Physical Mechanism Underlying Opinion Spreading

1
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).
2
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?

3
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.

4
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.

5
Evolution of mutually exclusive opinions
23
Time 0
34
Time 1
stable state (no one in local minority opinion)
Time 2
6
A. Opinion spread for initially
23
stable state 14
7
B. Opinion spread for initially
11
stable state 11
8
Phase Transition on square lattice
Phase 1
Phase 2
B
largest
size of the second largest cluster
100
A
critical initial fraction fc0.5
9
2 classes of network differ in degree k
distribution
Poisson distribution
Power-law distribution
(ii) scale-free
(i) Erdos-Rényi
µ
µ
Log P(k)
µ
Log k
10
Phase Transition for both classes of network
fc0.46
fc0.30
(b) scale-free
(a) Erdos-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?
11
Invasion Percolation
Example Inject water into 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
D. Wilkinson and J. F. Willemsen, J. Phys.
A 16, 3365 (1983).
12
Invasion Percolation
Trapped Region of size s P(s) s-1.89
S
Fractal dimension 1.84 s r1.84
S. Schwarzer et al., Phys. Rev. E. 59, 3262
(1999).
13
Clusters formed by minority opinion at fc
r
Cumulative distribution function
P(sgts) 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.
14
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.

15
Thank you!