1. OPINIONS DYNAMICS WITH RELATIVE AGREEMENT
An agent-based simulation approach
We model opinion dynamics in populations of agents with continuous opinion and uncertainty. The opinions and uncertainties are modified by random pair interactions. We propose a new model of interactions, called the relative agreement model, which is a variant of the bounded confidence model (Hegselmann and Krause, 2002). In this model, uncertainty as well as opinion can be modified by interactions. We aim at studying the behaviour of this model under several conditions.
Fully connected case…
The model of opinion dynamics (Deffuant et al., 2002)
Population of N agents
Each agent i has:
An opinion xi
An associated uncertainty ui
Pair-interactions between agents
During an interaction between agents i and j, let us consider opinion segments si=[xi-ui, xi+ui] and sj=[xj-uj, xj+uj]. We define the agreement of agent i with j (it is not symmetric) as the overlap of sI and sj, minus the non-overlapping part.
The overlap hij is given by: hij = min(xi+ui , xj+uj) - max(xi-ui , xj-uj)
The non-overlapping width is: 2.ui – hij
The agreement is then : hij - (2.ui – hij ) = 2.(hij – ui )
The relative agreement is the agreement divided by the length of segment si : 2.(hij – ui ) / 2.ui = hij/ui-1
If hij > ui, then the modification of xj and uj by the interaction with i is
δxj = µ.(hij/ui – 1).(xi-xj)
δuj = µ.(hij/ui – 1).(ui-uj)
where µ is a constant parameter which amplitude controls the speed of the dynamics.
If hij ≤ ui, there is no influence of i on j.
Homogeneous population in uncertainty
When every agent of the population is initialised with the same uncertainty U they tends to form clusters (cf. fig. left) the number of the clusters is close to w/2U (r2=0.99 cf. fig. right) w is the width of the initial distribution of the opinions
Introduction of extremists (heterogeneity in uncertainty)
x’j j i xi xj
hij
2ui - hij
Before interaction
After interaction
We now introduce extremists in our population: we suppose that agents at the extremes of the opinion distribution (proportion pe) are more confident (their uncertainty ue is lower). It follows different convergence cases in the population depending on the value of the uncertainty U of the other agents (the majority at the centre).
We observe then three different convergence types:
central convergence (top-left) where the majority converges at the centre and are not influenced by the extremes (typically for low values of U)
both extreme convergence (upon) where the majority is attracted by both of the extremes (for higher values of U)
single extreme convergence (on the left) where the majority is attracted by only one of the extreme (for high values of U and low extremists proportion in the population pe)
A more systematic exploration of the parameter space (on the left) gives us patterns of the average of convergence cases (central is yellow, both extreme orange and single extreme brown) depending on the uncertainty of the majority U in x-axis and on the extremists proportion pe in y-axis. Intermediate colours corresponds to a mixture of the previous convergence cases.
On a grid with Von Neumann’s neighbourhood…
Simulations on a grid with extremists. The conditional interactions lead to formation of similar opinion patterns on the network which are then reinforced with quickly few zones of disagreement.
We start from a situation where opinion are uniformly distributed on the grid ( agents). Then, we pick up at random the chosen number of extremists of both sides. The mechanisms at play when we focus on the simulation dynamics is the following:
Depending on the uncertainty of the majority U, the majority tends to converge rapidly to the centre. This convergence is observed locally.
Then local extremists tends to attract the majority if this latter has not converged too much quickly and kept isolated the extremists. In this latter case it results in a central convergence process.
Then if only one kind of extremists is kept connected to the majority, it results in a single extreme convergence.
Else if the two kinds of extremists are present and connected to the majority it results in a both extreme convergence
Finally, if the proportion of extremists is very low, the majority can still win the deal and it results in a central convergence (cf. fig. Left)
Further investigations have to be made concerning both a most systematic exploration of the parameter space in this case and the exploration of other graph topologies (scale-free networks, small-worlds)
F.Amblard*, G.Deffuant*, G.Weisbuch** - *Cemagref-LISC, **ENS-LPS :