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 - 2008 andrea.guazzini@gmail.com

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

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

“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

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

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

The Peace Mediator Effect Social Distance 0 Opinion Space 1 Selected with p=1/N xi xj Δxij 0 Affinity Space 1 xi xj 1 - ij Δxij αij dij = Δxij (1 - αij) 0 xi Social distances space (dij) 1 xi xj dij CSDC - Centre for the study of complex dynamics

“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 1 xj dij Agent Selection 0 xi Final Social distances space 1 xi xj dij Additive Noise CSDC - Centre for the study of complex dynamics

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

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

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

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  1  i tend to ignore j |∆Oij| < ∆Oc  Aij  0  i tend to relate itself with j Pecae Mediator Effect - Namur 15-04-2008 CSDC - Centre for the study of complex dynamics 12 12

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 0 0.5 1 : ∆OcPM = 0.5 : ∆OcN = 0.1 CSDC - Centre for the study of complex dynamics 13 13

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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