ex: in contrast to SIS model)

Difference from epidemic like models:

Difference from voter model:

Presence of intermediate state – coarsening & domain formation.

Plausible for studying situations where an individual does not require high personal investment to change opinion:Spread of buzz (Uzzi et al, forthcoming)

A “converted” individual can revert back.(in contrast to Threshold Model, Bass Model)

Baronchelli et al., PRE (2007).Castelló et al., EPJB (2009).Baronchelli, PRE (2011).Xie et al., PRE (2011).

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Model of social influence: Binary agreement model

Agents possess one of the following opinions at any given time:

A (vaccinations do not cause autism)

B (vaccinations cause autism)

A B (mixed / not sure)

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Model of social influence: Binary agreement model

Speaker Listener

A speaker is chosen at random.A random neighbor of the speaker is chosen as listener.

At each microscopic time step:

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Model of social influence: Binary agreement model

Opinion change:

Speaker ListenerAA BA

Case 1: If spoken opinion not on listener’s list

A

Speaker voices an opinion from his list

- he adds it

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Model of social influence: Binary agreement model

Opinion change:

Speaker ListenerA

BA

Case 2: If spoken opinion is on listener’s list

A

Speaker voices an opinion from his list

- both retain only spoken opinion

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Initial condition we care about: Small fraction p < 0.5 of nodes randomly chosen are committed to opinion A

Remaining fraction (1-p) of nodes have opinion B

Committed nodes are un-influencable i.e. never change opinion

Only absorbing state is the all A consensus state

Q. How long does it take to reach the all A consensusstate as a function of the committed minority fraction p ?

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Mean‐Field Equations

BABABABBAB

AABABABBAA

nnnnnndt

dn

nnnnnndt

dn

Baronchelli et al., PRE (2007).Castelló et al., EPJB (2009).Baronchelli, PRE (2011).

each individual can interact with all others (“complete graph”) number of individuals is large (N)

nA  = NA/N : density of individuals with opinion AnB  = NB/N : density of individuals with opinion BnAB= NAB/N : density of individuals with “mixed” opinion (AB)nA + nB + nAB = 1

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Mean‐Field Equations each individual can interact with all others (“complete graph”) number of individuals is large a small fraction of p individuals are committed to the initially minorityopinion A (committed individuals can never change their opinions) 

p = Nc/N: density of committed individuals with opinion A,         p < 0.5nA = NA/N : density of individuals with opinion A,                             nA (0)=0; nB = NB/N : density of individuals with opinion B,                             nB (0)=1‐p; nAB=NAB/N : density of individuals with “mixed” opinion (AB),p + nA + nB + nAB = 1

BBABABABBAB

ABAABABABBAA

pnnnnnnndt

dn

pnnnnnnndt

dn

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Xie et al., PRE (2011)

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Tipping point in Social Networksp: fraction of agents committed to opinion A

p 0.05 pc

(all‐A consensus)

(saddle point)

(B‐dominated, mixed)

An

Bn

A non-absorbing (B-dominated, mixed) stable fixed point exists;

All trajectories starting from initial conditionflow to the non-absorbing fixed point

10.0cp cpp 2.0

Bn

AnOnly all-A consensus fixed point exists

(all‐A consensus)

All trajectories flow to consensusfixed point.

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For

For steady state soln. is the all A consensus

Results on large complete graphs agree with mean-field resultsSharp transition from B-dominated mixed steady state to consensus

= 0.0979 steady state soln. is a mixed state

Tipping point in Social Networks

10.0cptipping pointfraction of committed agents

dens

ity o

f ini

tially

dom

inat

ing

opin

ion

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dens

ity o

f ini

tially

dom

inat

ing

opin

ion

fraction of committed agents fraction of committed agents

ER(sparse)

FC

tipping pointde

nsity

of i

nitia

lly d

omin

atin

g op

inio

n

Tipping point in Social Networks

Xie et al. (PRE, 2011)

scale-free(sparse)

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The meaning of “never” in finite networks with N >>1 nodes(using quasi-stationary approx./master-equation approach)

cNcc eTpp ~: )log(~: NTpp cc

Tipping point in Social Networks

Xie et al. (PRE, 2011)

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Social influencing and associated random-walk models: Asymptotic consensus times on the complete graph

50 100 150 200 250 300 350 400 450 50010

0

102

104

106

108

1010

1012

1014

1016

(a)

N

Tm

/ N

q=0.04q=0.06q=0.07q=0.08

50 100 150 200 250 300 350 400 450 50015

20

25

30

35

40

45

50

55

60

N

Tm

/ N

(b)

q=0.08q=0.09q=0.10q=0.12

cNcc eTpp ~:

Zhang et al. (Chaos, 2011)

)log(~: NTpp cc

Time spent in meta-stable state

cpp cpp

Time spent in state (nA,nB) before consensus

Time spent in meta-stable state

Time spent in state (nA,nB) before consensus

Tipping point in Social Networks

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Given knowledge of the graph topology, what non‐random committed node selection strategy, gives the lowest critical threshold ? 

How can we generalize our model to understand incentive mechanisms that drive opinion spread ?

influence of committed minorities in the presence of dedicated extremists

"Social Consensus through the Influence of Commited Minorities", arXiv:1102.3931,J. Xie, S. Sreenivasan, G. Korniss, W. Zhang, C. Lim, B. K. Szymanski, PRE (in press, 2011).

"Social Influencing and Associated Random Walk Models: Asymptotic Consensus Times on the Complete Graph", arXiv:1103.4659, W. Zhang, C. Lim, S. Sreenivasan, J. Xie, B.K. Szymanski, G. Korniss, Chaos (in press, 2011).

"The Naming Game in Social Networks: Community Formation and Consensus Engineering", Q. Lu, G. Korniss, and B.K. Szymanski, Journal of Economic Interaction and Coordination 4, 221 (2009).

Thanks to Qiming LuSupported by DTRA, ARL NS‐CTA, ONR

Ongoing/Future work