Influencing with committed minorities

Transcription

1 Ifluecig with committed miorities S. Sreeivasa, J. Xie W. Zhag, C. Lim G. Koriss,.K. Szymaski Supported by RL NS CT, ONR 1

2 Never doubt that a small group of thoughtful, committed, citizes ca chage the world. Ideed, it is the oly thig that ever has. Margaret Mead The role of iflexible miorities i the breakig of democratic opiio dyamics, Galam ad Jacobs, Physica 381, 366 (2007). (homogeeous mixig/mea field) 2

3 Q. Ca a committed set of miority opiio holders o a etwork, reverse the majority opiio? 3

4 Q. Ca a committed set of miority opiio holders o a etwork, reverse the majority opiio? (vacciatios cause autism) 4

5 Q. Ca a committed set of miority opiio holders o a etwork, reverse the majority opiio? (vacciatios do ot cause autism) (vacciatios cause autism) 5

6 Q. Ca a committed set of miority opiio holders o a etwork, reverse the majority opiio? pplicatios: Ifluecig public opiio o prevetative healthcare, Eradicatig hostile opiios i terrorist states. 6

7 Model of social ifluece: iary agreemet model (2-word Namig Game) Differece from epidemic like models: coverted idividual ca revert back. (i cotrast to Threshold Model, ass Model) Ifluecig is symmetric i both opiios. (for ex: i cotrast to SIS model) arochelli et al., PRE (2007). Castelló et al., EPJ (2009). arochelli, PRE (2011). Xie et al., PRE (2011). Differece from voter model: Presece of itermediate state coarseig & domai formatio. Plausible for studyig situatios where a idividual does ot require high persoal ivestmet to chage opiio: Spread of buzz (Uzzi et al, forthcomig) 7

8 Model of social ifluece: iary agreemet model gets possess oe of the followig opiios at ay give time: (vacciatios do ot cause autism) (vacciatios cause autism) (mixed / ot sure) 8

9 Model of social ifluece: iary agreemet model t each microscopic time step: speaker is chose at radom. radom eighbor of the speaker is chose as listeer. Speaker Listeer 9

10 Model of social ifluece: iary agreemet model Opiio chage: Speaker voices a opiio from his list Speaker Listeer Case 1: If spoke opiio ot o listeer s list - he adds it 10

11 Model of social ifluece: iary agreemet model Opiio chage: Speaker voices a opiio from his list Speaker Listeer Case 2: If spoke opiio is o listeer s list - both retai oly spoke opiio 11

12 Iitial coditio we care about: Small fractio p < 0.5 of odes radomly chose are committed to opiio Committed odes are u-ifluecable i.e. ever chage opiio Remaiig fractio (1-p) of odes have opiio Oly absorbig state is the all cosesus state Q. How log does it take to reach the all cosesus state as a fuctio of the committed miority fractio p? 12

13 Mea Field Equatios each idividual ca iteract with all others ( complete graph ) umber of idividuals is large (N ) = N /N : desity of idividuals with opiio = N /N : desity of idividuals with opiio = N /N : desity of idividuals with mixed opiio () + + = 1 d dt d dt arochelli et al., PRE (2007). Castelló et al., EPJ (2009). arochelli, PRE (2011). 13

14 Mea Field Equatios each idividual ca iteract with all others ( complete graph ) umber of idividuals is large a small fractio of p idividuals are committed to the iitially miority opiio (committed idividuals ca ever chage their opiios) p = N c /N: desity of committed idividuals with opiio, p < 0.5 = N /N : desity of idividuals with opiio, (0)=0; = N /N : desity of idividuals with opiio, (0)=1 p; =N /N : desity of idividuals with mixed opiio (), p = 1 d dt d dt 3 2 p p Xie et al., PRE (2011) 14

15 Tippig poit i Social Networks p: fractio of agets committed to opiio p 0.05 p c pc 0.10 p 0. 2 pc ( domiated, mixed) (saddle poit) (all cosesus) (all cosesus) o-absorbig (-domiated, mixed) stable fixed poit exists; ll trajectories startig from iitial coditio flow to the o-absorbig fixed poit Oly all- cosesus fixed poit exists ll trajectories flow to cosesus fixed poit. 15

16 Tippig poit i Social Networks Results o large complete graphs agree with mea-field results Sharp trasitio from -domiated mixed steady state to cosesus desity of iitially domiatig opiio For For fractio of committed agets tippig poit = steady state sol. is a mixed state steady state sol. is the all cosesus p c

17 Tippig poit i Social Networks FC ER (sparse) fractio of committed agets fractio of committed agets tippig poit 17 desity of iitially domiatig opiio desity of iitially domiatig opiio Xie et al. (PRE, 2011) scale-free (sparse)

18 Tippig poit i Social Networks The meaig of ever i fiite etworks with N >>1 odes (usig quasi-statioary approx./master-equatio approach) p p : T ~ c c e cn p p c : T ~ log( N) c Xie et al. (PRE, 2011) 18

19 Tippig poit i Social Networks Social ifluecig ad associated radom-walk models: symptotic cosesus times o the complete graph T m / N (a) Time spet i meta-stable state p p c q=0.04 q=0.06 q=0.07 q=0.08 T m / N Time spet i meta-stable state (b) p p c q=0.08 q=0.09 q=0.10 q= N p p : T ~ c c e Time spet i state (, ) before cosesus cn N p p c : T ~ log( N) c Time spet i state (, ) before cosesus Zhag et al. (Chaos, 2011) 19

20 Ogoig/Future work Give kowledge of the graph topology, what o radom committed ode selectio strategy, gives the lowest critical threshold? How ca we geeralize our model to uderstad icetive mechaisms that drive opiio spread? ifluece of committed miorities i the presece of dedicated extremists "Social Cosesus through the Ifluece of Commited Miorities", arxiv: , J. Xie, S. Sreeivasa, G. Koriss, W. Zhag, C. Lim,. K. Szymaski, PRE (i press, 2011). "Social Ifluecig ad ssociated Radom Walk Models: symptotic Cosesus Times o the Complete Graph", arxiv: , W. Zhag, C. Lim, S. Sreeivasa, J. Xie,.K. Szymaski, G. Koriss, Chaos (i press, 2011). "The Namig Game i Social Networks: Commuity Formatio ad Cosesus Egieerig", Q. Lu, G. Koriss, ad.k. Szymaski, Joural of Ecoomic Iteractio ad Coordiatio 4, 221 (2009). Thaks to Qimig Lu Supported by DTR, RL NS CT, ONR 20