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Physical Mechanism Underlying Opinion Spreading. Jia Shao Advisor: H. Eugene Stanley. J. Shao, S. Havlin, and H. E. Stanley, Phys. Rev. Lett. 103 , 018701 (2009). Questions. How do different opinions influence each other in human society? Why minority opinion can persistently exist?.

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Physical mechanism underlying opinion spreading
Physical Mechanism Underlying Opinion Spreading

Jia Shao

Advisor: H. Eugene Stanley

J. Shao, S. Havlin, and H. E. Stanley, Phys. Rev. Lett. 103, 018701 (2009).


Questions
Questions

  • How do different opinions influence each other in human society?

  • Why minority opinion can persistently exist?


Motivation
Motivation

US presidential elections

  • Public opinion is important.

  • Simulation of opinion formation is a challenging task, requiring reliable information on human interaction networks.

  • There is a gap between existing models and empirical findings on opinion formation.

Instead of using individual level

network, we propose a community

level network to study opinion

dynamics.


Outline
Outline

  • Part I: Empirical findings from presidential election

  • Part II: Invasion Percolation

  • Part III: We propose a dynamic opinion model which can explain the empirical findings


2008 presidential election

Part I

2008 Presidential Election

We study counties with f>fT

fDem

fT=0.8

fT=0.5

fT=0.3

http://www-personal.umich.edu/~mejn/election/2008/


Counties in southern new england

Part I

Counties in southern New England


Counties in southern new england1

Part I

Counties in southern New England


Counties in southern new england2

Part I

Counties in southern New England

Vote for Obama

Barnstable f=55%

Polymouth f=54%

Norfolk f=54%

Worcester f=57%

Fairfield f=52%

Litchfield f=47%

Windham f=53%

Kent f=56%


Counties in southern new england3

Part I

Counties in southern New England

County network of f>fT=0.6

Number of counties:17

Size of largest cluster: 10


County network of obama

Part I

County Network of Obama

Counties of fObama>fT

50% decrease


County network of mccain

Part I

County Network of McCain

Counties of fMcCain>fT

30% decrease


Clusters formed by counties when f t f c

Part I

Clusters formed by counties when fT=fc

Cluster Size

County network

Diameter

Fractal dimension: dl=1.56

=1.86

The phase transition in county network is

different from randomized county network.


Physical mechanism underlying opinion spreading

What class of percolation does this phase transition

belong to?


Invasion percolation

Part II percolation-like phase transition at

Invasion Percolation

Example: Inject water into earth layer containing oil

  • Invasion percolation describes the evolution of the front between two immiscible liquids in a random medium when one liquid is displaced by injection of the other.

  • Trapped region will

    not be invaded.

A: Injection Point

P. G. de Gennes and E. Guyon, J.Mech. 17, 403(1978).

D. Wilkinson and J. F. Willemsen, J. Phys. A 16, 3365 (1983).


Invasion percolation1

Part II percolation-like phase transition at

Invasion Percolation

Trapped Region of size s

P(s) ~ s-1.89

S

Fractal dimension: 1.51

s ~l1.51

S. Schwarzer et al., Phys. Rev. E. 59, 3262 (1999).


Physical mechanism underlying opinion spreading


Evolution of mutually exclusive opinions
Evolution of mutually exclusive opinions to the same universality of invasion percolation.

: =2:3

Time 0

: =3:4

Time 1

stable state

(no one in local

minority opinion)

Time 2


A opinion spread for initially
A. Opinion spread for initially to the same universality of invasion percolation.

: =2:3

stable state : =1:4


B opinion spread for initially
B. Opinion spread for initially to the same universality of invasion percolation.

: =1:1

stable state : =1:1


Phase transition on square lattice
“Phase Transition” on to the same universality of invasion percolation.square lattice

Phase 1

Phase 2

B

largest

size of the second

largest cluster

×100

A

critical initial fraction fc=0.5


2 classes of network differ in degree k distribution

Edges connect randomly to the same universality of invasion percolation.

Preferential Attachment

2 classes of network: differ in degree k distribution

Poisson distribution: Power-law distribution:

(i) Erdős-Rényi

(ii)scale-free

µ

µ

Log P(k)

µ

Log k


Phase transition for both classes of network
“Phase Transition” for both classes of to the same universality of invasion percolation.network

fc≈0.46

fc≈0.30

(b) scale-free

(a) Erdős-Rényi

µ

Q1: At fc , what is the distribution of cluster size“s”?

Q2: What is the average distance“r” between nodes belonging to the same cluster?


Clusters formed by minority opinion at f c
Clusters formed by minority opinion at to the same universality of invasion percolation.fc

r

Cumulative distribution function

P(s’>s) ~ s-0.89

r/s(1/1.84)

Const.

s ~ r1.84

PDF P(s) ~ s-1.89

Conclusion: “Phase transition” of opinion model belongs to the

same universality class of invasion percolation.


Summary
Summary to the same universality of invasion percolation.

  • There exists a phase transition in the county election network when changing fT.

  • The phase transition of county network can be mapped to oil-field-inspired physics problem, “invasion percolation”.

  • We propose a network model which can explain the empirical findings.


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