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, (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
: =2:3
Time 0
: =3:4
Time 1
stable state
(no one in local
minority opinion)
Time 2

6. A. Opinion spread for initially
: =2:3
stable state : =1:4

7. B. Opinion spread for initially
: =1:1
stable state : =1:1

8. “Phase Transition” on square latticeB
largest
size of the second
largest cluster
×100 A
critical initial fraction fc=0.5

9. 2 classes of network: differ in degree k distribution
Poisson distribution: Power-law distribution:
(i) Erdős-Rényi
(ii) scale-free
Edges connect randomly
Preferential Attachment
Log P(k)
Log k

10. “Phase Transition” for both classes of network
fc≈0.46
fc≈0.30
(b) scale-free
(a) Erdős-Rényi
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 A: Injection Point
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 Fractal dimension: 1.84 s ~r1.84
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
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.

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!