Opinion Dynamics of Skeptical Agents Read-Through

Opinion Dynamics of Skeptical Agents
by A. Tsang and K. Larson
Paper Read-Through
M2 Yu Matsuzawa

Title and publication notes
• Opinion Dynamics of Skeptical Agents
– Alan Tsang and Kate Larson
• Univ. of Waterloo, Canada
– AAMAS2014 (Proc. of the 13th International
Conference on Autonomous Agents and
Multiagent Systems, pp. 277-284), May 2014
Opinion Dynamics of Skeptical Agents 2

Shared Interest
• Opinion Dynamics
– Gradated (continuous) opinions in this paper
• Cf. Discrete opinions
• Cognitive Bias: Motivated cognition
– Leads subjects to skewed/irrational conclusion

Motivated Cognition
• Evaluation based on/affected by:
– Compatibility with the subject’s own beliefs
– Profitability to the subject
• 「自分にとって都合のいいように解釈する」
• In a case of opinion convergence/debate:
– Opinion/Evidence of the competitor diverges away
-> Less & less persuasive – “It must be flawed!”

Skepticism
• Key component of this research
• Disbelief/Refusal against incompatible opinion
due to motivated cognition
->Skepticism (Antonym: Trust)
• Summary: Opinion dynamics in networks of
agents with Skepticism/Trust mechanism

Opinion Dynamics Model
• Agents are embedded in undirected graphs
• Each agent 𝑖 has an opinion 𝑥𝑖 ∈ 0,1
• Influenced by neighbors (direct neighbor)
• For each neighbor 𝑗, an agent 𝑖 maintains
Trust value 𝑤𝑖,𝑗
– Represents weight of influence to 𝑖 from 𝑗

• Def. Trust function 𝑇:
– 𝑇 𝑥, 𝑥′ = exp −
𝑥−𝑥′ 2
ℎ
(Gaussian kernel)
– Used in Trusts 𝑤𝑖,𝑗 updating
– Bandwidth parameter ℎ represents empathy
• Higher ℎ means the population is more willing to be
persuaded by different opinion

• Opinion update function:
– 𝑥𝑖 ←
𝑤 𝑖,𝑖 𝑥 𝑖+∑𝑤 𝑖,𝑗 𝑥 𝑗
𝑤 𝑖,𝑖+∑𝑤 𝑖,𝑗
(Weighted average)
– 𝑤𝑖,𝑖 represents inertia of own opinion
• Trust update function:
– 𝑤𝑖,𝑗 ←
𝑤 𝑖,𝑗+𝑟𝑇(𝑥 𝑖,𝑥 𝑗)
1+𝑟
– 𝑟 represent learning rate
• Higher the 𝑟 is, faster the trust/distrust appear

• Opinions updated, then Trusts updated in
each iteration steps
• Majority of agents are initialized randomly
• The remainder represent extremists
– Initialized to 𝑥𝑖 = 0 𝑜𝑟 1
– Extremists’ opinions and trusts are NOT updated

Graph Models
• Classic BA model
• Homophily model based on ER random graph
• BA model
– Explanation snipped
– Parameter 𝑚 represents the number of edges
every new vertices have

Graph models
• Homophily model based on ER random graph
– In Erdos-Renyi random graph, every possible edge
has probability 𝑝 to be activated
– In this model, activation probability of possible
edge between 𝑖 and 𝑗 is:
• 1 − 𝑑 𝑝, where 𝑑 = 𝑥𝑖 − 𝑥𝑗
– If opinions of 𝑖 aligned with 𝑗’s, highly probable they are
connected
– Similar opinion, likely to be connected -> Homophily

Trust initialization
1. Uniform trust model
– 𝑤𝑖,𝑗 = 1, for every existing 𝑖 and 𝑗
– 𝑤𝑖,𝑖 = 𝑑𝑖, where 𝑑𝑖 represents 𝑖’s degree
2. Degree based trust model
– 𝑤𝑖,𝑗 = 𝑑𝑗/𝑑𝑖 , thus 𝑤𝑖,𝑖 = 1
3. Kernel based trust model
– 𝑤𝑖,𝑗 = 𝑇(𝑥𝑖, 𝑥𝑗), 𝑤𝑖,𝑖 = 1

Experimental designs
• Two experiments:
– Ability of extremists to influence the moderate
people
• 1-pole model
• BA model
– Conditions for opinions of the moderate people to
stratify and stabilize at multiple levels
• 2-pole model
• BA and Homophily model

• 200 agents
• In the first set of experiments:
– 10% of agents are extremists
– Fixed to 𝑥𝑖 = 1 (1-extremists) -> 1-pole model
• In the second set:
– 10% are 0-extremists, 10% are 1-extremists
-> 2-pole model

• 𝑟 = 1.5
– Changing 𝑟 did not affect results qualitatively
• Termination condition:
– No opinions changed by more than 𝜖 = 0.001
– Iterations reached maximum number 𝑡max = 500
• Rarely reached in practice
• Results are averaged over 25 replicated trials

First Experiment
• Investigation of the effect of extremists
• Measure: The mean opinion of the moderates
at the end of each experiments
– If completely unaffected -> hover around 0.5
– If completely affected -> near 1.0

Influence of Extremists
• Evolution of Opinions over the course of the
experiment
– Darker -> More
– Initially, the moderates converges to a common opinion,
regardless of how close to the pole
– After that the consensus gradually drifts to the pole

Effect of Empathy
• Mean Opinions at Convergence (uniform trust)
– Effect of empathy ℎ and BA parameter 𝑚
– Increasing ℎ has expected effect to polarization
– Increasing 𝑚 has no qualitative effect

Second Experiment
• Introducing 2-pole mode
• Measure: The mean of absolute differences of
each agents’ opinion from 0.5 at convergence
– If completely unaffected -> 0
– If completely affected -> 0.5

Type of convergence in 2-pole mode
• Deffuant’s characterization (2006)
• Type I: Moderate
• Type II: Polarized to one side
• Type III: Split in two pole
• Type IV: Fragmentation

Polarization in 2-pole mode
• Mean polarization at convergence (degree trust)
– Basic traits are the same as the first experiment
– Impact of extremists mitigated on highly connected graph
– Type I or II convergence

Effect of initialization
• So far the moderate population are initialized
uniformly at random
• Can initially divided population produce
separation(Type III) or fragmentation(IV)?
• Test with 𝐵𝑒𝑡𝑎 0.5,0.5
initialization
– http://www2.ipcku.kansai-u.ac.jp/~aki/pdf/beta1.htm

Divided initialization
• Evolution of opinions from 𝐵𝑒𝑡𝑎(0.5,0.5)
– Surprisingly, the result unchanged
– Even agents initialized to the opposite pole drawn to
the converged pole

Conditions for stratification
• There appear to be two main factors for Type
III(separation)/IV(fragmentation) to occur
->stratification (層を形成すること)
• Initial Trust
– Too much trusts are given to different opinions
• Homophily in graph structure

Kernel trust and Homophily model
• Evolution of opinions from 𝐵𝑒𝑡𝑎(0.5,0.5)
– Ends in Type IV convergence (fragmentation)

Kernel trust and Homophily model
• Mean polarization at convergence
– Under ℎ > 0.3, stratification occurs and
polarization decreases
– Higher empathy actually contributes to stratify

Discussion
• Hypothesis: High empathy agents are affected
by extremists of both poles
-> Preventing convergence to a single pole?
• Why the population can drift to consensus or
poles even if they are initially divided?
– Approximation of influence showed:
Unpolarized moderates can serve as bridge on
which influence will flow, starting an avalanche

Conclusion
• Introduction of robust Opinion Dynamics
model
• Combination of skepticism (Kernel trust) and
Homophily graph are the condition of opinion
stratification

Evaluation
• Modeling and Simulation centric research
– Normal in AAMAS?
– There is analytical approximation in discussion
-> Only shows the mechanism of influence-flow
within the model
– Not mentioning validity of the model itself,
according to real data

Evaluation
• Grand goal is unclear
– Condition of opinion stratification
(which is kind of unclear word as well,
i.e. diversification apart from extreme value?)
– Should be given more precisely?
– And early

Evaluation
• Related work and orientation of the research
is splendid
– Can learn the methodology
• Research lately published paper from the targeted
conference well (in this case AAMAS)
• Find some backbone research to stand upon

Inspiration
• Can we use some of the idea in this paper?
– Introducing cognitive bias into the model
– Initialization by Beta distribution
– Homophily graph model
• Easily construct innately-clustered network
• Possibly there is more common model for this?

Opinion Dynamics of Skeptical Agents Read-Through

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