Motivations. Opinions, influence networks and centrality. Small deliberative groups. Opinion Dynamics and the Evolution of Influence Networks

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1 Movaons Opnon Dynamcs and he Evoluon of drvers Francesco Bullo Deparmen of Mechancal Engneerng Cener for Conrol, Dynamcal Sysems & Compuaon Unversy of Calforna a Sana Barbara bg daa ncreasngle avalable quanave mehods n socal scences applcaons n markeng and (n)-secury dynamcal processes over socal neworks opnon dynamcs, nfo propagaon nework formaon and evoluon co-evoluonary processes SoCal Symp on Nework Economcs and Game Theory Krackhard s advce nework key novely: sequence of ssues Peng Ja Ana MrTabaabae Noah Fredkn SoCal NEGT / 6 Small delberave groups SoCal NEGT / 6 Opnons, nfluence neworks and cenraly Dynamcs and Formaon of Opnons small delberave groups are assembled n mos socal organzaon o deal wh sequences of ssues n parcular domans: udcal, legslave and execuve branches: grand ures, federal panels of udges, Supreme Cour sandng polcy bodes, congressonal commees advsory boards corporaons: board of drecors/rusees unverses: faculy meengs group properes may evolve over s ssue sequence accordng o naural socal processes ha modfy s nernal socal srucure Opnon formaon Dynamcs of and Socal Power refleced apprasal hypohess by Cooley, 9 ndvdual self-apprasal (e.g., self-confdence, self-eseem, self-worh) s nfluenced by he apprasal of oher ndvduals of her Socal nework for obesy sudy (Chrsaks and Fowler, 7) - varyng socal power and self-confdence - consan relave nerpersonal relaons a sablzaon of ndvduals levels of openness and closure o nerpersonal nfluences on her nal preferences, a sablzaon of ndvduals rankng of, and nfluence accorded o, oher members model by French ( 6), Harary ( 6), and DeGroo ( 7) mahemazaon by Fredkn, : possble sysemac changes: convex combnaons of opnons SoCal NEGT Nework cenraly cenraly measure of nework nodes, e.g., egenvecor cenraly by Bonacch, 97 / 6 Socal nework for male wre-aled manakns (Ryder e al. 8) SoCal NEGT / 6

2 Opnons, nfluence neworks and cenraly Opnons, nfluence neworks and cenraly Dynamcs and Formaon of Opnons convex combnaons of opnons Dynamcs and Formaon of Opnons convex combnaons of opnons model by French ( 6), Harary ( 6), and DeGroo ( 7) Dynamcs of and Socal Power refleced apprasal hypohess by Cooley, 9 Opnon formaon model by French ( 6), Harary ( 6), and DeGroo ( 7) Dynamcs of and Socal Power refleced apprasal hypohess by Cooley, 9 Opnon formaon ndvdual self-apprasal (e.g., self-confdence, self-eseem, self-worh) s nfluenced by he apprasal of oher ndvduals of her Socal nework for obesy sudy (Chrsaks and Fowler, 7) ndvdual self-apprasal (e.g., self-confdence, self-eseem, self-worh) s nfluenced by he apprasal of oher ndvduals of her Socal nework for obesy sudy (Chrsaks and Fowler, 7) mahemazaon by Fredkn, : mahemazaon by Fredkn, : - varyng socal power and self-confdence - consan relave nerpersonal relaons - varyng socal power and self-confdence - consan relave nerpersonal relaons Nework cenraly cenraly measure of nework nodes, e.g., egenvecor cenraly by Bonacch, 97 Socal nework for male wre-aled manakns (Ryder e al. 8) Nework cenraly cenraly measure of nework nodes, e.g., egenvecor cenraly by Bonacch, 97 Socal nework for male wre-aled manakns (Ryder e al. 8) SoCal NEGT / 6 The dynamcs of opnons SoCal NEGT / 6 The dynamcs of opnons DeGroo opnon dynamcs model y( + ) = W y() Opnons y R n Influence nework = row-sochasc W by P-F: lm y() = (w T y() where w s domnan lef egenvecor of W DeGroo opnon dynamcs model y( + ) = W y() Opnons y R n Influence nework = row-sochasc W by P-F: lm y() = (w T y() where w s domnan lef egenvecor of W Self-weghs W =: x Inerpersonal accorded weghs W Relave nerpersonal accorded weghs C, where W = ( x )C W (x) = dag(x)i n + dag( n x)c Self-weghs W =: x Inerpersonal accorded weghs W Relave nerpersonal accorded weghs C, where W = ( x )C W (x) = dag(x)i n + dag( n x)c SoCal NEGT / 6 SoCal NEGT / 6

3 The dynamcs of opnons The dynamcs of socal power and self-confdence DeGroo opnon dynamcs model y( + ) = W y() Opnons y R n Influence nework = row-sochasc W Refleced apprasal hypohess by Cooley, 9: ndvdual self-apprasal (e.g., self-confdence, self-eseem, self-worh) s nfluenced by he apprasal held by ohers of her by P-F: lm y() = (w T y() where w s domnan lef egenvecor of W Self-weghs W =: x Inerpersonal accorded weghs W Relave nerpersonal accorded weghs C, where W = ( x )C W (x) = dag(x)i n + dag( n x)c SoCal NEGT / 6 The dynamcs of socal power and self-confdence Refleced apprasal hypohess by Cooley, 9: ndvdual self-apprasal (e.g., self-confdence, self-eseem, self-worh) s nfluenced by he apprasal held by ohers of her Mahemazaon by Fredkn, : along a sequence of ssues, ndvdual dampens/elevaes self-wegh x accordng o her relave pror conrol SoCal NEGT 6 / 6 The dynamcs of socal power and self-confdence Refleced apprasal hypohess by Cooley, 9: ndvdual self-apprasal (e.g., self-confdence, self-eseem, self-worh) s nfluenced by he apprasal held by ohers of her Mahemazaon by Fredkn, : along a sequence of ssues, ndvdual dampens/elevaes self-wegh x accordng o her relave pror conrol x(s) W (x(s)) w(x(s)) self-apprasal nfluence nework socal power refleced apprasal mechansm x(s + ) = w(x(s)) self-apprasal = self-weghs relave conrol = socal power SoCal NEGT 6 / 6 SoCal NEGT 6 / 6

4 The dynamcal sysem The dynamcal sysem DeGroo dynamcs abou an ssue: y( + ) = W (x)y() Influence nework W (x) = dag(x)i n + dag( n x)c Refleced apprasal across ssues: DeGroo dynamcs abou an ssue: y( + ) = W (x)y() Influence nework W (x) = dag(x)i n + dag( n x)c Refleced apprasal across ssues: x(k + ) = w(x(k)) = F (x(k)) x(k + ) = w(x(k)) = F (x(k)) DeGroo-Fredkn dynamcs DeGroo-Fredkn dynamcs e, ( c c x x n = where c s he domnan lef egenvecor of C f x = e for all c x, oherwse e, ( c c x x n = where c s he domnan lef egenvecor of C f x = e for all c x, oherwse SoCal NEGT 7 / 6 The map and he egenvecor cenraly parameer SoCal NEGT 7 / 6 The map and he egenvecor cenraly parameer e, ( c c x x n = f x = e for all c x, oherwse F : n n locally Lpschz The verces {e } are fxed pons under F relave nerpersonal weghs C play role only hrough c c = approprae egenvecor cenraly (domnan lef egenvecor) e, ( c c x x n = f x = e for all c x, oherwse F : n n locally Lpschz The verces {e } are fxed pons under F relave nerpersonal weghs C play role only hrough c c = approprae egenvecor cenraly (domnan lef egenvecor) Lemma (Egenvecor cenraly) For any C row-sochasc, rreducble wh zero dagonal and c n, max{c }. c =. G(C) s wh sar opology and s he cener Lemma (Egenvecor cenraly) For any C row-sochasc, rreducble wh zero dagonal and c n, max{c }. c =. G(C) s wh sar opology and s he cener SoCal NEGT 8 / 6 SoCal NEGT 8 / 6

5 Problem: dynamcal sysem analyss and socologcal nerpreaon Man resuls for generc relave nerpersonal accorded weghs unque non-rval fxed pon: x = x (c neror of n convergence = forgeng nal condons for all non-rval nal condons, lm x(k) = lm w(x(k)) = x k k Exsence and sably of equlbra for he D-F model? Role of nework srucure and parameers? Condons of emergence of auocracy and democracy? Insgh no ron law of olgarchy by Mchels 9?.. accumulaon of socal power and self-apprasal fxed pon x Node Node 9.. > has same orderng of c socal power hreshold T such ha: x. c T or x... c T. Node... Node Node Node Node Node SoCal NEGT 9 / 6 Man resuls for generc relave nerpersonal accorded weghs SoCal NEGT / Man resuls..... for generc relave nerpersonal accorded weghs unque non-rval fxed pon: x = x (c neror of n convergence = forgeng nal condons for all non-rval nal condons, lm x(k) = lm w(x(k)) = x k k accumulaon of socal power and self-apprasal fxed pon x Node Node 9.. > has same orderng of c socal power hreshold T such ha: x. c T or x... c T... unque non-rval fxed pon: x = x (c neror of n convergence = forgeng nal condons for all non-rval nal condons, lm x(k) = lm w(x(k)) = x k k accumulaon of socal power and self-apprasal fxed pon x Node Node 9.. > has same orderng of c socal power hreshold T such ha: x. c T or x... c T... Node Node 9 Node Node Node Node 9 Node Node Node Node Node Node.. SoCal NEGT / 6.. SoCal NEGT / 6

6 Doubly-sochasc C : emergency of democracy Sar opology: emergency of auocracy c = [/, /, /] Lemma (Convergence o democracy) c = [/, /, /] Lemma (Convergence o auocracy) Iff C s doubly-sochasc:.6 x for all non-rval nal condons, lmk x(k) = lmk w (x(k)) = Iff graph has sar opology wh cener :.8 n n.... SoCal NEGT rankng and unqueness: elemenary seps and conraddcons monooncy: max and mn are nvaran max = argmax.... chrshld. Egenvecor cenraly score... Node max = argmax x (s), s x.. =. x () x convergence: Lyapunov funcon decreasng everywhere x 6= x x x V (x) = max ln mn ln x x Node / Node. SoCal NEGT exsence va Brower fxed pon heorem (F connuous on compac).. Node ssue N. Node 9. Egenvecor cenraly score vs. Equlbrum selfïwegh Node ssue... x x.... Proof mehods Egenvecor cenraly scores Equlbrum selfïweghs Node 9. Node.. ssue / 6 D-F on Krackhard s advce nework.. ssue N... ssue.6 Auocra appears n sar cener Exreme power accumulaon x x ssue for all nal non-rval condons, lmk x(k) = lms w (x(k)) = e.. No power accumulaon ssue.8 Unform socal power ssue here are no non-rval fxed pons of F x n n, he non-rval fxed pon of F s SoCal NEGT / 6 SoCal NEGT / 6

7 Ongong expermen Conrbuons and fuure work groups of subecs n a face-o-face dscusson opnon formaon on a sequence of ssues ssues n he doman of choce dlemmas: wha s your mnmum level of confdence (scored -) requred o accep a rsky opon wh a hgh payoff raher han a less rsky opon wh a low payoff groups under pressure o reach consensus, oher no On each ssue, each subec prvaely recorded (n followng emporal order): an nal opnon on he ssue pror o he group-dscusson, a fnal opnon on he ssue upon compleon of he group-dscusson (whch ranged from -7 mnues), and an allocaon of nfluence uns (under he nsrucon ha hese allocaons should represen her apprasals of he relave nfluence of each group member s opnon on her own opnon). Conrbuons a new perspecve and a novel dynamcal model for socal power, self-apprasal, nfluence neworks dynamcs and feedback n socology a new poenal explanaon for he emergence of auocracy see ron law of olgarchy by Mchels 9 Fuure work Robusness of resuls for dsnc models of opnon dynamcs Robusness of resuls for hgher-order models of refleced apprasal Reference: Opnon Dynamcs and The Evoluon of Socal Power n. SIAM Revew,, o appear Fundng: Insue for Collaborave Boechnology hrough gran W9NF-9-D- from he U.S. Army Research Offce SoCal NEGT / 6 Conrbuons and fuure work SoCal NEGT 6 / 6 Conrbuons a new perspecve and a novel dynamcal model for socal power, self-apprasal, nfluence neworks dynamcs and feedback n socology a new poenal explanaon for he emergence of auocracy see ron law of olgarchy by Mchels 9 Fuure work Robusness of resuls for dsnc models of opnon dynamcs Robusness of resuls for hgher-order models of refleced apprasal Reference: Opnon Dynamcs and The Evoluon of Socal Power n. SIAM Revew,, o appear Fundng: Insue for Collaborave Boechnology hrough gran W9NF-9-D- from he U.S. Army Research Offce SoCal NEGT 6 / 6