1. Facultés universitaires Notre Dame de la Paix à Namur (FUNDP)
Department of Mathematics
The Peace Mediator Effect The role of the openness of mind and the affnity in a continuous opinion model
Andrea Guazzini1 & Graziano Barnabei1
1 – CSDC – Centre for study of complex dynamics – University of Florence
Namur
15th – Avril

2. The Peace Mediator Effect
Summary
Introduction
Modelling psychological information processes
The Model
The social distance
The social temperature and the rule of interactions
Peace Mediator definition
The Diplomatics and the Auctoritas
Numerical Simulations
Convergence criterion
Dissemination strategy
Initial conditions
Results
System fragmentation and convergence time
Discussion and Conclusion
CSDC - Centre for the study of complex dynamics

3. The Peace Mediator Effect
Social systems
Dynamics of social systems is one of the fields of sociology that have interested physicists the most in the latest years.
Sociology
Psychology
Physics
Computer Science
CSDC - Centre for the study of complex dynamics

4. “Classical” agent information processes
The Peace Mediator Effect
“Classical” agent information processes -
Diff
Threshold
Variable j
subject
Control Parameter
Evolution Rule
CSDC - Centre for the study of complex dynamics

5. Agent information processing
The Peace Mediator Effect
Agent information processing -
Diff
Variable j
Threshold
subject
Control P. -
Diff
Control P.
Variable
Threshold -
Diff
Memory of the system
CSDC - Centre for the study of complex dynamics

6. The Peace Mediator Effect
The Model {
Deffuant et al.
Threshold mechanism A
Threshold mechanism B 1
αij - αc A 1 B -1
CSDC - Centre for the study of complex dynamics

7. The Peace Mediator Effect
Social Distance
Opinion Space
Selected with p=1/N xi xj
Δxij
Affinity Space xi xj
1 - ij
Δxij
αij
dij = Δxij (1 - αij)
xi Social distances space (dij) xi xj
dij
CSDC - Centre for the study of complex dynamics

8. “Social Temperature” and extraction rule
The Peace Mediator Effect
“Social Temperature” and extraction rule
Social Temperature
kt = 0.1
kt = 0.01 xi
0 xi Final Social distances space xj
dij
Agent Selection
0 xi Final Social distances space 1 xi xj
dij
Additive Noise
CSDC - Centre for the study of complex dynamics

9. The Peace Mediator Effect
Simulation results
Order Parameter
Nc Number of clusters
Control Parameters
Number of Peace Mediators
Spreading strategy of PM
Evolution of opinion
Affinity Matrix
CSDC - Centre for the study of complex dynamics

10. Social Temperature effect
The Peace Mediator Effect
Social Temperature effect
kt = 0.01
kt = 0.1
kt = 0.05
CSDC - Centre for the study of complex dynamics

11. Phase transition and critical features
The Peace Mediator Effect
Phase transition and critical features
Distribution of Opinion Shifts
Fractal Dimension of Trajectories
CSDC - Centre for the study of complex dynamics

12. Opinion threshold “∆Oc”: The concept of Openness of Mind
The Peace Mediator Effect
Opinion threshold “∆Oc”:
The concept of Openness of Mind
In psychological terms “Openness of Mind” embodies qualitative features that allow agents to ignore own perception of incompatibility between distant cognitions and consequently to relate theirself with agents at a given opinion distance.
Opinion threshold ( ∆Oc ) : defined as the outside limit of permissiveness of each agent to interacts with others in the opinion distance space.
Cases:
|∆Oij| ≥ ∆Oc  Aij   i tend to ignore j
|∆Oij| < ∆Oc  Aij   i tend to relate itself with j
Pecae Mediator Effect - Namur
CSDC - Centre for the study of complex dynamics 12 12

13. The Peace Mediator Effect
Peace-maker Effect:
“The Diplomat Effort”
PEACE-MEDIATORS:
In real life: Peace-operators that, focusing on long-term, establish equal power relationships and a stable status quo between parts in conflict.
In this model: Agent having function of approach for other agents with high opinion distance:
Higher ∆Oc : “The Diplomats”
Opinion Space
: ∆OcPM = 0.5
: ∆OcN = 0.1
CSDC - Centre for the study of complex dynamics 13 13

14. The Peace Mediator Effect
Peace-maker Effect:
“The Auctoritas”
Higher ij : “The Charismatic Auctoritas”
ij (  ) = [0 ; 0.5]
ij (  ) = 0.75
CSDC - Centre for the study of complex dynamics 14 14

15. The Peace Mediator Effect Numerical Simulation: Initial Conditions
Uniform distribution of Initial Opinions
Lower bound of opinion
Upper bound of opinion xi 1
Opinion Space
Initial Random α Matrix
Numerical simulations were computed according with the following parameters:
Number of Agents N = 100
Convergence Parameter μ = 0.5
Affinity Threshold αc = 0.5
Opinion Threshold ΔOc=0.5
Social Temperature Kt = 0.003
CSDC - Centre for the study of complex dynamics

16. Convergence Criterium
The Peace Mediator Effect
Convergence Criterium
When system is composed by more than ≈ 20 particles (agents) we know that dynamic of affinity convergence is slower than the dynamic of opinion. Thereby to estimate the equilibrium of the system we define the Convergence Criteria as follow:
The system is considered in the equilibrium state when:
C = 1 or
No Changes in the Affinity Matrix since 106 temporal iteration
CSDC - Centre for the study of complex dynamics

17. Diplomats Dissemination Strategy
The Peace Mediator Effect
Diplomats Dissemination Strategy
We have considered both the number of peace mediator and their distribution along the opinion space as factors under scrutiny.
Scenario I: Uniform Distribution
Scenario II: Gaussian Distribution
Scenario III: Bimodal Distribution 1
P(x) 1
Opinion Space 1
P(x) 1
Opinion Space 1
P(x) 1
Opinion Space
CSDC - Centre for the study of complex dynamics

18. Effect of Diplomats dissemination strategy on the system fragmentation
The Peace Mediator Effect
Effect of Diplomats dissemination strategy on the system fragmentation
CSDC - Centre for the study of complex dynamics

19. Effect of Diplomats dissemination strategy on the system fragmentation
The Peace Mediator Effect
Effect of Diplomats dissemination strategy on the system fragmentation
CSDC - Centre for the study of complex dynamics

20. Effect of Diplomats dissemination strategy on the convergence time
The Peace Mediator Effect
Effect of Diplomats dissemination strategy on the convergence time
CSDC - Centre for the study of complex dynamics

21. Auctoritas Dissemination Strategy
The Peace Mediator Effect
Auctoritas Dissemination Strategy 1
P(x)
Scenario I: Delta Distribution 1
Opinion Space 1
P(x)
Scenario II: Uniform Distribution 1
Opinion Space 1
P(x)
Scenario III: Gaussian Distribution 1
Opinion Space 1
P(x)
Scenario IV: Bimodal Distribution 1
Opinion Space
CSDC - Centre for the study of complex dynamics

22. The Peace Mediator Effect
Effect of Auctoritas dissemination strategy on the system fragmentation
CSDC - Centre for the study of complex dynamics

23. The Peace Mediator Effect
Effect of Auctoritas dissemination strategy on the system fragmentation
CSDC - Centre for the study of complex dynamics

24. Effect of Auctoritas dissemination strategy on the convergence time
The Peace Mediator Effect
Effect of Auctoritas dissemination strategy on the convergence time
CSDC - Centre for the study of complex dynamics

25. Diplomats Vs Auctoritas
The Peace Mediator Effect
Diplomats Vs Auctoritas
CSDC - Centre for the study of complex dynamics

26. Diplomats Vs Auctoritas
The Peace Mediator Effect
Diplomats Vs Auctoritas
CSDC - Centre for the study of complex dynamics

27. Diplomats Vs Auctoritas
The Peace Mediator Effect
Diplomats Vs Auctoritas
CSDC - Centre for the study of complex dynamics

28. The Peace Mediator Effect
Conclusions
The role of Diplomats and Auctoritas can be crucial for obtaining the final consensus.
For large population (e.g. For a small fraction of PM) heterogeneous system assure an higher probability to obtain final consensus
When diplomats density overcome 20%, the probability of obtaining a single cluster in the final state is the same as in the homogeneous system, which is however characterized by a shorter convergence time. So in this case it is advisable to “educate” the population (e.g. increase its mean openness of mind) rather than using diplomats.
Spreading strategy doesn’t has a crucial role for diplomats goals.
CSDC - Centre for the study of complex dynamics

29. The Peace Mediator Effect
Conclusions
For auctoritas both the number and the dissemination strategy affect the probability to obtain the final consensus state.
The best dissemination strategy is to insert the auctoritas in the “middle” of the opinion space as so as to minimize the distances from all the agents.
For the “delta” dissemination, when auctoritas density is equal to 20% the probability of obtaining a single cluster in the final state become 1 and we observe a phase transition for the system.
Finally, auctoritas effect appears as more incisive than diplomatics one, both for the probability of consensus and for the convergence time.
CSDC - Centre for the study of complex dynamics

30. Convergence time of the system
The Peace Mediator Effect
Convergence time of the system
Convergence time scaling
Different regimes for the dynamics of “opinion” and “affinity”
Analytical approximation reproduces experimental data
CSDC - Centre for the study of complex dynamics