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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)
“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”.
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