The Dynamics of Persuasion Sid Redner, Santa Fe Institute (physics.bu.edu/~redner)

Transcription

1 The Dynamics of Persuasion Sid Redner, Santa Fe Institute (physics.bu.edu/~redner) CIRM, Luminy France, January 5-9, 2015 T. Antal, E. Ben-Naim, P. Chen, P. L. Krapivsky, M. Mobilia, V. Sood, F. Vazquez, D. Volovik + support from Modeling Consensus: introduction to the voter model voter model on complex networks voting with some confidence majority rule Modeling Discord & Diversity: 3-state voter models strategic voting bounded compromise dynamics of social balance Axelrod model lecture 1 lecture 2 lecture 3

2 Persuasion Dynamics Real People People as interacting atoms

3 Voter Model Clifford & Sudbury (1973) Holley & Liggett (1975) 0. Binary voter variable at each site i 1. Pick a random voter 2. Assume state of randomly-selected neighbor individual has no self-confidence & adopts neighbor s state

4 Voter Model Example update: 3/4 Clifford & Sudbury (1973) Holley & Liggett (1975) 0. Binary voter variable at each site i 1. Pick a random voter 2. Assume state of randomly-selected neighbor individual has no self-confidence & adopts neighbor s state

5 Voter Model Example update: 3/4 1/4 0. Binary voter variable at each site i 1. Pick a random voter 2. Assume state of randomly-selected neighbor individual has no self-confidence & adopts neighbor s state Clifford & Sudbury (1973) Holley & Liggett (1975) proportional rule

6 Voter Model Example update: 3/4 1/4 0. Binary voter variable at each site i 1. Pick a random voter 2. Assume state of randomly-selected neighbor individual has no self-confidence & adopts neighbor s state 3. Repeat 1 & 2 until consensus necessarily occurs in a finite system Clifford & Sudbury (1973) Holley & Liggett (1975) proportional rule

7 Voter vs. Ising Models Voter model: proportional rule 3/4 Clifford & Sudbury (1973) Holley & Liggett (1975) consensus inevitable in a finite system 1/4 Kinetic Ising model: majority rule at T=0 Glauber (1963) 1 consensus not inevitable in a finite system

8 Voter Evolution vs. Ising Evolution random initial condition: Voter Dornic et al. (2001) droplet initial condition: random initial condition: Ising droplet initial condition:

9 Lattice Voter Model: 3 Basic Properties

10 1. Final State (Exit) Probability E(ρ 0 ) Evolution of a single active link (homogeneous network): 1/2 1/2

11 1. Final State (Exit) Probability E(ρ 0 ) Evolution of a single active link (homogeneous network): 1/2 1/2

12 1. Final State (Exit) Probability E(ρ 0 ) = ρ 0 Evolution of a single active link (homogeneous network): 1/2 1/2 average magnetization is conserved! ρ 0 1 ρ 0 ρ 0 0 1

13 2. Spatial Dependence of 2-Spin Correlations flip rate: w i = 1 2 h 1 i z X j2hii j i 1-spin correlations: dh i i dt = 2h i w i i = h i i + 1 z X h j i j

14 2. Spatial Dependence of 2-Spin Correlations flip rate: w i = 1 2 h 1 i z X j2hii j i 1-spin correlations: dh i i dt = 2h i w i i = h i i + 1 z 2-spin correlations: X h j i j h i (t)i = I i (t) e t for h i (t=0)i = i,0 dh i dt j i = 2h i j (w i + w j )i = 2h i X j i + 1 2d k2hii h k j i + X k2hji h i k i

15 2. Spatial Dependence of 2-Spin Correlations (infinite system) Equation for 2-spin correlation function: dh i dt j i = 2h i j (w i + w j 2 = r 2 c 2 (r,t) c(r =0,t)=1; c(r > 0,t =0)=0 c(r,t) d>2 c(r,t) d<2 1 (a/r) d 2 r r early time

16 2. Spatial Dependence of 2-Spin Correlations (infinite system) Equation for 2-spin correlation function: dh i dt j i = 2h i j (w i + w j 2 = r 2 c 2 (r,t) c(r =0,t)=1; c(r > 0,t =0)=0 c(r,t) d>2 c(r,t) d<2 1 (a/r) d 2 r intermediate early time r

17 2. Spatial Dependence of 2-Spin Correlations (infinite system) Equation for 2-spin correlation function: dh i dt j i = 2h i j (w i + w j 2 = r 2 c 2 (r,t) c(r =0,t)=1; c(r > 0,t =0)=0 c(r,t) d>2 c(r,t) d<2 1 (a/r) d 2 Dt late intermediate early time r r

18 2. Spatial Dependence of 2-Spin Correlations (infinite system) Equation for 2-spin correlation function: dh i dt j i Asymptotic solution: c(r, t) = 2h i j (w i + w j )i 1 1 ( a r )d 2 1 ( a Dt ) d 2 1 ln r ln a 1 ln Dt ln 2 d 2 d =2 = r 2 c 2 (r,t) c(r =0,t)=1; c(r > 0,t =0)=0 c(r,t) d>2 c(r,t) d<2 1 (a/r) d 2 Dt late intermediate early time r r

19 3. System Size Dependence of Consensus Time Liggett (1985), Krapivsky (1992) Dt c(r, t)r d 1 dr = N dimension consensus time 1 N 2 2 N ln N >2 N

20 Lattice Voter Model: 3 Basic Properties 1. Final State Probability E(ρ 0 ) = ρ 0 Evolution of a single active link: 1/2 1/2 average magnetization conserved 2. Two-Spin Correlations c(r,t) d>2 1 (a/r) d 2 = r 2 c(r,t) c(r =0,t)=1 c(r>0,t=0) = 0 d<2 Dt r late time intermediate early 3. Consensus Time R p Dt c(r,t) dr=n dimension consensus time 1 N² 2 N ln N >2 N

21 Voter Model on Complex Networks magnetization on regular networks 1/2 average magnetization conserved! 1/2 magnetization not conserved on complex networks high degree; few nodes changes rarely Suchecki, Eguiluz & San Miguel (2005) low degree; many nodes changes often flow from high degree to low degree

22 New Conservation Law Sood & SR (2005) low degree changes often high degree changes rarely to compensate the different rates: degree-weighted 1st moment:! P P k kn k k k kn k = P k kn k k µ 1 conserved! µ 1 = av. degree n k = frac. nodes of degree k k = frac. on nodes of degree k

23 Conservation Law for Voter Model Transition probability adjacency matrix x y x y P[η η x ]= y η η x state of system state after flip at x η(x) state at x (0,1) Density change: voter at x degree-weighted moments: A xy Nk x [Φ(x, y) + Φ(y, x)] neighbor at y Φ(x, y) η(x)[1 η(y)] = x y η(x) = [1 2η(x)] P[η η x ] } {{ } ±1 at x ω n = 1 Nµ n x k n x η(x) change in weighted first moment: ω 1 = x,y A xy Nk x k x [η(y) η(x)] =0 conserved!

24 Exit Probability on Complex Graphs E(ω) =ω Extreme case: star graph N nodes: degree 1 1 node: degree N 0 0 = 1 µ 1 k kn k k = Final state: all 1 with prob. 1/2! 0 0 0

25 Voter Model on Complex Networks complete bipartite graph a sites degree b Sucheki, Eguiluz & San Miguel (2005) Sood & SR (2005) Antal, Sood & SR (2005) two-clique graph b sites degree a c N=10000, C links/node t< c = ρ b ρ b c = ρ a ρ a

26 Consensus Time Evolution Equation warmup: complete graph T ( ) av. consensus time starting with density T ( ) = R( )[T ( + d )+dt] +L( )[T ( d )+dt] +[1 R( ) L( )][T ( )+dt]

27 Consensus Time Evolution Equation warmup: complete graph T ( ) av. consensus time starting with density T ( ) = R( )[T ( + d )+dt] +L( )[T ( d )+dt] +[1 R( ) L( )][T ( )+dt] R R( ) prob( ) 0 ρ 1 = (1 )

28 Consensus Time Evolution Equation warmup: complete graph T ( ) av. consensus time starting with density T ( ) = R( )[T ( + d )+dt] +L( )[T ( d )+dt] +[1 R( ) L( )][T ( )+dt] 0 L R 1 R( ) prob( ) L( ) prob( ) = (1 )

29 Consensus Time Evolution Equation warmup: complete graph T ( ) av. consensus time starting with density T ( ) = R( )[T ( + d )+dt] +L( )[T ( d )+dt] +[1 R( ) L( )][T ( )+dt] 0 L 1 R L R 1 R( ) prob( ) L( ) prob( ) = (1 )

30 Consensus Time on Complete Graph T ( ) = R( )[T ( + d )+dt] +L( )[T ( d )+dt] +[1 R( ) L( )][T ( )+dt] continuum limit: T = N (1 ) solution: T ( )= N [ ln + (1 ) ln(1 )]

31 Consensus Time on Heterogeneous Networks T ({ k }) av. consensus time starting with density k on nodes of degree k T ({ k })= + k k + 1 R k ({ k })[T ({ + k })+dt] L k ({ k })[T ({ k })+dt] k R k ({ k }) = prob( k + k ) = 1 N x 1 k x y P (,, ) = n k (1 k) [R k ({ k })+L k ({ k })] [T ({ k })+dt] L k ({ k })=n k k (1 )

32 Consensus Time on Heterogeneous Networks continuum limit: k ( k) T k + + k 2 k 2Nn k 2 T 2 k = 1

33 Voter Model on Complex Networks configuration model k 1 n k k 2.5, µ 1 =8 ρ ρ ρ ω

34 Consensus Time on Heterogeneous Networks continuum limit: k ( k) T k + + k 2 k 2Nn k 2 T 2 k = 1 now use k k and k = k = kn k µ 1 to give 2 T 2 = Nµ2 1/µ 2 (1 ) same as T = N (1 ) with effective size N e = Nµ 2 1/µ 2 µ n = X k k n n k

35 Consensus Time for Complex Networks with n k k ν 8 N > 3 >< N/ ln N =3 T N / N e = N µ2 1 µ 2 N 2( 2)/( 1) 2 < < 3 (ln N) 2 =2 >: O(1) < 2 fast consensus

36 Consensus Time for Complex Networks with n k k ν 8 N > 3 >< N/ ln N =3 T N / N e = N µ2 1 µ 2 N 2( 2)/( 1) 2 < < 3 (ln N) 2 =2 >: O(1) < 2 fast consensus

37 Consensus Time for Complex Networks with n k k ν 8 N > 3 >< N/ ln N =3 T N / N e = N µ2 1 µ 2 N 2( 2)/( 1) 2 < < 3 (ln N) 2 =2 >: O(1) < 2 fast consensus

38 Consensus Time for Complex Networks with n k k ν 8 N > 3 >< N/ ln N =3 T N / N e = N µ2 1 µ 2 N 2( 2)/( 1) 2 < < 3 (ln N) 2 =2 >: O(1) < 2 fast consensus lnvasion process: T N N > 2, N ln N =2, N 2 < 2.

39 Confident Voter Model motivation: Centola (2010) related work: Dall Asta & Castellano (2007) unsure P M P M P M P M confident (a) (b)

40 Confident Voter Model motivation: Centola (2010) related work: Dall Asta & Castellano (2007) unsure P M P M P M P M confident (a) (b)

41 Confident Voter Model motivation: Centola (2010) related work: Dall Asta & Castellano (2007) unsure P M P M P M P M confident (a) (b) extremal

42 Confident Voter Model motivation: Centola (2010) related work: Dall Asta & Castellano (2007) unsure P M P M P M P M confident (a) (b) marginal

43 Simplest case: 2 internal states M 1 P 1 unsure densities P₀, P₁, M₀, M₁, with P₀+P₁+ M₀+ M₁=1 M 0 P 0 confident basic processes: M 1 P 1! P 0 P 1 or M 0 M 1 M 0 P 0! M 0 P 1 or M 1 P 0 P 0 P 1! P 0 P 1 or P 0 P 0 M 0 M 1! M 0 M 1 or M 0 M 0 M 1 P 0! M 1 P 1 or P 0 P 0 M 0 P 1! M 1 P 1 or M 0 M 0 rate equations/mean-field limit: P 0 = M 0 P 0 + M 1 P 1 + P 0 P 1 P 1 = M 0 P 0 M 1 P 1 P 0 P 1 +(M 1 P 0 M 0 P 1 ) similarly for M₀, M₁

44 special soluble case: symmetric limit P 0 + P 1 = M 0 + M 1 = 1 2 P 0 = M 0 P 0 + M 1 P 1 + P 0 P 1 P 1 = M 0 P 0 M 1 P 1 P 0 P 1 +(M 1 P 0 M 0 P 1 ) P 0 = P 1 = P P 0 = (P 0 + )(P 0 ) 1 4 ± = 1 4 ( 1± 5) 0.309, solution: P 0 (t) + P 0 (t) = P 0(0) + P 0 (0) e ( + )t

45 near symmetric limit: P 0 = , M 0 = , P 1 =M 1 =0 densities P₀(t) P 0 M₀(t) P 1 P₁(t) M 1 M₁(t) M O(1) time ln N

46 near symmetric limit: composition tetrahedron P₀ M₀ P₁ M₁

47 Consensus Time in Two Dimensions 10 6 T N N

48 Consensus Time Distribution P(T N ) P(T N ) T T droplets stripes

49 T N Island N Stripe N 3/ N two time scales control approach to consensus see also Spirin, Krapivsky, SR (2001), Chen & SR (2005) Ising model Majority vote model

50 Majority rule Galam (1999), Krapivsky & SR (2003), Slanina & Lavicka (2003), Chen & SR (2005) 1. Pick a random group of G spins (with G odd). 2. All spins in G adopt the majority state. 3. Repeat until consensus necessarily occurs Basic questions: 1. Which final state is reached? 2. What is the time until consensus?

51 Mean-field theory (for G=3) where p n = q n = 3 N 3. N 2 n 2 n 3 N 3. N 1 n 1 n r n =1 p n q n T n mean time to m = 1 starting from n plus spins = p n (T n+1 + δt)+q n (T n 1 + δt)+r n (T n + δt)

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58 The Dynamics of Persuasion Sid Redner, Santa Fe Institute (physics.bu.edu/~redner) CIRM, Luminy France, January 5-9, 2015 T. Antal, E. Ben-Naim, P. Chen, P. L. Krapivsky, M. Mobilia, V. Sood, F. Vazquez, D. Volovik + support from Modeling Consensus: introduction to the voter model voter model on complex networks voting with some confidence majority rule Modeling Discord & Diversity: 3-state voter models strategic voting bounded compromise dynamics of social balance Axelrod model lecture 1 lecture 2 lecture 3

59 Discord & Diversity If people are reasonable, why is consensus hard to reach? Possibilities: insufficient communication appreciable diversity stubbornness. your favorite mechanism Models: Three voting states: 0 +; + noninteracting Vazquez & SR (2004) Strategic voting: ideology vs. strategy Volovik, Mobilia & SR (2009) Bounded confidence: compromise only when close Deffuant, Neau, Amblard & Weisbuch (2000) Hegselmann & Krause (2002) Ben Naim, Krapivsky & SR (2003) Social balance: Axelrod model: dynamics of positive/negative links Antal, Krapivsky & SR (2005, 2006) many features, many traits Axelrod (1997) Castellano, Marsili & Vespignani (2000) Vazquez & SR (2007)

60 Three Voting States: state voter at each site: Pick a random voter 2. Assume state of neighbor if compatible 3. Repeat until either consensus or frozen final state (N 0,N )! (N 0 ±1,N 1) prob. (N 0,N + )! (N 0 ±1,N + 1) prob. N 0 N N 2 = 0 N 0 N + N 2 = 0 + compatible + + incompatible

61 Evolution in Composition Triangle ρ 0 consensus ρ + 1 2p x 2p y ρ_ frozen p y p x p x p y p x = 0 p y = 0 +

62 Evolution in Composition Triangle ρ 0 consensus ρ + 1 2p x 2p y ρ_ frozen p y p x p x p y p x = 0 p y = 0 +

63 The Phase Diagram F (, + ) = prob. to reach frozen state starting from (, + ) recursion: F (, + )=p x [F (, + )+F ( +, + )] = p y [F (, + )+F (, + + )] =[1 2(p x + p y )] F (, + ) continuum F (, F (, =0 F (, 0) = 0 F (0, + )=0 F ( +, 1 + )=1 F (, + )= X n odd 2(2n + 1) n(n + 1) p + ( + + ) n Pn F ( 0 )=1 1 (1 0 ) 2 p 1+(1 0 ) 2 symmetric limit _ +

64 The Phase Diagram F (, + ) = prob. to reach frozen state starting from (, + ) recursion: F (, + )=p x [F (, + )+F ( +, + )] = p y [F (, + )+F (, + + )] =[1 2(p x + p y )] F (, + ) continuum limit: F (, + )= X F ( 0 F (, F (, n odd 2(2n + 1) n(n + 1) =0 p + ( + + ) n Pn F (, 0) = 0 F (0, + )=0 F ( +, 1 + )=1 1 (1 0 ) 2 p _ 1+(1 0 ) 2 _ 0 0 in each region, prob. to reach specified state is > 50% + symmetric limit + _ frozen +

65 Phase Diagram & Final State Probabilities 0 0 in each region, prob. to reach specified state is > 50% F P₀ 0.6 _ _ 0.2 P P+ _ p 0 0 moral: extremism promotes deadlock

66 Strategic Voting vote for first choice? vote against last choice? UK elections Canadian elections Conservative Liberal NDP % vote Liberal Conservative Labor (a) year % of vote (b) year

67 evolution of the densities a, b, c: Strategic Voter Model ȧ = T (b + c ḃ = T (c + a ċ = T (a + b 2a) + r AC ac + r AB ab 2b) + r BA ba + r BC bc 2c) + r CA ca + r CB cb

68 evolution of the densities a, b, c: ȧ = T (b + c Strategic Voter Model temperature 2a) + r AC ac + r AB ab ḃ = T (c + a 2b) + r BA ba + r BC bc ċ = T (a + b 2c) + r CA ca + r CB cb

69 evolution of the densities a, b, c: ȧ = T (b + c Strategic Voter Model temperature strategic voting 2a) + r AC ac + r AB ab ḃ = T (c + a 2b) + r BA ba + r BC bc ċ = T (a + b 2c) + r CA ca + r CB cb

70 evolution of the densities a, b, c: strategic voting rates: two natural choices Strategic Voter Model temperature ȧ = T (b + c ḃ = T (c + a ċ = T (a + b ȧ = T (1 ḃ = T (1 r AB = r BA = r = const. strategic voting 2a) + r AC ac + r AB ab 2b) + r BA ba + r BC bc 2c) + r CA ca + r CB cb r = r 0 [(a + b)/2 c] 3a)+rac 3b)+rbc ċ = T (1 3c) rc(1 c) 8 >< +r B minority 0 C minority >: r A minority too drastic better in c < sector b a c < b < a < c

71 Strategic Voter Model ċ =(1 3c)T r 0 2 c(1 c)(1 3c) 3r 0 2 (c c )(c c +)(c c 3 ) c 3 = 1 3 c ± = ± p 1 8x 0 x 0 T r 0 apple c(t) c3 c(0) c 3 3 apple c(t) c+ c(0) c + + apple c(t) c(0) c c = e 3(c + c )r 0 t/2 ± = 1 c 3 c ± 3 = + = c + c (c 3 c + )(c 3 c )

72 Strategic Voter Model mean-field phase diagram a location of fixed point 0.35 c < b < c* b a < c T/r 0

73 Simulations of Strategic Voter Model 40 A B C 10 3 time in minority % vote time time (c) 10 0 (b) N 60 % vote Liberal Conservative Labor (a) year

74 Bounded Compromise Model 1 1 x 1 x 2 x 1 x 2 x 1 + x 2 2 If x 2 x 1 < 1 compromise x 1 x 2 If x 2 x 1 > 1 no interaction

75 The Opinion Distribution P(x,t) = probability that agent has opinion x at time t Fundamental parameter: Δ<1: consensus Δ>1: fragmentation Δ the diversity (initial opinion range) w e t/2 P (x, t) t = ZZ x 1 x 2 <1 dx 1 dx 2 P (x 1,t)P (x 2,t) x 1 2 (x 1+x 2 ) (x x 1 ) gain by averaging opinions loss by interaction same as Maxwell model for inelastic collisions & inelastic collapse phenomena Ben-Naim and Krapivsky (2000) Baldassarri, Marconi, Puglisi (2001) Ben-Naim, Krapvisky, SR (2003)

76 Early time evolution (for Δ=4.2) integrate master equation rather than simulate! t=9 t=6 t=3 t= x (opinion)

77 Early time evolution (for Δ=4.2) integrate master equation rather than simulate! t=9 t=6 t=3 t= x (opinion)

78 Early time evolution (for Δ=4.2) integrate master equation rather than simulate! t=9 t=6 t=3 t= x (opinion)

79 Early time evolution (for Δ=4.2) integrate master equation rather than simulate! t=9 t=6 t=3 t= x (opinion)

80 Fragmentation Sequence final opinion 10 5 minor x 0 5 major (diversity)

81 Fragmentation Sequence final opinion 10 5 minor x 0 5 major (diversity)

82 Fragmentation Sequence final opinion 10 5 minor x 0 5 major 10 minor (diversity)

83 Birth of Extremists t =0-1 (1 + ϵ) (1 + ϵ) 1 separation: w = ϵ = e t sep/2 w e t/2 m ϵe t 1 m(t sep ) ϵ

84 Fragmentation Sequence final opinion 10 5 minor x 0 5 major 10 minor major (diversity)

85 Fragmentation Sequence final opinion 10 5 x central minor major (diversity)

86 A Possible Realization 1993 Canadian Federal Election year PQ NDP L PC SC R/CA

87 The Dynamics of Persuasion Sid Redner, Santa Fe Institute (physics.bu.edu/~redner) CIRM, Luminy France, January 5-9, 2015 T. Antal, E. Ben-Naim, P. Chen, P. L. Krapivsky, M. Mobilia, V. Sood, F. Vazquez, D. Volovik + support from Modeling Consensus: introduction to the voter model voter model on complex networks voting with some confidence majority rule Modeling Discord & Diversity: 3-state voter models strategic voting bounded compromise dynamics of social balance Axelrod model lecture 1 lecture 2 lecture 3

88 Dynamics of Social Balance friend enemy Investigate dynamical rules that promote evolution to a balanced state

89 Socially Balanced States you you you you friendly link Hollywood divorce Brad Jen Clint Dina Clint Dina Clint Dina unfrustrated/balanced frustrated/imbalanced Social Balance a friend of my friend an enemy of my enemy a friend of my enemy an enemy of my friend } } is my friend; is my enemy.

90 Static properties of signed graphs: Balanced states on complete graph must either be utopia: only friendly links bipolar: two mutually antagonistic cliques Cartwright & Harary (1956)

91 Two Natural Evolution Rules Local Triad Dynamics: reduce imbalance in one triad by single update i p p: amity parameter 1 j k 1 p (a) Balance transition as a function of p (b) Global Triad Dynamics: reduce global imbalance by single update Outcome unknown Tantalizing connections to spin glasses & jamming phenomena

92 Local Triad Dynamics on Arbitrary Networks (social graces of the clueless) 1. Pick a random imbalanced (frustrated) triad 2. Reverse a single link so that the triad becomes balanced probability p: unfriendly friendly; probability1-p: friendly unfriendly i p 1 j k 1 p Fundamental parameter p: p=1/3: flip a random link in the triad equiprobably p>1/3: predisposition toward tranquility p<1/3: predisposition toward hostility

93 The Evolving State rate equation for the density of friendly links ρ: + in 1 + in 1 + in 3 dρ dt = 3ρ 2 (1 ρ)[p (1 p)] + (1 ρ) 3 = 3(2p 1)ρ 2 (1 ρ)+(1 ρ) 3 ρ(t) ρ + Ae Ct 1 1 ρ 0 1+2(1 ρ0 ) 2 t p<1/2; p =1/2; rapid approach to frustrated steady state slow relaxation to utopia 1 e 3(2p 1)t p>1/2. rapid attainment of utopia

94 Simulations for a Finite Society T N (a) p< 1 2, T N e N N T N T N (b) (c) 10 N p = 1 2, T N N 4/3 p> 1 2, T N ln N 2p N

95 Fate of a Finite Society p<1/2: effective random walk picture balance (N 3 /6 balanced triads) ±N N D N 2 v N # balanced triads balance T N e vl N /D e N 2 p>1/2: inversion of the rate equation u e 3(2p 1)t N 2 T N u=1-ρ, the unfriendly link density ln N 2p 1

96 p=1/2 naive rate equation estimate: u 1 ρ t 1/2 N 2 T N N 4 incorporating fluctuations as balance is approached: ln u 1/2 U = Lu + Lη L t + L t 1/4 N 2 equating the 2 terms in U: T N L 2/3 N 4/3 ln t N 4/3 N 4

97 Possible Application I: Long Beach Street Gangs Figure 2. Gang Network Nakamura, Tita, & Krackhardt (2007) gang relations cool with hate _: Gang _: Ally Relation _: Enemy Relation Figure 3. Gang Network with Violent Incidents violence frequency low incidence high incidence _: Gang _: Number of Violent Attacks (Thickness is proportional to the number)

98 Possible Application I: Long Beach Street Gangs Figure 2. Gang Network Nakamura, Tita, & Krackhardt (2007) gang relations 1,3 cool with hate 3,1 _: Gang _: Ally Relation _: Enemy Relation Figure 3. Gang Network with Violent Incidents violence frequency low incidence high incidence _: Gang _: Number of Violent Attacks (Thickness is proportional to the number)

99 Possible Application II: A Historical Lesson GB AH GB AH GB AH F G F G F G R 3 Emperor s League I R Triple Alliance 1882 I R German-Russian Lapse 1890 I GB AH GB AH GB AH F G F G F G R I R I R I French-Russian Alliance Entente Cordiale 1904 British-Russian Alliance 1907

100 Axelrod Model Axelrod (1997) You: car cuisine recreation politics abode BMW SUV Ford Trabant bicycle meat vegetarian vegan fast weekly dieting skiing hiking fishing hunting running F features leftist rightist anarchist apathetic fascist suburb house city apartment homeless pied-à-terre city house q traits Me: car cuisine recreation politics abode BMW SUV Ford Trabant bicycle meat vegetarian vegan fast weekly dieting skiing hiking fishing hunting running leftist rightist anarchist apathetic fascist suburb house city apartment homeless pied-à-terre city house

101 Axelrod Model Axelrod (1997) You: car cuisine recreation politics abode BMW SUV Ford Trabant bicycle meat vegetarian vegan fast weekly dieting skiing hiking fishing hunting running F features leftist rightist anarchist apathetic fascist suburb house city apartment homeless pied-à-terre city house q traits Me: car cuisine recreation politics abode BMW SUV Ford Trabant bicycle meat vegetarian vegan fast weekly dieting skiing hiking fishing hunting running leftist rightist anarchist apathetic fascist suburb house city apartment homeless pied-à-terre city house

102 Axelrod Model Axelrod (1997) You: car cuisine recreation politics abode BMW SUV Ford Trabant bicycle meat vegetarian vegan fast weekly dieting skiing hiking fishing hunting running F features leftist rightist anarchist apathetic fascist suburb house city apartment homeless pied-à-terre city house q traits Me: car cuisine recreation politics abode BMW SUV Ford Trabant bicycle meat vegetarian vegan fast weekly dieting skiing hiking fishing hunting running leftist rightist anarchist apathetic fascist suburb house city apartment homeless pied-à-terre city house

103 Axelrod s Simulation F=5, q=10, 10x10 lattice t=0 t=200 t=400 t=800

104 Axelrod Model Castellano, Marsili & Vespignani (2000) simulations on 150 x 150 square lattice 1.0 small (F,q): consensus large (F,q): fragmentation fraction in largest cluster 0.8 F=10 1st-order transition L = 50 L = 100 L = 150 <s max >/L <s max >/L L = 50 L = 100 L = q q

105 Axelrod Model Castellano, Marsili & Vespignani (2000) simulations on 150 x 150 square lattice 1.0 small (F,q): consensus large (F,q): fragmentation fraction in largest cluster <s max >/L F=10 1st-order transition <s max >/L F=2 2nd-order L = 50 L = 100 L = q L = 50 L = 100 L = q

106 Axelrod Model Castellano, Marsili & Vespignani (2000) simulations on 150 x 150 square lattice fraction of active links F=10 q= t co 10 4 q=100! n a (t) L t

107 Axelrod Model with F=2 on 4-regular graph P m fraction of links with m common features m =0, 2 inactive; m = 1 active q = q c 1 4 q = q c 1 4 k bond densities P 0 P 2 P 1 time long time scale P 1 (t) k = T 200 k = time k =3 k =5 k =7 q = q c

108 Master Equations for Bond Densities P 0 = z 1 h z P 1 P P 1 i direct processes P 1 = P 1 z + z 1 h z P 1 P 2 = P 1 z + z 1 z P 1 P (1 + )P 1 + P 2 i 1 2 P 1 P 2

109 direct process for P : choose random link 1 N 1 = 1 2 P 1! P 1 1 = 2 P 1/L t 1/N = P 1 z

110 Master Equations for Bond Densities P 0 = z 1 h z P 1 P P 1 i indirect processes P 1 = P 1 z + z 1 h z P 1 P (1 + )P 1 + P 2 i P 2 = P 1 z + z 1 z P P 1 P 2

111 direct process for P : choose random link 1 N 1 = 1 2 P 1! P 1 1 = 2 P 1/L t 1/N = P 1 z indirect processes for P : a 1 b 1 a 1 b 1 a1 a 1 b 1 b

112 direct process for P : choose random link 1 N 1 = 1 2 P 1! P 1 1 = 2 P 1/L t 1/N = P 1 z indirect processes for P : a 1 b 1 a 1 b 1 a1 a 1 b 1 b

113 direct process for P : choose random link 1 N 1 = 1 2 P 1! P 1 1 = 2 P 1/L t 1/N = P 1 z indirect processes for P : a 1 b 1 a 1 b 1 a1 a 1 b 1 b

114 direct process for P : choose random link 1 N 1 = 1 2 P 1! P 1 1 = 2 P 1/L t 1/N = P 1 z indirect processes for P : a 1 b 1 a 1 b 1 a1 a 1 b 1 b rate 1 2 P 1! 1 2 P 1(q 1) 1 mean field

115 Master Equations for Bond Densities P 0 = z 1 h z P 1 P P 1 i d z 1 z P 1 dt x P 0 y P 1 P 2 =1 P 0 P 1 P 1 = P 1 z + z 1 h z P 1 P (1 + )P 1 + P 2 i P 2 = P 1 z + z 1 z P P 1 P 2

116 Master Equations for Bond Densities x 0 = x y y 0 = 1 1 +( 1)x 3+ 2 y d z 1 z P 1 dt x P 0 y P 1 P 2 =1 P 0 P 1

117 Master Equations for Bond Densities x 0 = x y y 0 = 1 1 +( 1)x 3+ 2 y y d z 1 z P 1 dt x P 0 y P 1 P 2 =1 P 0 P 1 x+y=1 y =0 x =0 x

118 x =0 q c q

119 q = q c 1 4 k 10 1 k = 1 k = k =3 P 1 (t) k = T time k =7 t 1 (q c q) 1/2

120 q = q c 1 4 k 10 1 k = 1 k = k =3 P 1 (t) k = T 200 t (q c q) 1/ time k =7 t 1 (q c q) 1/2

121 Axelrod Model with F=2 transition between steady state (q<q ) & fragmented static state (q>q ) c c q<q : very slow approach to steady state with c time scale ' (q c q) 1/2 long transient in which P 1 ' (q c q) before steady state is reached

122 Some Closing Thoughts Voter Model well characterized, but: consensus route incompletely understood on complex graphs generalizations, role of heterogeneity, role of internal beliefs, data-driven models Models of Diversity & Discord bifurcation sequence in bounded compromise role of competing social interactions mostly unknown mathematical understanding of Axelrod model lacking data-driven models Notes: physics.bu.edu/~redner: click the slides from selected talks link