Dynamics of Voter Models on Heterogeneous Networks

Dynamics of Voter Models on Heterogeneous Networks

Sid Redner (physics.bu.edu/~redner) Complex Network Program SAMSI Aug. 29-Sept. 1, 2010

T. Antal (Edinburgh), V. Sood (NBI)NSF DMR0535503 & DMR0906504

Dynamics of Voter Models on Heterogeneous Networks

Sid Redner (physics.bu.edu/~redner) Complex Network Program SAMSI Aug. 29-Sept. 1, 2010

T. Antal, V. Sood

new conservation law & fixation probabilitiestwo time-scale route to consensus short consensus time

Voting on complex networks

The classic voter model3 basic results

Partisan voting can truth be reached?

N. Gibert (Paris)N. Masuda (Tokyo)

T. Antal (Edinburgh), V. Sood (NBI)NSF DMR0535503 & DMR0906504

Classic Voter Model Clifford & Sudbury (1973) Holley & Liggett (1975)


!

" ! !

!

0. Binary voter variable at each site i


!

" ! !

!

0. Binary voter variable at each site i1. Pick a random voter


!

" ! !

!

0. Binary voter variable at each site i1. Pick a random voter2. Assume state of randomly-selected neighbor

individual has no self-confidence & adopts neighbor’s state


!

" ! !

!


!

" ! !

!

3/4

Example update:

2. Assume state of randomly-selected neighborindividual has no self-confidence & adopts neighbor’s state


!

" ! !

!


!

" ! !

!

3/4

Example update:

!

" " !

!

1/4



!

" ! !

!


!

" ! !

!

3/4

Example update:

!

" " !

!

1/4

proportional rule



!

" ! !

!

0. Binary voter variable at each site i

3. Repeat 1 & 2 until consensus necessarily occurs in a finite system

1. Pick a random voter

!

" ! !

!

3/4

Example update:

!

" " !

!

1/4

proportional rule


random initial condition, 256 x 256 square: Voter Model Evolution Dornic et al. (2001)

t=4 t=16 t=64 t=256

random initial condition, 256 x 256 square: Voter Model Evolution Dornic et al. (2001)

t=4 t=16 t=64 t=256

droplet initial condition:t=4 t=16 t=64 t=256

no surface tension

Voter Model & Cousins

Voter Model: Tell me how to vote

lemming



lemming

Invasion Process: I tell you how to vote

persuasive “friend”



lemming

Link Dynamics: Pick two disagreeing agents and change one at random




identical on regular lattices, distinct on random graphsSuchecki, Eguiluz & San Miguel (2005), Castellano (2005), Sood & SR (2005)


lemming

Link Dynamics: Pick two disagreeing agents and change one at random



Lattice Voter Model: 3 Basic Properties


Evolution of a single active link:

1/2

1/2

average magnetization conserved

1. Final State (Exit) Probability E(!0)



1/2

1/2


1. Final State (Exit) Probability E(!0) = !0



1/2

1/2



2. Two-Spin Correlations !c2(r, t)

!t= !

2c2(r, t)c2(r=0, t) = 1c2(r>0, t=0) = 0



1/2

1/2



late time

early

d>2c(r,t)

r

1!(a/r)d!2

c(r,t) d<2

Dt

rintermediate


!t= !

2c2(r, t)c2(r=0, t) = 1c2(r>0, t=0) = 0



1/2

1/2



late time

early

d>2c(r,t)

r

1!(a/r)d!2

c(r,t) d<2

Dt

rintermediate


!t= !

2c2(r, t)c2(r=0, t) = 1c2(r>0, t=0) = 0

dimension consensus time

1 N2

2 N ln N

>2 N

3. Consensus Time! !

Dt

c(r, t)rd"1 dr = N

Voter Model on Complex Networks

V. Sood & SR, PRL 94, 178701 (2005);

C. Castellano, D. Vilon, A. Vespignani, EPL 63, 153 (2003)

T. Antal, V. Sood, SR, PRE 77, 041121 (2008)K. Suchecki, V. M. Eguiluz, M. San Miguel, EPL 69, 228 (2005)


illustrative example: complete bipartite graph

degree a

a sitesdegree b

b sites

V. Sood & SR, PRL 94, 178701 (2005);





degree a

a sitesdegree b

b sites

V. Sood & SR, PRL 94, 178701 (2005);



dN↑,a =N↓,aN↑,b −N↑,aN↓,b

(a + b)b

dN↑,b =N↓,bN↑,a −N↑,bN↓,a

(a + b)a

rate equation


magnetization not conserved


degree a

a sitesdegree b

b sites

V. Sood & SR, PRL 94, 178701 (2005);



dN↑,a =N↓,aN↑,b −N↑,aN↓,b

(a + b)b

dN↑,b =N↓,bN↑,a −N↑,bN↓,a

(a + b)a

rate equation

Subgraph densities:

!a,b(t) =1

2[!a,b(0) ! !b,a(0)] e!2t +

1

2[!a(0) + !b(0)]

"

1

2[!a(0) + !b(0)]

ρa = N↑,a/a, ρb = N↑,b/b dt = 1/(a + b)


low degree; picked often import often


high degree; few nodes change rarely



high degree; few nodes change rarely


“flow” from high degree to low degree

Invasion Process on Complex NetworksCastellano, AIP Conf Proc 779, 114 (2005)

low degree; picked often export often


high degree; change often



high degree; change often


“flow” from low degree to high degree


η = {1, 1, 0, 0, . . . , 1} system stateηx = system state when voter at x flipsη(x) = state of voter at x

Formal Approach for Conservation Law

flip rate: P[η → ηx] =�

y

Axy

Z [Φ(x, y) + Φ(y, x)]



Φ(x, y) ≡ η(x)[1− η(y)]Axy = adjacency matrix


y

Axy

Z [Φ(x, y) + Φ(y, x)]





y

Axy

Z [Φ(x, y) + Φ(y, x)]

2 connected nodes in different states





y

Axy

Z [Φ(x, y) + Φ(y, x)]

choose x, choose neighbor of x with prob.

choose y (neighbor of x), choose of x with prob.

choose link & update x with prob.

Z ≡

Nkx VMNky IPNµ1 LD

(Nkx)−1

(Nky)−1

(Nµ1)−1

2 connected nodes in different states


�∆η(x)� = [1− 2η(x)]P[η → ηx] =�

y

Axy

Z [η(y)− η(x)]


�∆η(x)� = [1− 2η(x)]P[η → ηx] =�

y

Axy

Z [η(y)− η(x)]

degree-weighted moments

�ωm� ≡ 1Nµm

�

x

kmx η(x)=

1µm

�

k

km nk ρk


�∆η(x)� = [1− 2η(x)]P[η → ηx] =�

y

Axy

Z [η(y)− η(x)]

link dynamics

�∆ω0�=�∆ρ�=�

x,y

Axy

Nµ1[η(y)−η(x)] = 0

invasionprocess∆�ω−1� =

�

x,y

Axy

Nkxky[η(y) − η(x)] = 0

∆�ω1� =�

x,y

Axy

Nkxkx [η(y) − η(x)] = 0 voter

model

degree-weighted moments

�ωm� ≡ 1Nµm

�

x

kmx η(x)=

1µm

�

k

km nk ρk

Exit Probability on Complex NetworksE(!) = !Voter model:

Exit Probability on Complex Networks

Extreme case: star graph0

0

0

0

0

00

0

0

0

0

0

1

N nodes: degree 11 node: degree N

E(!) = !Voter model:

Exit Probability on Complex Networks

Extreme case: star graph0

0

0

0

0

00

0

0

0

0

0

1

N nodes: degree 11 node: degree N

Final state: all 1 with prob. 1/2!

ω =1µ1

�

k

k nkρk =12

E(!) = !Voter model:

Byproduct: Voter Model Fixation ProbabilityWhat is the probability that a single mutant “takes over” a population?

Byproduct: Voter Model Fixation ProbabilityWhat is the probability that a single mutant “takes over” a population?

00

0

0

0

00

0

0

0

0

0

1

1 consensus

fixation probability∝ node degree

00

0

0

0

00

0

0

0

0

0

1

0 consensus

Invasion Process Fixation Probability

fixation probability∝ 1/(node degree)

Route to Consensus on Complex Graphs

complete bipartite graph

degree a

a sitesdegree b

b sites

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ρ b

ρa

t<!

1

Route to Consensus on Complex Graphs

complete bipartite graph

degree a

a sitesdegree b

b sites

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ρ b

ρa

t<!

1

c = 1

c = 100

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ρ b

ρa

two-clique graph

c

N=10000, C links/node

Consensus Time Evolution Equation

T (ρ) = R(ρ)[T (ρ + dρ) + dt]+L(ρ)[T (ρ− dρ) + dt]+[1−R(ρ)− L(ρ)][T (ρ) + dt]

warmup: complete graphT (ρ) ≡ av. consensus time starting with density ρ

A Guide to First-Passage Processes (CUP, 2001)

Consensus Time Evolution Equationwarmup: complete graphT (ρ) ≡ av. consensus time starting with density ρ

R(ρ) ≡ prob(↓↑→↑↑)= ρ(1− ρ)ρ


1

R

ρ0



R(ρ) ≡ prob(↓↑→↑↑)L(ρ) ≡ prob(↑↓→↓↓)

= ρ(1− ρ)ρ


1

RL

0



R(ρ) ≡ prob(↓↑→↑↑)L(ρ) ≡ prob(↑↓→↓↓)

= ρ(1− ρ)ρ


1

RL1−R−L

0


Consensus Time on Complete Graph

continuum limit: T �� = − N

ρ(1− ρ)

T (ρ) = −N [ρ ln ρ + (1− ρ) ln(1− ρ)]

solution:


Consensus Time on Heterogeneous Networks

T ({ρk}) ≡ av. consensus time starting with density ρk

on nodes of degree k

T ({ρk}) =�

k

Rk({ρk})[T ({ρ+k }) + dt]

+�

k

Lk({ρk})[T ({ρ−k }) + dt]

+�1−

�

k

[Rk({ρk}) + Lk({ρk})]�[T ({ρk}) + dt]

Lk({ρk}) = nkρk(1− ω)Rk({ρk}) = prob(ρk → ρ+k )

=1N

��

x

1kx

�

y

P (↓ ,−− , ↑)

= nkω(1− ρk)

continuum limit:�

k

�(ω − ρk)

∂T

∂ρk+

ω + ρk − 2ωρk

2Nnk

∂2T

∂ρ2k

�= −1


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ρ

ω

ρ6

ρ11

(Molloy-Reed) Configuration Modelnk ! k!2.5, µ1 = 8

continuum limit:�

k

�(ω − ρk)

∂T

∂ρk+

ω + ρk − 2ωρk

2Nnk

∂2T

∂ρ2k

�= −1

now use ρk → ω ∀k∂∂ρk

= ∂ω∂ρk

∂∂ω = knk

µ1

∂∂ωand


continuum limit:�

k

�(ω − ρk)

∂T

∂ρk+

ω + ρk − 2ωρk

2Nnk

∂2T

∂ρ2k

�= −1

now use ρk → ω ∀k∂∂ρk

= ∂ω∂ρk

∂∂ω = knk

µ1

∂∂ωand

T �� = − N

ρ(1− ρ)same as

Neff = N µ21/µ2with effective size

to give∂2T

∂ω2= − Nµ2

1/µ2

ω(1− ω)


Consensus Time for Power-Law Degreenk ! k

!!Distribution

TN !

!

"

"

"

#

"

"

"

$

N ! > 3,N/ lnN ! = 3,N (2!!4)/(!!1) 2 < ! < 3,(lnN)2 ! = 2,O(1) ! < 2.

Voter model: TN ∼ Neff = Nµ21/µ2

Consensus Time for Power-Law Degreenk ! k

!!Distribution

TN !

!

"

"

"

#

"

"

"

$

N ! > 3,N/ lnN ! = 3,N (2!!4)/(!!1) 2 < ! < 3,(lnN)2 ! = 2,O(1) ! < 2.

Voter model: TN ∼ Neff = Nµ21/µ2

lnvasion process: TN ∼ Neff = Nµ1 µ−1

TN ∼

N ν > 2N lnN ν = 2N3−ν ν < 2

Partisan Voting and Truth N. Masuda, N. Gibert, SRPRE 82, 010103(R) (2010)

prefers truth& in T state

density T+

prefers false& in F state

density F−

prefers false& in T state

density F+

prefers truth& in F state

density T−

Partisan Voting and Truth N. Masuda, N. Gibert, SRPRE 82, 010103(R) (2010)

prefers truth& in T state

density T+

prefers false& in F state

density F−

prefers false& in T state

density F+

prefers truth& in F state

density T−

T

T fraction F fraction

+F_

+T F_ fraction that believe the truth

1. Pick voter, pick neighbor (as in usual voter model);

2a. If initial voter becomes concordant by adopting neighboring state, change occurs with rate 1+ε; 1+ε

2b. If initial voter becomes discordant by adopting neighboring state, change occurs with rate 1-ε. 1-ε

partisan voting update:

Rate Equations

S = T+ + F+

∆ = T+ − F+

T− = T − T+

F− = F − F+

T+ = (1 + �)T− [T+ + F−]− (1− �)T+ [T− + F+]

T− = (1− �)T+ [T− + F+]− (1 + �)T− [T+ + F−]

F+ = (1 + �)F− [F+ + T−]− (1− �)F+ [F− + T+]

F− = (1− �)F+ [F− + T+]− (1 + �)F− [F+ + T−]

S=0

1−T Δ=0

2T−1

T

S

ΔT−1

T

(2T−1)/20

� < 2T − 1

truth consensus

1−T

T

(2T−1)/20 TT−1

S

Δ

2T−1� > 2T − 1

partisanship

Flow Diagram S = T+ + F+

∆ = T+ − F+

Summary & OutlookVoter model:

paradigmatic, soluble, (but hopelessly naive)

Voter model on complex networks:new conservation lawtwo time-scale route to consensusfast consensus for broad degree distributions

Future: “churn” rather than consensusheterogeneity of real peoplepositive and negative social interactions

Extension to Partisanship:partisanship forestalls consensus to the truth

Crass Commercialism

1. Aperitifs2. Diffusion3. Collisions4. Exclusion5. Aggregation

6. Fragmentation7. Adsorption 8. Spin Dynamics9. Coarsening10. Disorder

11. Hysteresis12. Population Dynamics13. Diffusion Reactions14. Complex Networks

to appear this October

TABLE OF CONTENTS

Aimed at graduate students, this book explores some of the core

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

kinetics of aggregation, fragmentation, and adsorption, where basic

phenomenology and solution techniques are emphasized. The

following chapters cover kinetic spin systems, by developing both a

discrete and a continuum formulation, the role of disorder in

non-equilibrium processes, and hysteresis from the non-equilibrium

perspective. The concluding chapters address population dynamics,

chemical reactions, and a kinetic perspective on complex networks.

The book contains more than 200 exercises to test students'

understanding of the subject. A link to a website hosted by the

authors, containing an up-to-date list of errata and solutions to

some of the exercises, can be found at

www.cambridge.org/9780521851039.

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.

Cover illustration: Snapshot of a collision cascade in a perfectly elastic, initiallystationary hard-sphere 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.

A Kinetic View of

St

at

ist

ica

lP

hy

sic

sKrapivskyRednerBen-N

aim

Pavel L. Krapivsky Sidney Redner Eli Ben-Naim

A Kinetic View of

Statistical Physics

KRAP

IVSK

Y: S

tatis

tical

Phy

sics

PPC

C M

Y

BL

K

Dynamics of Voter Models on Heterogeneous Networks

Documents

Transcript of Dynamics of Voter Models on Heterogeneous Networks