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!