Physical Mechanism Underlying Opinion Spreading Jia Shao Shlomo Havlin, H. Eugene Stanley J. Shao,...

Physical Mechanism Underlying Opinion Spreading

Jia Shao

Shlomo Havlin, H. Eugene Stanley

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

Questions

• How can we understand, and model the coexistence of two mutually exclusive opinions?

• Is there a simple physical mechanism underlying the spread of mutually exclusive opinions?

Motivation

• Coexistence of two (or more) mutual exclusive opinions are commonly seen social phenomena.

Example: US presidential elections • Community support is essential for a small

population to hold their belief.

Example: the Amish people • Existing opinion models fail

to demonstrate both.

What do we do?

• We propose a new opinion model based on local configuration of opinions (community support), which shows the coexistence of two opinions.

• If we increase the concentration of minority opinion, people holding the minority opinion show a “phase transition” from small isolated clusters to large spanning clusters.

• This phase transition can be mapped to a physical process of water spreading in the layer of oil.

Evolution of mutually exclusive opinions

Time 0

Time 1

Time 2

: =2:3

: =3:4

stable state(no one in local minority opinion)

A. Opinion spread for initially : =2:3

stable state : =1:4

B. Opinion spread for initially: =1:1

stable state : =1:1

“Phase Transition” on square lattice

critical initial fraction fc=0.5

size of the second largest cluster ×100

Phase 1 Phase 2

A

B

larg

est

2 classes of network: differ in degree k distribution

Poisson distribution: Power-law distribution:

P(k

)

k

Poisson distribution

Scale Free

Log(

P(k

))Log(k)

Power-lawdistribution

Edges connect randomly Preferential Attachment

(ii) scale-free (i) Erdős-Rényi

( ) ~ exp( )!

ktP k t

k ( ) ~P k k

µ Log

P(k

)Log k

µ µ

“Phase Transition” for both classes of network

fc≈0.46

Q1: At fc , what is the distribution of cluster size “s”? Q2: What is the average distance “r” between nodes b

elonging to the same cluster?

(a) Erdős-Rényi (b) scale-free

fc≈0.30

µ

Invasion Percolation

• 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.

Example: Inject water into layer containing oil

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

Invasion PercolationTrapped Region of size s P(s) ~ s-1.89

Fractal dimension: 1.84 s ~r1.84

S

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

Clusters formed by minority opinion at fc

Cumulative distribution function P(s’>s) ~ s-0.89

PDF P(s) ~ s-1.89

r/s(1/1.84) Const.

s ~ r1.84

Conclusion: “Phase transition” of opinion model belongs to the same universality class of invasion percolation.

r

Summary

• Proposed an opinion model showing coexistence of two opinions.

• The opinion model shows phase transition

at fc.

• Linked the opinion model with an oil-field-inspired physics problem, “invasion percolation”.

Thank you!