Mixing beliefs among interacting agents
Guillaume Deffuant , David Neau , Frederic Amblard
and Ge´rard Weisbuch
Laboratoire d’Inge´nierie pour les Syste`mes Complexes (LISC)
Cemagref - Grpt de Clermont-Ferrand
24 Av. des Landais - BP50085
F-63172 Aubie`re Cedex (FRANCE)
Laboratoire de Physique Statistique
de l’Ecole Normale Supe´rieure,
24 rue Lhomond, F 75231 Paris Cedex 5, France.
email:weisbuch@physique.ens.fr
ABSTRACT. We present a model of opinion dynamics where agents adjust continuous opinions
on the occasion of random binary encounters whenever their difference in opinion is below a
given threshold. High thresholds yield convergence of opinions towards an average opinion,
but low thresholds result in several opinion clusters: members of the same cluster share the
same opinion but do not adjust any more with members of other clusters.
. Laboratoire associe´ au CNRS (URA 1306), a` l’ENS et aux Universite´s Paris 6 et Paris 7

2 Applications of Simulation to Social Sciences
1. Introduction
Most models about opinion dynamics [FOL 74], [ART 94], [ORL 95], [LAT 97],
[GAL 97], [WEI 99], are based on binary opinions which social actors update under
either social influence or according to their own experience. One issue of interest
concerns the importance of the binary assumption: what would happen if opinionwere
a continuous variable such as the worthiness of a choice (a utility in economics), or
some belief about adjustment of a control parameter? In some European countries, the
Right/Left political choices were often considered as continuous and were represented
for instance by the geometrical position of the seat of a deputy in the Chamber.
Binary opinion dynamics under imitation processes have been well studied, and
we expect that in most cases the attractor of the dynamics will display uniformity of
opinions, either 0 or 1, when interactions occur across the whole population. This is
the “herd” behaviour often described by economists [FOL 74], [ART 94], [ORL 95].
Clusters of opposite opinions appear when the dynamics occurs on a social network
with exchanges restricted to connected agents. Clustering is reinforced when agents
diversity is introduced, for instance diversity of influence [LAT 97], [GAL 97], [WEI
99].
The a priori guess for continuous opinions is also homogenisation, but towards
average initial opinion [LAS 89]. The purpose of this paper is to present results
about continuous opinion dynamics when convergent opinion adjustments only pro-
ceed when opinion difference is below a given threshold. We will give results con-
cerning homogeneous mixing across the whole population and mixing across a social
network. Preliminary results about binary vectors of opinions will also be presented.
2. Complete Mixing
2.1. The basic model
Let us consider a population of agents with continuous opinions . At each
time step any two randomly chosen agents meet. They re-adjust their opinion when
their difference of opinion is smaller in magnitude than a threshold . Suppose that
the two agents have opinion and and that ; opinions are then adjusted
according to:
Where is the convergence parameter taken between 0 and 0.5 during the simulations.
The rationale for the threshold condition is that agents only interact when their
opinion are already close enough; otherwise they do not even bother to discuss. The
reason for such behaviour might be for instance lack of understanding, conflicts of
interest or social pressure. Although there is no reason to suppose that openness to
discussion, here represented by threshold , is constant across population or even

Sociology 3
symmetrical on the occasion of a binary encounter, we will always take it as a constant
simulation parameter in the present paper (We conjecture that the results we get would
remain similar provided that the distribution of accross the whole population is sharp
rather than uniform).
The evolution of opinions can be mathematically predicted in the limit case of
small values of [NEA 00]. Density variations of opinions obeys the follow-
ing dynamics:
This implies that starting from an initial distribution of opinions in the population, any
local higher opinion density is amplified. Peaks of opinions increase and valleys are
depleted until very narrow peaks remains among a desert of intermediate opinions.
2.2. Results
Figures 1 and 2 obtained by computer simulations, display the time evolution of
opinions among a population of agents for two values of the threshold .
Initially opinions were randomly generated across a uniform distribution on [0,1]. At
each time step a random pair is chosen and agents re-adjust their opinion according to
equation 1 and 2 when their opinions are closer than . Convergence of opinions is
observed, but uniformity is only achieved for the larger value of .
0
0.2
0.4
0.6
0.8
1
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
opinions
Figure 1. Time chart of opinions ( ). One time unit
corresponds to sampling 1000 pairs of agents.
Another way to follow agents opinion dynamics is to plot final opinions as a func-
tion of initial opinions. The plot on figure 3 shows how final opinions “reflect” initial

4 Applications of Simulation to Social Sciences
0
0.2
0.4
0.6
0.8
1
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
opinions
Figure 2. Time chart of opinions ( ). One time unit
corresponds to sampling 1000 pairs of agents.
opinions for . One can notice that some agents with initial opinions roughly
equidistant from final peaks of opinions can end up in either peak: basin of attractions
in the space of opinions overlap close to the clusters frontiers. The overlap observed
when is strongly reduced when (not represented here): agents then
have more time to make up their mind since opinions are changing 10 times more
slowly and their final opinion are those of the nearest peak.
A large number of simulations were carried out and we found that the qualitative
dynamics mostly depend on the threshold . and only influence convergence time
and the width of the distribution of final opinions (when a large number of different
random samples are made). controls the number of peaks of the final distribution
of opinions as shown in figure 4. The maximum number of peaks, , decreases
as a function of . A rough evaluation of based on a minimal distance of
between peaks (all other intermediate opinions being attracted by one of the peaks),
plus a minimal distance of of extreme peaks from 0 and 1 edges gives ,
in accordance with the observations of figure 4.
The finiteness of the population allows some slight variations of the number of
peaks according to random samplings for intermediate values of . These size effects
were confirmedwhen studying larger and smaller population sizes. In the intermediate
regions one also observes small populations of “wings” (a few percent) in the vicinity
of extreme opinions 0 and 1 (we call wings asymmetric peaks with a vertical bound
of either 0 or 1).

6 Applications of Simulation to Social Sciences
3. Social Networks
The literature on social influence and social choice also considers the cases when
interactions occur along social connections between agents [FOL 74] rather than ran-
domly across the whole populations. Apart from the similarity condition, we now add
to ourmodel a condition on neighborhood: agents only interact if they are directly con-
nected through a social pre-existing relation. Although one might certainly consider
the possibility that opinions on certain unimportant subjects could be influenced by
complete strangers, we expect important decisions to be influenced by advice taken
either from professionals (doctors for instance) or from socially connected persons
(e.g. through family, business or clubs). Facing the difficulty of inventing a credi-
ble instance of a social network as in the literature on social binary choice [WEI 99],
we adopted the standard simplification and we carried out our simulations on square
lattices: square lattices are simple, allow easy visualisation of opinion configurations
and contain many short loops, a property that they share with real social networks.
We then started from a 2 dimensional network of connected agents on a square
grid. Any agent can only interact with his four connected neighbours ( N, S, E andW).
We then use the same initial random sampling of opinions from 0 to 1 and the same
basic interaction process between agents as in the previous sections. At each time
step a pair is randomly selected among connected agents and opinions are updated
according to equations 1 and 2 provided of course that their distance is less than .
The results are not very different from those observed with non-local opinion mix-
ing described in the previous section, at least for the larger values of ( , all
results displayed in this section are equilibrium results at large times). As seen on fig-
ure 5 the lattice is filled with a large majority of agents which have reached consensus
around apart from isolated agents which have “extremists” opinions closer
to 0 or 1: the importance of extremists is the most noticeable difference with the full
mixing case described in the previous section.
Interesting differences are noticeable for the smaller values of as observed
in figure 6. When several values are possible for clusters of converging opinions,
consensus can only be reached on connected clusters of agents.
For connectivity 4 on a square lattice, only one cluster can percolate [STA 94]
across the lattice which then has homogeneous opinion for all the agents that belong
to it. Otherwise, non percolating clusters have homogeneous opinions inside the clus-
ter and these opinions correspond to groups of non-connected clusters with similar but
not exactly equal opinions as observed on the histogram (figure 7) and on the pattern
of opinions on the lattice (figure 6). The differences of opinions between group of
clusters relates to , but the actual values inside a small cluster fluctuates from cluster
to cluster because homogenisation occurred independently among the different clus-
ters: the resulting opinion depends on fluctuations of initial opinions and histories
from one cluster to the other. The same increase of fluctuations compared with the
full mixing case is observed from sample to sample with the same parameter values.

Sociology 7
Figure 5. Display of final opinions of agents connected on a square lattice of size
29x29 ( after 100 000 iterations) . Note the percolation of the large
cluster of homogeneous opinion and the presence of isolated “extremists”.
Figure 6. Display of final opinions of agents connected on a square lattice of size
29x29 ( after 100 000 iterations) . One still observes a large per-
colating cluster of homogeneous opinion and the presence of smaller non-percolating
clusters with similar but not equal opinions.

8 Applications of Simulation to Social Sciences
0
100
200
300
400
500
600
0 20 40 60 80 100
histogram of opinions
Figure 7. Histogram of final opinions corresponding to the pattern observed on figure
6 . The 101 bins noted from 0 to 100 correspond to a hundred times the final opinions
which vary between 0 and 1.0. Note the high narrow peak at 76 corresponding to the
percolating cluster and the smaller and wider peaks corresponding to non-percolating
clusters.
The qualitative results obtained with 2D lattices should be observed with any con-
nectivity, either periodic random or small world since they are related with the perco-
lation phenomenon [STA 94].
4. Vector opinions
4.1. The model
Another direction for investigation are vectors of opinions. Usually people have
opinions on different subjects, which can be represented by vectors of opinions. In ac-
cordance with our previous hypotheses, we suppose that one agent interacts concern-
ing different subjects with another agent according to some distance with the other
agent’s vector of opinions. In order to simplify the model, we revert to binary opin-
ions. An agent is characterised by a vector of binary opinions about the complete set
of subjects, shared by all agents. We use the notion of Hamming distance between
binary opinion vectors (the Hamming distance between two binary opinion vectors is
the number of different bits among the two vectors). Here, we only treat the case of
complete mixing; any couple of agents might interact and adjust opinions according
to how many opinions they share. The adjustment process occurs when agents agree
on at least subjects (i. e. they disagree on or less subjects). The rules
for adjustment are the following: all equal opinions are conserved; when opinions

Sociology 9
on a subject differ, one agent (randomly selected from the pair) is convinced by the
other agent with probability . Obviously this model has connections with popula-
tion genetics in the presence of sexual recombination when reproduction only occurs
if genome distance is smaller than a given threshold; such a dynamics results in the
emergence of species (see [HIG 91]).
We are again interested to figure out how opinion vectors cluster. In fact clusters
of opinions here play the same role as biological species in evolution. A first guess is
that vector opinions dynamics might be intermediate between the binary opinion case
and continuous opinions.
4.2. Results
We observed once again that and only modify convergence times towards
equilibrium; the most influencial factors are threshold and the number of sub-
jects under discussion. Most simulations were done for . For ,
convergence times are of the order of 10 million pair iterations. For :
– When , the radius of the hyperspace, convergence towards a single opinion
occurs (the radius of the hyperspace is twice its diameter which is equal to 14, the
maximum distance in the hyperspace).
– Between and a similar convergence is observed for more than 99.5
per cent of the agents with the exception of a few clustered or isolated opinions distant
from the main peak by roughly 7.
– For , one observes from 2 to seven significant peaks (with a population
larger than 1 per cent) plus some isolated opinions.
– For a large number (around 500) of small clusters is observed (The num-
ber of opinions is still smaller than the maximum number of opinions distant by 2).
The same kind of results is obtained with larger values of : two regimes, uni-
formity of opinions for larger values and extreme diversity for smaller values are
separated by one value for which a small number of clusters is observed (e.g for
, , seems to scale in proportion to ).
Since there is no a priori reference opinion as in the previous cases of continuous
opinions, information about the repartition of opinions is obtained from the histogram
of distances among couples of opinions represented on figure 8. The results of the two
next figures were obtained by averaging over 200 samples.
After all agents have been involved in 1000 possible exchanges of opinions on av-
erage, most pairs of opinions are different. One important result is that polarisation of
opinions (opposite vectors of opinions) is not observed. The clustering process rather
results in orthogonalisation of opinions with an average distance around 6: opinion
vectors have no correlation, positive or negative, whatsoever. Similar results were
observed concerning distances of binary strategies in the minority game [MAR 97].

10 Applications of Simulation to Social Sciences
0
20000
40000
60000
80000
100000
120000
0 2 4 6 8 10 12
t=500
t=15 000
Figure 8. Histogram of distances in vector opinions for agents and thus
500 000 pairs of agents. ( ), times are given in 1000 iterations units.
We were also interested in the populations of the different clusters. Figure 9 repre-
sent these populations at equilibrium (iteration time was 12 000 000) in a log-log plot
according to their rank-order of size. No scaling law is obvious from these plots, but
we observe the strong qualitative difference in decay rates for various thresholds .
0.1
1
10
100
1000
1 3 5 10 30
d=2
d=3
d=4
Figure 9. Log-log plot of average populations of clusters of opinions arranged ac-
cording to decreasing order. for agents ( ).

Sociology 11
5. Conclusions
We thus observe than when opinion exchange is limited by similarity of opin-
ions among agents, the dynamics yield isolated clusters among initially randomly dis-
tributed opinions. Exchange finally only occurs inside clusters as a the result of the ex-
change dynamics; initially all agents were communicating either directly or indirectly
through several directly connected agents. The concertation process as described here
is sufficient to ensure clustering even in the absence of difference in private interests
or in experience about the external world.
We have studied three very basic models and observed the same clustering be-
haviour, at least for some parameter regimes, which make us believe that the observed
clustering is robust and should be observed in more complicated models, not to men-
tion political life! Many variations and extensions can be proposed, including of
course further opinion selection according to experience with some external world
(“social reinforcement learning”). An interesting extension would be a kind of histor-
ical perspective where subjects (or problems) would appear one after the other: posi-
tion and discussions concerning an entirely new problem would then be conditioned
by the clustering resulting from previous problems.
Acknowledgments: We thank Jean Pierre Nadal and John Padgett and themembers
of the IMAGES FAIR project, Edmund Chattoe, Nils Ferrand and Nigel Gilbert for
helpful discussions. This study has been carried out with financial support from the
Commission of the EuropeanCommunities, Agriculture and Fisheries (FAIR) Specific
RTD program, CT96-2092, ”Improving Agri-Environmental Policies : A Simulation
Approach to the Role of the Cognitive Properties of Farmers and Institutions”. It does
not necessarily reflect its views and in no way anticipates the Commission’s future
policy in this area.
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12 Applications of Simulation to Social Sciences
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