Strategic voting (>2 states) long time-scale switching. Partisan voting selfishness vs. collectiveness ultraslow evolution. T. Antal, V.

Transcription

1 Dynamics of Heterogeneous Voter Models Sid Redner (physics.bu.edu/~redner) Nonlinear Dynamics of Networks UMD April 5-9, 21 T. Antal (BU Harvard), N. Gibert (ENSTA), N. Masuda (Tokyo), M. Mobilia (BU Leeds), V. Sood (BU NBI), D. Volovik (BU) NSF DMR53553 & DMR9654 The classic voter model 3 basic results Voting on complex networks new conservation law two time-scale route to consensus short consensus time Strategic voting (>2 states) long time-scale switching Partisan voting selfishness vs. collectiveness ultraslow evolution T. Antal, V. Sood M. Mobilia, D. Volovik N. Masuda, N. Gibert

2 Clifford & Sudbury (1973) Classic Voter Model Holley & Liggett (1975) Example update: 3/4. 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 Clifford & Sudbury (1973) Classic Voter Model Holley & Liggett (1975) Example update: 3/4 1/4. 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 proportional rule

4 Voter Model Evolution random initial condition: Dornic et al. (21) t=4 t=16 t=64 t=256 droplet initial condition: t=4 t=16 t=64 t=256

5 Lattice Voter Model: 3 Basic Properties 1. Final State (Exit) Probability E(ρ ) = ρ 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 c(r,t) c 2 (r,t) t = 2 c 2 (r,t) d<2 Dt r c 2 (r =,t)=1 c 2 (r>,t=) = late time intermediate early 3. Consensus Time dimension consensus time 1 N 2 2 N ln N >2 N

6 Voter Model on Complex Networks C. Castellano, D. Vilon, A. Vespignani, EPL 63, 153 (23) K. Suchecki, V. M. Eguiluz, M. San Miguel, EPL 69, 228 (25) V. Sood, SR, PRL 94, (25); T. Antal, V. Sood, SR, PRE 77, (28) illustrative example: complete bipartite graph a sites degree b b sites degree a dn a = dn b = pick site on a sublattice pick on a pick on b sublattice a [ a Na N b a + b a b N a b N ] b a b b [ b Nb N a a + b b a N b a N ] a b 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() ρ b,a ()] e 2t [ρ a() + ρ b ()] 1 2 [ρ a() + ρ b ()] magnetization not conserved

7 Voter Model on Complex Networks high degree; few nodes changes rarely low degree; many nodes changes often flow from high degree to low degree

8 New Conservation Law low degree changes often high degree changes rarely to compensate different rates, consider: degree-weighted ω = 1 kn k ρ k conserved! 1st moment: µ 1 k µ 1 = av. degree n k = frac. nodes of degree k ρ k = frac. on nodes of degree k

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

10 Route to Consensus on Complex Graphs 1.8 t< c =1.6.6 ρ b ρ b c = ρ a ρ a complete bipartite graph two-clique graph a sites degree b b sites degree a c N=1, C links/node

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

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

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

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

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

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

17 Consensus Time on Heterogeneous Networks continuum limit: k (ω ρ k ) T + ω + ρ k 2ωρ k 2 T ρ k 2Nn k ρ 2 k = 1

18 Molloy-Reed Scale-Free Network 1 n k k 2.5, µ 1 =8 ρ ρ.4.2 ρ ω

19 Consensus Time on Heterogeneous Networks continuum limit: k (ω ρ k ) T + ω + ρ k 2ωρ k 2 T ρ k 2Nn k ρ 2 k = 1 now use ρ k ω k and to give ρ k = ω ρ k ω = kn k µ 1 ω 2 T ω 2 = Nµ2 1/µ 2 ω(1 ω) same as T = N ρ(1 ρ) with effective size N eff = Nµ 2 1/µ 2

20 Consensus Time for Power-Law Degree Distribution n k k ν T N N eff = N µ2 1 µ 2 N ν > 3 N/ ln N ν =3 N 2(ν 2)/(ν 1) (ln N) 2 ν =2 O(1) ν<2 2 <ν<3 ] fast (<N) consensus

21 Strategic Voting 6 British election results since 183 Canadian election results since Conservative Liberal NDP/CC % vote 4 2 % of vote 4 2 Liberal Conservative Labor (a) year (b) year

22 D. Volovik, M. Mobilia, SR Strategic Voter Model EPL 85, 481 (29) randomly-selected voter changes to any other state equiprobably (rate T) majority-minority interaction: minority preferentially changes to majority (rate r) rate equations (A, B majority; c minority): A = T (B + c 2A) + r Ac Ḃ = T (c + A 2B) + rbc ċ = T (A + B 2c) r (A + B)c

23 Phase Portrait a c a regular 3-state voter model b b c

24 Phase Portrait a c a strategic 3-state voter model b c < b < b a < c

25 Slow Switching 4 A B C 35 % vote 3 (c) time

26 6 British election results since 183 Canadian election results since Conservative Liberal NDP/CC % vote 4 2 % of vote 4 2 Liberal Conservative Labor (a) year (b) year

27 happy democrat density D h Partisan Voter Model sad democrat density D s partisan voting update: N. Masuda, N. Gibert, SR arxiv: sad republican density R s 1. Pick voter, pick neighbor (as in usual voter model); 2a. If initial voter becomes happy by adopting neighboring state, change occurs with rate 1+ε; 1+ε 2b. If initial voter becomes unhappy by adopting neighboring state, change occurs with rate 1-ε. 1-ε happy republican density R h

28 Partisan Voter Model: Mean-Field Limit rate equations: Ḋ h = 2D h D s + (1 + ) D s R s (1 ) D h R h Ḋ s = 2D h D s + (1 ) D h R h (1 + ) D s R s and R D

29 Symmetric Case: D=R=½ H D h + R h = density of happy voters D h R h = D h ( 1 2 R s)=ρ 1 2 = density democratic voters 1 2 S H 1 C C Δ= 1/2 1/2 H= Δ

30 Consensus Time on Finite Graphs 1 6 ε=.15 ε= d T N 1 4 ε=.15 ε= complete graph N

31 Summary & Outlook Voter model: paradigmatic, soluble, (but hopelessly naive) Voter model on complex networks: new conservation law meandering route to consensus fast consensus for broad degree distributions Extensions: strategic voting minority suppressed partisan voting selfishness forestalls consensus Future: churn rather than consensus heterogeneity of real people positive and negative social interactions social balance

32 Crass Commercialism phenomena in non-equilibrium statistical physics. It focuses on the development and application of theoretical methods to help students develop their problem-solving skills. The book begins with microscopic transport processes: diffusion, collision-driven phenomena, and exclusion. It then presents the basic phenomenology and solution techniques are emphasized. The following chapters cover kinetic spin systems, both from a discrete and a continuum perspective; the role of disorder in nonequilibrium processes; hysteresis from the non-equilibrium perspective; the kinetics of chemical reactions; and the properties of complex networks. The book contains 2 exercises to test students' understanding of the subject. A link to a website hosted by the authors, containing supplementary material including solutions to some of the exercises, can be found at Pavel L. Krapivsky is Research Associate Professor of Physics at Boston University. His current research interests are in strongly interacting many-particle systems and their applications to kinetic spin systems, networks, and biological phenomena. Sidney Redner is a Professor of Physics at Boston University. His current research interests are in non-equilibrium statistical physics, and its applications to reactions, networks, social systems, biological phenomena, and first-passage processes. Eli Ben-Naim is a member of the Theoretical Division and an affiliate of the Center for Nonlinear Studies at Los Alamos National Laboratory. He conducts research in statistical, nonlinear, and soft condensed-matter physics, including the collective dynamics of interacting particle and granular systems. A Kinetic View of S tat i s t i c a l P h y s i c s kinetics of aggregation, fragmentation and adsorption, where the Krapivsky, Redner and Ben-Naim Aimed at graduate students, this book explores some of the core A Kinetic View of Statistical Physics Cover illustration: Snapshot of a collision cascade in a perfectly elastic hardsphere gas in two dimensions due to a single incident particle. Shown are the cloud of moving particles (red) and the stationary particles (blue) that have not yet experienced any collisions. Figure courtesy of Tibor Antal. to appear this October Paul L. Krapivsky, Sid Redner and Eli Ben-Naim TABLE OF CONTENTS 1. Aperitifs 2. Diffusion 3. Collisions 4. Exclusion 5. Aggregation 6. Fragmentation 7. Adsorption 8. Spin Dynamics 9. Coarsening 1. Disorder 11. Hysteresis 12. Population Dynamics 13. Diffusion Reactions 14. Complex Networks