The Dynamics of Consensus and Clash Annual CNLS Conference 2006 Question: How do generic opinion dynamics models with (quasi)-ferromagnetic interactions evolve? Models: Voter model on heterogeneous graphs Bounded compromise Axelrod model of cultural diversity V. Sood E. Ben-Naim & P. Krapivsky F. Vazquez Results: Heterogeneous voter model: fast consensus Bounded compromise: self-similar fragmentation Axelrod model: non-monotonicity and long time scale
Voter Model Liggett (1985) 0. Binary spin variable at each site i,! =±1 1. Pick a random spin 2. Assume state of randomly-selected neighbor each individual has zero self-confidence and adopts state of randomly-chosen neighbor 3. Repeat 1 & 2 until consensus necessarily occurs i Example update step: 3/4 1/4 encoded by flip rate w ( {σ} i {σ} i ) = 1 2 ( 1 σ i z ) σ k k nn i
1. Final State (Exit) Probabilities E ± (ρ 0 ) = ρ 0 Equation of Motion for single spin: mean spin s i = σ i : ds i dt = 2 σ iw i = s i + 1 z k nn i s k ṁ i ṡ i = 0 ρ 0 1 ρ 0 ρ 0 0 1
2. Spatial Dependence of 2-Spin Correlations Equation for 2-spin correlation function: d(s i s j ) dt Asymptotic solution: c(r,t) c(r, t) = 2 σ i σ j (w i + w j ) c 2(r, t) t 1 1 ( a r )d 2 1 ( a Dt ) d 2 d 2 1 ln r ln a 1 ln Dt ln a d = 2 d>2 d<2 c(r,t) (infinite system) = 2 c 2 (r, t) c(r =0, t) = 0; c(r, t=0) = δ(r) 1 (a/r) d 2 Dt later time r r early time
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
Voter Model on Heterogeneous Graphs illustrative example: complete bipartite graph pick site on a sublattice pick! on a Castellano et al (2003) Suchecki et al (2004) Sood & SR (2005) pick "on b sublattice a sites degree b b sites degree a dn a = dn b = a [ a Na a + b a b [ b Nb a + b b N b b N a a N a a N b b b N b b a N a a ] ] Subgraph densities: ρ a = N a /a, ρ b = N b /b dt = 1/(a + b) ρ a,b (t) = 1 2 [ρ a,b(0) ρ b,a (0)] e 2t + 1 2 [ρ a(0) + ρ b (0)] Exit probabilities: 1 2 [ρ a(0) + ρ b (0)] N.B.: magnetization is not conserved E + = 1 E = 1 2 [ρ a(0) + ρ b (0)].
Exit probabilities E + = 1 E = 1 2 [ρ a(0) + ρ b (0)] Extreme case: star graph Initial state: 1 plus, N minus + Final state: all + probability 1/2!
Route to Consensus trajectory of a single realization 1 0.8 t < 1 1 0.8 c = 100 0.6 0.6! b! b 0.4 0.4 0.2 0.2 c = 1 0 0 0.2 0.4 0.6 0.8 1! a 0 0 0.2 0.4 0.6 0.8 1! a complete bipartite graph two-clique graph a sites degree b b sites degree a c K 10000 K 10000
Mean Consensus Time pick site on the a sublattice pick! on a pick "on b sublattice consensus time from new state continuum limit: T (ρ a, ρ b ) = a a + b (1 ρ a)ρ b [T (ρ a + 1 a, ρ b) + δt] + a a + b ρ a(1 ρ b )[T (ρ a 1 a, ρ b) + δt] + b a + b (1 ρ b)ρ a [T (ρ a, ρ b + 1 b ) + δt] + b a + b ρ b(1 ρ a )[T (ρ a, ρ b 1 b ) + δt] + (1 ρ a ρ b +2ρ a ρ b )[T (ρ a, ρ b ) + δt], Nδt = (ρ a ρ b )( a b )T N (ρ a, ρ b ) 1 ( 1 2 (ρ a + ρ b 2ρ a ρ b ) a 2 a + 1 ) b 2 b T N (ρ a, ρ b )
assuming ρ a = ρ b and ρ = (ρ a + ρ b )/2: equation of motion for T becomes: ( 1 1 ρ(1 ρ) 4 a + 1 ) 2 T = 1 b with solution: T ab (ρ) = 4ab a + b [(1 ρ) ln(1 ρ) + ρ ln ρ)] implications: a = O(1), b = O(N) (star graph), T = O(1) a = O(N), b = O(N) (symmetric graph), T = O(N)
Power-Law Degree Distribution Network n j = fraction of nodes with degree j µ m = j jm n j = m th moment of degree distribution ω = 1 µ 1 j jn jρ j = degree-weighted up spin density Basic result: T N (ω) = N µ2 1 µ 2 [(1 ω) ln(1 ω) + ω ln ω] For power-law network: (n j j ν ) N ν > 3, N/ ln N ν = 3, T N N (2ν 4)/(ν 1) 2 < ν < 3, quick (ln N) 2 ν = 2, O(1) ν < 2. ]consensus!
Bounded Compromise Model Deffuant et al (2000) 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
The Opinion Distribution Basic observable: P(x,t) = probability that agent has opinion x Fundamental parameter: ", the initial opinion range Master equation: P (x, t) t = x 1 x 2 <1 [ δ ( dx 1 dx 2 P (x 1, t)p (x 2, t) x x 1 + x 2 2 ) δ(x x 1 ) ] "<1: eventual consensus ">1: disjoint parties
Early time evolution (for "=4) t=9 t=6 t=3 t=0.5 4 2 0 2 4 x
Time Evolution (for "=10) P(x,t) 2.0 1.5 1.0 0.5 10 2 10 1 10 0 10 1 10 2 0.0 0 2 4 6 8 10! early time 10 3 0 2 4 6 8 10! late time
Bifurcation Sequence 10 5 minor x 0 central 5 major 10 0 2 4 6 8 10!
Axelrod Model culture = (car, food, politics, job,...) [ q states [ SUV economy sports luxury F features Axelrod (1997) Castellano et al (2000) Klemm et al (2003) Vazquez & SR (2006) Typical interaction: (SUV, steak, GOP, cop) (SUV, vegan, dem, lawyer) (SUV, steak, GOP, cop) (SUV, steak, dem, lawyer) Basic question: consensus or cultural fragmentation?
A Minimalist (ersatz Mean-Field) Description P m (t) fraction of links with m shared features Master equation: dp m (t) dt = 2 η+1 total activity of indirect bonds + 2η η+1 gain [ m 1 F F 1 k=1 indirect interaction direct interaction loss P m 1 m F P m [ k F P k m+1 F ] gain -- gain + P m+1 + λ(1 m 1 F [ loss λ(1 m F ) + m F m m+1 i ] j )P m 1 P m ] k η + 1 = coordination number λ = prob. that i & k share 1 feature not shared by j = (q 1) 1
Special Case: F=2, varying q dp 0 dt dp 1 dt dp 2 dt = Formal Solution: τ = η η+1 P 1 = P 1 η+1 + η [ λp 0 + 1 ] 2 P 1 η+1 P 1 = P 1 η+1 + η η+1 P 1 1 4λ(η 1) [ ln + 1 λ S ln [ λp 0 1+λ [ λ 2 P 1 P 2 qualitatively similar to general (F,q) 2 P 1 + P 2 ] ( ) ( ) S + ηλ(1 λ) 2 2 ln 1 ± 1 λ ( ( S 1 λ)( S ± )] ) ( S + 1 + λ)( S ) S = 4ηλ (1 + λ) 2 = 2η(1 + λ) 2 P 1 S τ = t/[2(1 + η)(1 + λ)] ]
Dynamical Analysis dp 0 dx dp 1 dx = λp 0 + 1 2 P 1 ( = 1 1 ) + (λ 1)P 0 η ( ) 3 + λ 2 dx = η η+1 P 1dt P 1 P 2 = 1 P 0 P 1 P 1 uncorrelated initial condition P 0 =0 q > q c q < q c P 1 =0 P 0
Simulation Results q = q c 1 4 q c = 10.898... 1 bond densities 0.8 0.6 0.4 0.2 P 0 P 2 0 10 0 10 1 10 2 10 3 P 1 time
Outlook & Open Questions 1. Heterogeneous voter model: fast consensus What is the route to consensus? Role of fluctuations? Role of the correlations? Application to real voting?
History of US Presidential Elections 0.8 1st/(1st + 2nd) 0.7 Adams/Jackson/ Clay/Crawford 1824 0.6 0.5 Hayes/Tilden 1872 Roosevelt/ Landon 1936 Bush/Gore 2000 0.4 0.3 1820 1860 1900 1940 1980
Outlook & Open Questions 1. Heterogeneous voter model: fast consensus What is the route to consensus? Role of fluctuations? Role of the correlations? Application to real voting? permanence/impermanence 2. Bounded Compromise: fragmentation a natural outcome Is threshold an appropriate mechanism for fragmentation?
A Possible Realization 1993 Canadian Federal Election year BQ NDP L PC SC R/CA 1979 26 114 136 6 1980 32 147 103 1984 30 40 211 1988 43 83 169 1993 54 9 177 2 52 1997 44 21 155 20 60
Outlook & Open Questions 1. Heterogeneous voter model: fast consensus What is the route to consensus? Role of fluctuations? Role of the correlations? Application to real voting? permanence/impermanence 2. Bounded Compromise: fragmentation a natural outcome Is threshold the right mechanism for lack of consensus and fragmentation? 3. Axelrod model: slow non-monotonic dynamics Spatially local interactions? more complex than coarsening Why is the dynamics so slow? Why is there non-monotonicity?