A Logical Approach to Extreme Opinion Diffusion
Enzo Battistella and Laurence Cholvy
ONERA, 2 avenue Edouard Belin, 31055 Toulouse, France
Keywords:
Opinion Diffusion, Logic, Modelisation, Simulation.
Abstract:
This paper focuses on diffusion of extreme opinions among agents which influence each other. In this work,
opinions are modeled as formulas of the propositional logic or equivalently, as sets of propositional interpreta-
tions. Moreover, we assume that any agent changes its opinion by merging the opinions of its influencers, from
most to least influential. We propose a first definition of extreme opinions and extremism. We then consider
degrees of extremism. Formal studies of these definitions are made as well as simulations.
1 INTRODUCTION
Understanding the dynamics of the diffusion of opin-
ions and especially of extremism is a tremendous
question in Multi-Agent System and Artificial Intel-
ligence communities. See for instance (Crawford
et al., 2013; Christoff and Hansen, 2015; Cholvy,
2016; Grandi et al., 2015; Hafizoglu and Sen, 2012;
Jager and Amblard, 2004; Tsang and Larson, 2014;
Christoff and Grossi, 2017) for the study of opinion
diffusion and (Chau et al., 2014; Deffuant et al., 2002;
Sureda et al., 2017) for the study of extremism diffu-
sion.
Opinions are usually represented by a single real
value between 0 (or -1) and 1 corresponding to the
adequacy to a given affirmation: the closer to 1 an
agent’s opinion is, the more this agent agrees with the
affirmation. However some works, mainly in Artifi-
cial Intelligence community, represent an opinion by
a single binary vector ((Grandi et al., 2015; Christoff
and Grossi, 2017)) corresponding to adequacies to
several affirmations. For instance, if the two affirma-
tions are “Canada will host Winter Olympics in 2026”
and “there will be acroski trials”, then the vector (1, 0)
represents the opinion of an agent which thinks that
Canada will host Winter Olympics in 2026 but that
there will not be acroski trials.
In the present paper, we consider the model of
Influence-Based Opinion Diffusion Structure (IODS)
defined in (Cholvy, 2016) in which opinions are rep-
resented by propositional logic formulas or, equiv-
alently, sets of binary vectors. There, the set
{(1, 1), (0, 1), (0, 0)} represents the opinion of an
agent which thinks that, if the hosting country is
Canada then there will be acroski trials.
As for extremism, when opinions are represented
by a single real value, it is obviously defined by hav-
ing an opinion which is close to 0 (or -1) or to 1.
Moreover, in such models, agents can easily be clas-
sified from most to least extremist. But for IODS, this
is not so obvious and definitions of extreme opinions
and extremists have to be proposed. This is the aim of
the present paper.
A difficulty we will face is to define structures of
influence that are relevant in some real cases. This dif-
ficult task, aiming at characterizing social networks,
has been paid a huge attention to. One of the initiators
of this domain was Milgram (Milgram, 1967) who
proposed the theory of six degrees of separation or
“small world” phenomenon. In response to this the-
ory, several authors in graph theory defined models
to describe this kind of networks (Easley and Klein-
berg, 2010; Prettejohn et al., 2002; Watts and Stro-
gatz, 1998; Watts, 1999). The difficulty dwells on
adapting these models to IODS.
This paper is organized as follows.
Section 2 recalls the definition of IODS. In section
3, we list some properties of extremism diffusion in-
spired from sociology. In section 4, we define extrem-
ism so that extremists are people with rather strong
opinions. Then, we study some properties about this
definition. In section 5, we introduce degrees of ex-
tremism and we study some properties of this defi-
nition. In section 6, we present and comment several
simulations. Finally, section 7 presents some perspec-
tives of this work.
Battistella, E. and Cholvy, L.
A Logical Approach to Extreme Opinion Diffusion.
DOI: 10.5220/0006530900170028
In Proceedings of the 10th International Conference on Agents and Artificial Intelligence (ICAART 2018) - Volume 2, pages 17-28
ISBN: 978-989-758-275-2
Copyright © 2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
17
2 THE UNDERLYING DIFFUSION
MODEL
We consider a finite propositional language L. The set
of its interpretations is denoted 2
L
. If ϕ is a formula
of L, Mod(ϕ) denotes the set of models of ϕ i.e., the
set of interpretations in which ϕ is true. A multi-set
of formulas {ϕ
1
, ..., ϕ
n
} equipped with a total order
s.t. ϕ
i
ϕ
i+1
(i = 1...n 1) is called an ordered
multi-set of formulas and denoted ϕ
1
... ϕ
n
.
Definition 1 (Importance-Based Merging Operator).
An Importance-Based Merging Operator is a function
which associates a formula µ and a non-empty or-
dered multi-set of consistent formulas ϕ
1
... ϕ
n
with a formula denoted
µ
(ϕ
1
... ϕ
n
) so that:
Mod(
µ
(ϕ
1
... ϕ
n
)) = Min
d,ϕ
1
...ϕ
n
Mod(µ)
with:
w
d,ϕ
1
...ϕ
n
w
0
iff [D(w, ϕ
1
), ..., D(w, ϕ
n
)]
lex
[D(w
0
, ϕ
1
), ..., D(w
0
, ϕ
n
)]
[D(w, ϕ
1
), ..., D(w, ϕ
n
)] is a vector which k
th
ele-
ment is D(w, ϕ
k
)
D(w, ϕ) = min
w
0
Mod(ϕ)
d(w, w
0
), w and w
0
being
two interpretations and d is a pseudo-distance
between interpretations (ie., ww
0
d(w, w
0
) =
d(w
0
, w) and d(w, w
0
) = 0 = w = w
0
).
lex
is a lexicographic comparison of vectors
of reals defined by: [v
1
, ..., v
n
]
lex
[v
0
1
, ..., v
0
n
]
iff (i) k v
k
= v
0
k
or (ii) k v
k
< v
0
k
and j <
k v
j
= v
0
j
Some simple distances d which can be used for
instantiating the previous definition are: d
D
, the dras-
tic distance, (d
D
(w, w
0
) = 0 iff w = w
0
, 1 otherwise);
d
H
, the Hamming distance (d
H
(w, w
0
) = m iff w and
w
0
differ on m propositional letters).
Definition 2 (Influence-Based Opinion Diffusion
Structure ). An Influence-Based Opinion Diffusion
Structure (IODS) is a quadruplet S = (A,µ, B, In f )
where:
A = {1, ..., n} is a finite set of agents.
µ is a consistent formula of L.
B is a function which associates any agent i of A
with a consistent formula of L denoted for short
B
i
such that B
i
|= µ.
In f is a function which associates any agent i
of A with a non-empty set of agents {i
1
, ..., i
n
i
}
equipped with a total order
i
s.t. i
k
i
i
k+1
for
k = 1...(n
i
1). The agents of In f (i) will be called
influencers of i. The influencer i
1
will be called
the first influencer or the main influencer of i. For
short, we denote In f (i) = {i
1
i
...
i
i
n
i
}.
A is the finite set of agents. The formula µ rep-
resents the information which is true in the world. It
is called integrity constraint. For any agent i, the for-
mula B
i
represents its initial opinion. We assume that
agents are rational and thus that B
i
is consistent and
satisfies the integrity constraint µ. In f (i) is the non-
empty set of agents which influence agent i. These
influencers are ranked so that j
i
k means that, ac-
cording to i, its own opinion is more influenced by js
opinion than by ks opinion.
Definition 3 (Influence-Based Opinion Sequence).
Let S = (A, µ, B, In f ) be an IODS and i A with
In f (i) = {i
1
i
...
i
i
n
i
}. The Influence-Based Opin-
ion Sequence of i, denoted (B
s
i
)
sN
, is defined by:
(i) B
0
i
= B
i
(ii) s > 0, B
s
i
=
µ
(B
s1
i
1
... B
s1
i
n
i
)
The Influence-Based Opinion Sequence (or Opin-
ion Sequence for short) of agent i, (B
s
i
)
sN
, represents
the history of is opinion evolution. This evolution is
done according to operator
µ
: is opinion at step s is
the result of
µ
applied to the ordered multi-set of its
influencers’ opinions: B
s1
i
1
... B
s1
i
n
i
.
Example 1. Consider a language with propositional
letters a and b. Let S = (A, µ, B, In f ) be an IODS
with: A = {1, 2, 3}, µ is a tautology, B
1
= ¬a, B
2
=
a b, B
3
= ¬b, In f
1
= {1}, In f
2
= {2
2
1}, In f
3
=
{3
3
2}. Table 1 shows the evolution of the agents
opinions.
Table 1: Opinion evolution in example 1.
s = 0 s = 1 s 2
i = 1 ¬a ¬a ¬a
i = 2 a b ¬a b ¬a b
i = 3 ¬b a ¬b a ¬b
Finally, let us recall three definitions which will
be useful in the rest of the paper.
Definition 4 (Self-confident agent). Let
S = (A, µ, B, In f ) and i
1
A. i
1
is self-confident
iff In f (i
1
) = {i
1
i
i
2
i
...
i
i
n
i
} with n
i
1.
Definition 5 (Dogmatic agent). Let S = (A, µ, B, In f )
and i A. i dogmatic iff In f (i) = {i}.
Definition 6 (Sphere of Influence of an Agent). Let
S = (A, µ, B, In f ) and i A. The sphere of influence
of i is defined by: Sphere(i) =
S
k1
Sphere
k
(i) with
Sphere
1
(i) = { j
1
: In f ( j
1
) = {i ...}}
Sphere
k
(i) = { j
k
: In f ( j
k
) = { j
k1
...} and j
k1
Sphere
k1
(i)}
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
18
3 PROPERTIES OF EXTREMISM
DIFFUSION
In this section, we recall some observations made
in the sociology litterature (Bronner, 2009), (Atran,
2015) about extremism diffusion and we translate
them in our setting as properties. Notice that these
observations are not supposed to be general rules but
only tendancies. In the two following sections, we
will check if these properties are satisfied or not by
the definitions we will introduce.
The first observation about extremism is that in
general extremist people are close-minded, sure of
their opinions and listen exclusively people with sim-
ilar beliefs. But, on an other hand, extremism can and
often arise in groups of friends with moderate opin-
ions. Atran highlighted the fact that extremist people
have often been introduced to extremism by a social
connection.
We have retained the following five properties for
our study. The first property states that if all the in-
fluencers of an agent are extremist, then after updat-
ing its opinion, the agent is extremist. The second
states that if all the influencers of an agent are moder-
ate (i.e. not extremist), then after updating its opinion
the agent is moderate. The third one states that ex-
tremist agents are more certain of their opinions than
moderate ones. The fourth states that extremist agents
are more influential in the sense that an agent influ-
enced by extremist agents should be ”more prone” to
become extremist than moderate. The fifth property,
called persistence of discredited beliefs, states that an
opinion will persist even if the evidence that made the
opinion emerge is removed.
S being the IODS (A, µ, B, In f ), these properties
are formalized as follows:
Definition 7.
(P1) S satisfies (P1) iff i A, t N, if j
In f (i), j is extremist at step t, then i is extremist at
step t + 1.
(P2) S satisfies (P2) iff i A, t N, if j
In f (i), j is moderate at step t, then i is moderate
at step t + 1.
(P3) S satisfies (P3) iff i A, j A, t N,
if i is extremist at t and j is moderate at t then
| Mod(B
t
i
) |<| Mod(B
t
j
) |.
(P4) S satisfies (P4) iff t N, there are more
agents i which are extremist at step t + 1 and
which main influencers were extremist at step t
than agents i which are moderate at step t + 1 and
which main influencers were extremist at step t.
(P5) S satisfies (P5) iff t N, i A, if i is ex-
tremist at step t + 1, its main influencer was ex-
tremist at step t and moderate at step t + 1, then i
remains extremist at step t + 2.
The previous definitions are applicable only if we
have a binary definition of extremism stating whether
a given agent is extremist or not. In case we have a
definition stating only if an agent i is more extremist
than an other agent j at some step t, denoted d
t
extr
(i)
d
t
extr
( j), we adapt the properties as follows:
The first property states that if all the influencers
of an agent are very extremist at some step, then after
the opinion updating the agent should be very extrem-
ist. The second one says symmetrically that if all the
influencers of an agent are very moderate, then after
the opinion updating the agent should be very moder-
ate. The third expresses the fact that an agent should
be more certain than less extremist ones in the sense
of an agent’s opinion should have less models than
the ones of less extremist agents. The fourth states
that the extremist agents should be more influential in
the sense that any agent influenced by more extremist
agents should become more extremist. The fifth prop-
erty, called Persistence of discredited beliefs, states
that an opinion will persist even if the evidence that
made the opinion emerge is removed. At these prop-
erties we can add two reality constraints stating that
an agent cannot have its degree of extremism increase
”too much” above (respectively decrease ”too much”
below) the maximal degree of extremism of its influ-
encers.
S being the IODS (A, µ, B, In f ) and N
being the
set of integers strictly greater than 0, these properties
are formalized as follows:
Definition 8.
(Q1) Given L
extr
N
. S satisfies (Q1) iff i
A, t N, if j In f (i), d
t
extr
( j) L
extr
, then
d
t+1
extr
(i) L
extr
.
(Q2) Given L
mod
N
. S satisfies (Q2) iff i
A, t N, if j In f (i), d
t
extr
( j) L
mod
, then
d
t+1
extr
(i) L
mod
.
(Q3) S satisfies (Q3) iff i, j A, t N, if
d
t
extr
(i) d
t
extr
( j) then | Mod(B
t
i
) |≥| Mod(B
t
j
) |.
(Q4) S satisfies (Q4) iff i A, t N, if j
In f (i), d
t
extr
(i) d
t
extr
( j) then d
t
extr
(i) d
t+1
extr
(i).
(Q5) Given L
extr
N
. S satisfies (Q5) iff i A
such that its first influencer j satisfies d
t
extr
( j)
L
extr
, d
t+1
extr
( j) L
extr
and d
t+1
extr
(i) L
extr
then
d
t+2
extr
(i) L
extr
.
(Q6) Given Lim N. S satisfies (Q5) iff i A,
t N, d
t+1
extr
(i) max
jIn f (i)
d
t
extr
( j) Lim .
(Q7) Given Lim N. S satisfies (Q5) iff i A,
t N, min
jIn f (i)
d
t+1
extr
( j) d
t
extr
(i) Lim .
A Logical Approach to Extreme Opinion Diffusion
19
4 EXTREME OPINIONS AND
EXTREMIST AGENTS
In this section, we consider that extreme opinions are
strong opinions i.e. formulas which have “few” mod-
els. Moreover, we adopt a binary definition of extrem-
ism by considering that people are divided into two
distinct groups: the extremist ones and the moderate
ones.
4.1 Definitions
Definition 9 (Extreme Opinions). Let R be a given
integer closer to 1 than to 2
|L|
. An opinion o is extreme
iff 1 ≤| Mod(o) |≤ R.
The choice of the threshold R will depend on the
application. But R has to be much smaller than the
number of interpretations in the language. Moreover,
inconsistent opinions are not considered as extreme.
For instance, consider that the two letters of the lan-
guage are a, b. If R = 1 then a b, a ¬b, ¬a b,
¬a ¬b are the extreme opinions.
As for extremist agents, we define them as agents
which opinions are extreme. Moreover, an agent is
moderate when it is not extremist. This leads to the
following definition:
Definition 10 (Extremist, Moderate). At step s, agent
i is extremist iff B
s
i
is an extreme opinion. Otherwise
it is moderate.
Example 2. Consider two propositional letters a, b
and assume that at a given step t agents opinions are:
B
t
i
= a b, B
t
j
= a, B
t
k
= a b. If R = 1 then only k is
extremist. If R = 2 then j and k are extremist.
4.2 Formal Analysis
In the following, we prove some results on extrem-
ism diffusion and in particular we focus on properties
described in section 3.
The following proposition gives a description of
extremist agents from a syntactical point of view.
Proposition 1. Assume that the propositional letters
are a
1
, ..., a
n
. Consider an IODS S = (A, µ, B, In f ).
Agent i is extremist at step t iff B
t
i
W
N
k=1
l
1,k
... l
n,k
with k [1, N], p [1, n], l
p,k
a literal correspond-
ing to a
p
(l
p,k
= a
p
or l
p,k
= ¬a
p
) and N R.
Thus, the opinion of an extremist agent is equiva-
lent to a disjunction of less than R conjunctions of all
the literals.
The following proposition states that an agent
which main influencer is extremist at some step will
be extremist at the next step.
Proposition 2. In an IODS S = (A, µ, B, In f ), for i
A with In f (i) = { j
i
...}, for t N, if j is extremist
at step t, then i is extremist at step t + 1.
As a consequence, a self-confident agent (see def-
inition 4) which is extremist at some step will remain
extremist ever after:
Proposition 3. In an IODS S = (A, µ, B, In f ), for i
A with In f (i) = {i
i
...}, if t N, i is extremist at t,
then s t, i is extremist at s.
The following proposition states that an agent
which k-th influencer has an opinion consistent with
the merging of the ones of the previous influencers at
step t and which k-th influencer is extremist at step t
will be extremist at step t + 1.
Proposition 4. In an IODS S = (A, µ, B, In f ), for i
A with In f (i) = { j
1
i
...
i
j
k
i
...
i
j
n
}, for t N,
if
µ
(B
t
j
1
i
...
i
B
t
j
k1
) B
t
j
k
is consistent and j
k
is
extremist at step t, then i is extremist at step t + 1.
More generally, an agent will be extremist at step
t + 1 iff for some k, the merging of the k first influ-
encers’ opinions at step t has less than R models.
Proposition 5. In an IODS S = (A, µ, B, In f ), for i
A with In f (i) = { j
1
i
...
i
j
k
i
...}. Let t N.
If k N, such that | Mod(
µ
(B
t
j
1
... B
t
j
k
)) |≤ R
then, i is extremist at t + 1. Otherwise, it is moderate
at step t + 1.
The following proposition states that a self-
confident extremist agent spreads extremism in its
sphere of influence.
Proposition 6. Let S = (A, µ, B, In f ) an IODS and i
A extremist at step t with In f (i) = {i
i
...}. t
0
t,
s t
0
, j Sphere(i) j is extremist at step s.
The last proposition shows which are the proper-
ties formalized in section 3 that are satisfied.
Proposition 7. Any IODS satisfies (P1), (P3) and
(P4). (P2) and (P5) are not always satisfied.
5 EXTENSION TO DEGREES OF
EXTREMISM
In this section, we extend the notion of extreme opin-
ions according to a more relative point of view.
5.1 Definitions
Definition 11 (Degree of Extremism of Opinions).
An opinion ϕ is extreme at degree
2
N
−|Mod(ϕ)|
2
N
1
and
moderate at degree 1
2
N
−|Mod(ϕ)|
2
N
1
with N the num-
ber of propositional letters.
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
20
Example 3. For instance, if we consider the letters a
and b then a b is extreme at degree 1 and a ¬b is
extreme at degree 1/3.
Definition 12 (Degree of Extremism of Agents). At
step s, the degree of extremism of agent i is the degree
at which B
s
i
is extreme. It is denoted d
s
ext
(i). Similarly,
at step s, the degree of moderation of agent i is the
degree at which B
s
i
is moderate. It is denoted d
s
mod
(i).
Definition 13. At step s, agent i is more extremist than
agent j iff d
s
ext
(i) d
s
ext
( j). At step s, agent i is more
moderate than agent j iff d
s
mod
(i) d
s
mod
( j).
Example 4. An agent which current opinion is a b
is more extremist (i.e. less moderate) than an agent
which current opinion is a b.
5.2 Formal Analysis
This first proposition states that an agent is more ex-
tremist at step t (t > 0) than its first influencer at step
t 1.
Proposition 8. Consider an IODS S = (A, µ, B, In f ),
and an agent i A with In f (i) = { j
i
...}. Then
t > 0, d
t
ext
(i) d
t1
ext
( j).
In particular, the degree of extremism of a self-
confident agent can only increase.
The following proposition is a generalization of
the previous one. It states that an agent is more ex-
tremist at some step t than its k-th influencer at the
previous step if the opinions of this influencer is con-
sistent with the merging of the opinions of the previ-
ous ones.
Proposition 9. Consider an IODS S = (A, µ, B, In f )
and an agent i with In f (i) = { j
1
i
...
i
j
n
}. Then,
t > 0, k [1, n], if
µ
(B
j
1
... B
j
k
) B
t
j
k
is
consistent and j
k
is an extremist agent at step t then
d
t+1
ext
(i) d
t
ext
( j
k
)
The following proposition states that an agent will
be extremist at degree at least d at step t + 1 if one
of its influencers j has an opinion at step t such that
during the opinion updating process, the merging of
the previous influencers’ opinions has less than 2
N
(2
N
1)d models at minimal distance from B
t
j
with N
the number of letters in the language.
Proposition 10. In an IODS S = (A, µ, B, In f ) with
A = {a
1
, ..., a
N
}, for i A, t N, with In f (i) = { j
1
i
...
i
j
n
}, i is of degree of extremist at least d at step
t + 1 iff k [1, n], | Mod(
µ
(B
t
j
1
... B
t
j
k
)) |≤
2
N
(2
N
1)d.
The following proposition states that a self-
confident extremist agent spreads extremism in its
sphere of influence.
Proposition 11. Let S = (A, µ, B, In f ) an IODS and
i A with In f (i) = {i
i
...}. t N, t
0
t, s t
0
,
j Sphere(i) d
s
ext
( j) d
t
ext
(i).
The last proposition shows which are the proposi-
tions formalized in section 3 that are satisfied.
Proposition 12. Any IODS satisfies (Q1), (Q3), (Q4)
and (Q7). (Q2), (Q5) and (Q6) are not always satis-
fied.
6 EXPERIMENTS
In this section, we focus on simulations realized with
Netlogo. First, we address the question of generating
IODS corresponding to real social networks. For that,
we review some propositions made in graph theory
during the last decades. Then, we adapt them to our
context by using a notion of distances between opin-
ions.
6.1 Graph Theory Bases
One of the most used models of graph is the one of
Erd
¨
os-R
´
enyi. It is a model of random graph (see
(Easley and Kleinberg, 2010)).
Definition 14 (Erd
¨
os-R
´
enyi Graph). Given a number
of nodes n and an integer m. An Erd
¨
os-R
´
enyi Graph
is any graph obtained by selecting randomly m edges
among the 2
n
possible ones.
Another model of graph that is widely used is the
model of Watts-Strogatz. This model has been made
to describe the phenomenon of Small-World or ”six
degrees of separation” highlighted by Milgram (Mil-
gram, 1967). This psychologist established through
an experiment the theory that a message can be trans-
mitted from one person to one another by passing by
an average of six friends. The Small-World theory is
commonly formalized (Easley and Kleinberg, 2010;
Prettejohn et al., 2002; Watts and Strogatz, 1998) as
follows:
Definition 15 (Small-World). A graph G is said
Small-World if it satisfies:
(i) G is connected.
(ii) G is sparse: the average degree of the nodes k is
low compared to the number of nodes n, k n.
(iii) G is decentralized: the maximal degree of the
nodes k
max
is low compared to the number of
nodes n, k
max
n.
(iv) The characteristic path L (the average number
of nodes traversed by a short path between two
nodes) is close to the one of a random graph with
A Logical Approach to Extreme Opinion Diffusion
21
the same number of nodes n and the same average
degree k, L L
random
ln(n)
ln(k)
.
(v) The clustering coefficient C (the probability that
two nodes i and j are connected given that they
share a common neighbor) is high compared to
the one of a random graph with the same number
of nodes n and the same average degree k, C
C
random
k
n
.
One can notice that Erd
¨
os-R
´
enyi graphs as ran-
dom graphs have low characteristic paths by defini-
tion.
The following model, from (Easley and Klein-
berg, 2010) and adapted from a model generally at-
tributed to Watts and Strogatz, define Small-World
graphs:
Definition 16 (Rank-Based Friendship Graph).
Given a number of nodes n, a threshold r, an expo-
nent q and a dimension d, the nodes are randomly
distributed in a space of dimension d. Rank-Based
Friendship Graph is obtained by going as follows:
For each node i, we rank the other nodes accord-
ing to their distances to i and we break ties with a
chosen method. There will be an edge from a node
j to the node i with probability
1
Z.rank
i
( j)
q
, rank
i
( j)
being the rank of j in i’s neighbors and Z a coefficient
of normalization, Z =
n
i=1
1
rank
i
( j)
q
=
n
i=1
1
i
q
.
6.2 Distances between Opinions
Considering a pseudo-distance d between two inter-
pretations, there are many ways to characterize how
close two opinions are. (Eiter and Mannila, 1997)
highlights several pseudo-distances between sets of
points, we adapt some to opinions as follows:
Definition 17 (Pseudo-Distances on opinions). Let o
1
and o
2
be two propositional formulas.
Sum of minimum distances: d
summin
(o
1
, o
2
) =
1
2
(
wMod(o
1
)
D(w, o
2
) +
wMod(o
2
)
D(w, o
1
))
Hausdorff distance: d
Hau
(o
1
, o
2
) =
max(max
wMod(o
1
)
D(w, o
2
), max
wMod(o
2
)
D(w, o
1
))
Surjection distance d
S
(o
1
, o
2
) =
min
ηF
(o
1
,o
2
)
(w
1
,w
2
)η
d(w
1
, w
2
) with F
(o
1
,o
2
)
the set of surjections from the larger set between
Mod(o
1
) and Mod(o
2
) on the other.
Link distance: d
l
(o
1
, o
2
) =
min
RL
(o
1
,o
2
)
(w
1
,w
2
)R
d(w
1
, w
2
) with L
(o
1
,o
2
)
the
set of linkings. A linking R Mod(o
1
)×Mod(o
2
)
satisfies w
1
Mod(o
1
), w
2
Mod(o
2
),
(w
1
, w
2
) R and w
2
Mod(o
2
),
w
1
Mod(o
1
), (w
1
, w
2
) R.
But, as we consider relations of influence that
are not symmetric, dropping the property of symme-
try and considering weaker functions than pseudo-
distances will also give interesting IODS to study. We
thus define Difference functions as follows:
Definition 18 (Difference of Opinions).
d
max
(o
1
, o
2
) = max
w
1
Mod(o
1
),w
2
Mod(o
2
)
d(w
1
, w
2
)
d
min
(o
1
, o
2
) = min
w
1
Mod(o
1
),w
2
Mod(o
2
)
d(w
1
, w
2
)
d
maxmin
(o
1
, o
2
) =
max
w
1
Mod(o
1
)
min
w
2
Mod(o
2
)
d(w
1
, w
2
)
d
sum
(o
1
, o
2
) =
w
1
Mod(o
1
),w
2
Mod(o
2
)
d(w
1
, w
2
)
In the following we will indifferently refer to the
previously defined functions as distances by lack of a
better term.
6.3 Models
Here we adapt the previous models of graphs to IODS
and explain how we construct them for our simula-
tions. In the following we take an integrity constraint
being a tautology.
The first model we adapt is the one of the random
graph defined by Erd
¨
os and R
´
enyi. The following
definition shows how we construct Erd
¨
os and R
´
enyi
IODS. Notice that we add a parameter, the number of
self-confident agents,which is an interesting variable
to study.
Definition 19 (Erd
¨
os-R
´
enyi-Based IODS). Given the
parameters num-letters, num-nodes, num-links and
num-self-confident, the IODS is constructed as fol-
lows:
We begin by creating num-nodes agents, each of
them has a random opinion in a language of num-
letters letters. Then, we create num-links relations
of influence by choosing randomly an influencer and
an influenced agent (potentially the same). The influ-
encers are ordered according to the order of creation
of the relation of influence, the sooner a relation of
influence would have been created the more influenc-
ing it is. Finally, each agent with no influencers will
become dogmatic and, if necessary, we add relations
of self-influence until we have num-self-confident self-
confident agents (dogmatic agents included). We pick
randomly an agent and if it does not already influ-
ence himself we make it self-confident by putting it as
its main influencer (the order of the other influencers
remains unchanged).
This second model adapts the model of Rank-
Based Friendship by considering a distance between
opinions instead of a physical distance as for the
graph model.
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
22
Definition 20 (Rank-Based Influenceship IODS).
Given the parameters num-letters, num-nodes,
opinions-distance, q and num-self-confident, the
IODS is constructed as follows:
We begin by creating num-nodes agents, each of
them has a random opinion in a language of num-
letters letters. Then, we fill a matrix with the distances
between every couple of agents according to the dis-
tance between their opinions and computed with the
distance opinions-distance. For each agent i we have
a list l
i
of all the agents (i included) sorted accord-
ing to their distances to i. If two agents j
1
and j
2
are at the same distance of i, then the tie will be ran-
domly solved. Each agent j will be an influencer of i
with probability
1
Z.rank
i
( j)
q
, rank
i
( j) being the rank
of j in l
i
and Z being a coefficient of normalization,
Z =
numnodes
i=1
1
rank
i
( j)
q
=
numnodes
i=1
1
i
q
. The in-
fluencers of i are ordered as in l
i
. Finally, if neces-
sary, we add relations of self-influence such as we
have num-self-confident self-confident agents (dog-
matic agents included). We pick randomly an agent
and if it is not already self-confident we make it so by
putting it as its main influencer (the order of the other
influencers remains unchanged).
The third model is a variant of the previous one,
here an agent will be influenced by the m agents that
have the closest opinions from its own one for a given
integer m.
Definition 21 (Deterministic Rank-Based Influence-
ship IODS). Given the parameters num-letters, num-
nodes, opinions-distance, m and num-self-confident,
the IODS is constructed as follows:
We begin by creating num-nodes agents, each of
them has a random opinion in a language of num-
letters letters. Then, we fill a matrix with the distances
between every couple of agents according to the dis-
tance between their opinions and computed with the
distance opinions-distance. For each agent i, we con-
serve the m closest agents to i to be its influencers.
If two agents j
1
and j
2
are at the same distance of
i, then the tie will be randomly solved. The influ-
encers of i are ordered according to their distances
to i. Finally, we add relations of self-influence such
as we have num-self-confident self-confident agents
(dogmatic agents included). We pick randomly an
agent and if it is not already self-confident we make
it so by putting it as its main influencer (the order of
the other influencers remains unchanged).
The fourth model is a generalization of the Rank-
Based Influenceship in which we have in addition
to the distance between opinions a physical distance
along a circle. The influencers of an agent i are or-
dered according to the distance between their opin-
ions and the one of i.
Definition 22 (Opinions and Physical Rank-Based
Influenceship IODS). Given the parameters num-
letters, num-nodes, opinions-distance, r, q and num-
self-confident, the IODS is constructed as follows:
We begin by creating num-nodes agents, each of
them has a random opinion in a language of num-
letters letters. We fill a matrix with the distances be-
tween every couple of agents according to the dis-
tance between their opinions and computed with the
distance opinions-distance. For each agent i we have
a list l
i
of all the agents (i included) sorted accord-
ing to their distances to i. If two agents j
1
and j
2
are at the same distance of i, then the tie will be ran-
domly solved. Each agent j will be an influencer of i
with probability
1
Z.rank
i
( j)
q
, rank
i
( j) being the rank
of j in l
i
and Z a coefficient of normalization Z =
numnodes
i=1
1
rank
i
( j)
q
=
numnodes
i=1
1
i
q
. At the previ-
ous influencers we add influencers that are physically
close. Indeed, all the agents will be placed on a cir-
cle. The agents that are separated on the circle from
an agent i by less than r agents will influence i. The
influencers of i are ordered as in l
i
(according to the
distance between opinions). Finally, if necessary, we
add relations of self-influence such as we have num-
self-confident self-confident agents (dogmatic agents
included). We pick randomly an agent and if it is not
already self-confident we make it so by putting it as
its main influencer (the order of the other influencers
remains unchanged).
To study and compare the results between the dif-
ferent models and distances, we carried out several
simulations with the same settings. Furthermore, in
order to do comparable and reproducible experiments
we chose some values of seeds for the random oper-
ations in Netlogo. Seeds allow to have the same re-
sults in the same order for random operations when
we repeat the simulations. The values we study are
the number of extremist agents for R = 1, the average
number of models per agents which is proportional to
the average degree of moderation and the number of
dogmatic agents.
In the simulations we present here, we have
taken the following values: seed {0, 100, 200}, num-
letters {3, 4, 5, 6}, num-nodes {10, 60, 110, 160, 210},
num-self-confident {0, 50, 100, 150, 200, 210}. For
the three models using ranks we tested d
summin
, d
Hau
,
d
max
, d
min
, d
maxmin
, d
sum
. with the drastic pseudo-
distance between interpretations. For the Erd
¨
os-
R
´
enyi-Based model we took num-links varying from
10 to 2000 with an increment of 50. For the Rank-
A Logical Approach to Extreme Opinion Diffusion
23
Average Number of Models per Agents Proportion of extremist agents (blue) and dogmatic agents (red)
Figure 1: Example: Erd
¨
os-R
´
enyi-Based model seed=0, num-nodes =210, num-self-confident=0.
Based Influenceship and the Opinions and Physical
Rank-Based Influenceship models we took the values
{1, 2, 3, 4, 5} for q, for the Deterministic Rank-Based
Influenceship model we took m in {1, 2, 3, 4} and
for the Opinions and Physical Rank-Based Influence-
ship model we took r in {1, 2, 3, 4, 5}. The pseudo-
distance used to compute the Importance-based Merg-
ing operator is the drastic one.
First of all, with the drastic pseudo-distance, we
can notice that the different distances we used have
particular behaviors. d
summin
(o
1
, o
2
) [0, 2
numletters
]
and d
summin
(o
1
, o
2
) = 0 iff o
1
o
2
. d
summin
favors
relations of influence between agents which opin-
ions have models very close according to D, in av-
erage. d
Hau
(o
1
, o
2
) is 0 or 1 and d
Hau
(o
1
, o
2
) = 0
iff o
1
o
2
. d
Hau
favors relations of influence be-
tween agents which opinions have no models very
far from one another. So, in the case of the dras-
tic pseudo-distance, it favors relations between agents
that have the same opinion. So, for the number of
letters and agents we will consider, as such a case
is unlikely the relations of influence will be mostly
random. d
max
(o
1
, o
2
) is 0 or 1 and d
max
(o
1
, o
2
) = 0
iff w, Mod(o
1
) = Mod(o
2
) = {w}. So, as hav-
ing two agents with only one model and the same
model is very unlikely for the number of letters and
agents we consider, d
max
favors random relations of
influence and it will be interesting to compare the re-
sults obtained with this distance and the ones obtained
with the other distances. d
min
(o
1
, o
2
) is 0 or 1 and
d
min
(o
1
, o
2
) = 0 iff w Mod(o
1
) Mod(o
2
). Then,
for a given agent i, d
min
favors relations of influence
that are from agents that share a model with is opin-
ion but that are otherwise random. d
maxmin
(o
1
, o
2
) is
0 or 1 and d
maxmin
(o
1
, o
2
) = 0 iff o
1
|= o
2
. So, d
maxmin
favors relations of influence from an agent i to an
agent j such that all the models of B
j
are models of
B
i
. d
sum
(o
1
, o
2
) [0, 2
numletters
] and d
sum
(o
1
, o
2
) = 0
iff Mod(o
1
) = Mod(o
2
) = {w}. d
sum
favors relations
of influence from agents which opinions have the less
models.
For the Erd
¨
os-R
´
enyi-Based model (see figure 1),
we have several peaks of the average number of mod-
els and of the number of dogmatic agents, corre-
sponding to having low num-links. Indeed, in these
cases, there are potentially more agents that are not
influenced by other agents and that keep their initial
opinions. Furthermore, we can see that the dogmatic
agents are almost the only agents that are not extrem-
ist and thus contribute the more to the average number
of models. So, the diffusion of extremism depends
a lot on the ratio between num-links and num-nodes,
the more there are relations of influence the more the
agents will become extremist. We can only notice that
the peaks of average number of models are higher and
higher according to the increasing of the number of
letters. Another experiment in which we took 200
agents and much more relations of influence (up to
7000) showed that for more than 2000 there are very
few simulations with non-extremist agents.
For the Rank-Based Influenceship model (see fig-
ure 2), we have several plateaus higher and higher
according to the increasing of the number of letters.
Furthermore, there are cases with very low numbers
of extremist agents and without very much dogmatic
agents. Then, we can notice that there are big dif-
ferences according to the distance we use. Indeed, a
thorougher analysis highlights that the biggest peaks
are with d
summin
and then with d
Haus
and d
maxmin
. d
max
and d
min
cause some lesser peaks when q gets bigger
(more than 3) and d
sum
causes very small peaks for
q = 5. For q = 1 almost all the agents are extremist
whatever the distance we use. It can be explained by
the fact that the lesser q is the more likely relations
of influence are to be created, furthermore for q high
enough the distance used matter less then even d
sum
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
24
Average Number of Models per Agents Proportion of extremist agents (blue) and dogmatic agents (red)
Figure 2: Example: Rank-Based Influenceship model seed=0, num-letters=5,num-nodes=210, num-self-confident=0.
Average Number of Models per Agents Proportion of extremist agents (blue) and dogmatic agents (red)
Figure 3: Example: Deterministic Rank-Based Influenceship model seed=0, num-nodes=210, num-self-confident=0.
that in the other cases spread extremism may be used
to create an IODS where they may remain some mod-
erate agents. But, according to (Easley and Klein-
berg, 2010) in the case of graphs, the Rank-Based
Friendship generates graphs the closest of reality for
q = 1. Furthermore, when the number of agents in-
creases, the average number of models decreases be-
cause more relations of influence may be created. One
can notice that with this model d
summin
and d
Haus
par-
ticularly favor moderation. So, having influencers
with opinions for which each model is close of one
of us model or for which each model is not far of any
of our model favor moderation. But, we can notice
that with d
Haus
agents are much less dogmatic than
with d
summin
.
For the Deterministic Rank-Based Influenceship
model (see figure 3), extremism spreads more and
more when m gets bigger. Moreover, this time there
is much more differences according to the distance
we used because the ranking is more important in
the choice of the influencers than before. Then, only
d
summin
keeps many non-extremist agents when m is
at its highest. Indeed, this distance characterizes the
best the similarity between opinions, the first agents in
the ranking of an agent i actually have opinions that
share many models with the one of i and it often is i
itself. Thus, when m = 1, we have almost only dog-
matic agents with d
summin
. When m gets higher than 3
only models with d
summin
keep moderate agents.
For the Opinions and Physical Rank-Based Influ-
enceship model (see figure 4), we have very few non-
extremist agents even with d
summin
and even less when
r increases. It is due to the fact that here there can-
not be any dogmatic agent (contrary to the case of
the Rank-Based Influenceship) and that an agent may
have influencers with very different opinions (con-
trary to the case of the Deterministic Rank-Based In-
fluenceship).
In all the simulations, the number of letters does
not affect the proportion of extremist agents. The
number of nodes affects the proportion of extrem-
ist agents for the Erd
¨
os-R
´
enyi-Based model because
of our definition of the model, in fact it is the ra-
tio between the number of agents and the number of
relations of influence that truly matters. It also has
an influence for Rank-Based Influenceship model and
the Opinions and Physical Rank-Based Influenceship
model because it increases the average number of in-
A Logical Approach to Extreme Opinion Diffusion
25
Average Number of Models per Agents Proportion of extremist agents
Figure 4: Example: Opinions and Physical Rank-Based Influenceship model seed=0, num-letters=5, num-nodes=210, num-
self-confident=0.
fluencers.
For summarizing, we have that for the models of
generation of IODS, the ones which spread extrem-
ism the less are the Rank-Based Influenceship when
q is very high and the Erd
¨
os-R
´
enyi-Based when num-
links is much lower than num-nodes. But, those mod-
els have many dogmatic agents, on the other hand,
the Deterministic Rank-Based Influenceship spreads
extremism very little with d
summin
and a small m and
without many dogmatic agents. For the distances, it
is d
summin
that spreads extremism the less because it
favors relations of influence from agents with opin-
ions sharing many models and it spreads extremism
less than d
max
(the random one). At the opposite, d
sum
spreads extremism very well by creating hubs, agents
with very few models that influence a lot of agents.
d
min
spreads extremism a little less because it is less
random, there is a constraint on one model. So, with
the Importance-Based Merging Operator, the extrem-
ism spreads very well when the most extremist agents
are very influential and much less when agents are in-
fluenced by agents with opinions similar to its own in
the sense of they share many models. So, what makes
that an agent remains moderate is the fact that he is in-
fluenced by agents which opinions share many mod-
els between them and that he does not have too many
influencers. Having many self-confident agents fa-
vor extremism spreading with the Erd
¨
os-R
´
enyi-Based
model as it increases the average number of influ-
encers but in the other models it favors moderation.
Indeed, in this case the agents keep opinions close to
their initial ones and so agents’ influencers keep close
opinions.
We can notice that, in every simulation, we
reached the convergence very quickly in general in
less than 5 updates.
It would have been interesting to test the mod-
els for much larger numbers of agents to increase the
probabilities we have deemed negligible in our study
of the distances for instance. Indeed, the Small-World
phenomenon is considered interesting for very large
number of nodes i.e. billions of nodes (see (Watts,
1999)) but the computation time that would be needed
only for models of thousands of agents is very impor-
tant.
Furthermore, other simulations with Hamming
pseudo distance both for the computation of the dis-
tances between opinions and the update of the opin-
ions gave similar results. Notwithstanding, extremism
spreads slightly much, in average 0.8 less models per
agents and 9% less extremist agents. This can mainly
be explain by the fact that Min
d
H
,ϕ
1
ϕ
2
Mod(µ) con-
tains generally less models than Min
d
D
,ϕ
1
ϕ
2
Mod(µ)
as the second one keeps all the models of ϕ
1
if ϕ
1
and ϕ
2
are inconsistent. The only type of IODS that
spreads less extremism in this case is the Rank-Based
Influenceship model, in average there are 2 more
models per agents and 10% less extremist agents.
But, it can be explained by the fact that there are
twice more (15% more) dogmatic agents, the ham-
ming pseudo-distance allows a more accurate rank-
ing of the agents and thus, it is less likely that agents
with very different opinions influence an agent. It ap-
pears that this accuracy is all the more significant that
the number of letters is important. However, the first
agents in the rankings do not change a lot, so the De-
terministic Rank-Based Influenceship model spreads
more extremism. Another noticeable difference are
for d
Hau
and d
summin
which spread extremism much
less in the three Rank-Based Influenceship models.
What we can notice is that the more relations of in-
fluence there are, the more extremism spreads. And,
the more influencers of agents have close opinions,
the less extremism spreads. This result can be in-
terpreted as follows: When someone makes its own
mind by taking into account the opinions of many
people it considers as reliable or experts on the matter
and with different opinions then, it will be very sure
ICAART 2018 - 10th International Conference on Agents and Artificial Intelligence
26
of its new opinion as it is a compromised between the
opinions of many experts. And so, this person will
become extremist according to our definition.
7 CONCLUDING REMARKS
This paper focused on modelling extreme opinion dif-
fusion when opinions are modelled as propositional
formulas. It can be extended according to several di-
rections.
First, we could add a dynamic aspect in the dif-
ferent types of IODS, by changing the relations of in-
fluence through time as it is often done in the usual
models (Crawford et al., 2013; Christoff and Hansen,
2015; Chau et al., 2014; Deffuant et al., 2002). It will
be especially interesting with the rank-based models
where the ranking of the influencers is based on the
distances between opinions. Since opinions change
through time these distances also change and comput-
ing new rankings could be done.
Another aspect that may be complexified is the
definition of extreme opinions. In the definitions we
considered here, the main parameter is the number
of the models of an opinion. They do not take into
account what opinions are about. For instance, ac-
cording to this model, the opinion using pesticides is
safe is as extreme as growing tomatoes is easy. Tak-
ing the domain into account would allow us to dis-
tinguish some sensitive letters and to use them for a
more refined definition of extremism. In the agricul-
ture domain, having a strong position towards pesti-
cides (pro or cons) is obviously more noticeable than
having a strong position towards tomatoes.
Similarly, we could consider a more complex def-
inition of extremism which would define extremist
agents as agents which opinions are close to some ref-
erential extreme opinions. In the agriculture example,
an agent which thinks that with caution, using pesti-
cides is safe is more extremist than an agent which
thinks with water, growing tomatoes is easy because
its opinion is not far from the sensitive opinion using
pesticides is safe.
ACKNOWLEDGEMENTS
We thank the anonymous reviewers whose comments
helped us to improve the paper.
REFERENCES
Atran, S. (2015). On youth, violent extremism and promot-
ing peace. https://www.unisa.edu.au/Global/EASS/
MnM/Publications/Address UN Security Council
Scott Atran.pdf.
Bronner, G. (2009). La pens
´
ee extr
ˆ
eme. Comment les
hommes ordinaires deviennent des fanatiques. Edi-
tions Deno
¨
el, Paris.
Chau, H., Wong, C., Chow, F., and Fung, C. (2014). Social
judgment theory based model on opinion formation,
polarization and evolution. Physica A, 415:133–140.
Cholvy, L. (2016). Diffusion of opinion and influence.
In Scalable Uncertainty Management - 10th Interna-
tional Conference, SUM 2016, Nice, France, Septem-
ber 21-23, 2016, Proceedings, pages 112–125.
Christoff, Z. and Grossi, D. (2017). Stability in binary opin-
ion diffusion. In Logic, Rationality, and Interaction
- 6th International Workshop, LORI 2017, Sapporo,
Japan, September 11-14, Proceedings, 2017, pages
166–180.
Christoff, Z. and Hansen, J. U. (2015). A logic for diffusion
in social networks. J. Applied Logic, 13(1):48–77.
Crawford, C., Brooks, L., and Sen, S. (2013). Opposites
repel: the effect of incorporating repulsion on opin-
ion dynamics in the bounded confidence model. In
International conference on Autonomous Agents and
Multi-Agent Systems, AAMAS ’13, Saint Paul, MN,
USA, May 6-10, 2013, pages 1225–1226.
Deffuant, G., Amblard, F., Weisbuch, G., and Faure, T.
(2002). How can extremism prevail? A study based on
the relative agreement interaction model. J. Artificial
Societies and Social Simulation, 5(4).
Easley, D. A. and Kleinberg, J. M. (2010). Networks,
Crowds, and Markets - Reasoning About a Highly
Connected World. Cambridge University Press.
Eiter, T. and Mannila, H. (1997). Distance measures for
point sets and their computation. Acta Inf., 34(2):109–
133.
Grandi, U., Lorini, E., and Perrussel, L. (2015). Proposi-
tional opinion diffusion. In Proceedings of the 2015
International Conference on Autonomous Agents and
Multiagent Systems, AAMAS 2015, Istanbul, Turkey,
May 4-8, 2015, pages 989–997.
Hafizoglu, F. M. and Sen, S. (2012). Analysis of opinion
spread through migration and adoption in agent com-
munities. In PRIMA 2012: Principles and Practice of
Multi-Agent Systems - 15th International Conference,
Kuching, Sarawak, Malaysia, September 3-7, 2012.
Proceedings, pages 153–167.
Jager, W. and Amblard, F. (2004). A dynamical perspec-
tive on attitude change. In Proceedings of NAACSOS
(North American Association for Computational So-
cial and Organizational Science) Conference, Pitts-
burgh, USA, June 22-25.
Milgram, S. (1967). The small world problem. Psychology
Today, 2:60–67.
Prettejohn, B. J., Berryman, M. J., and McDonnell, M. D.
(2002). Methods for generating complex networks
with selected structural properties for simulations: A
A Logical Approach to Extreme Opinion Diffusion
27
review and tutorial for neuroscientists. Frontiers in
Computational Neuroscience, 5(11).
Sureda, C., Gaudou, B., and Amblard, F. (2017). An
agent-based simulation of extremist network forma-
tion through radical behavior diffusion. In Proceed-
ings of the 9th International Conference on Agents
and Artificial Intelligence - Volume 1: ICAART,, pages
236–243. INSTICC, ScitePress.
Tsang, A. and Larson, K. (2014). Opinion dynamics of
skeptical agents. In International conference on Au-
tonomous Agents and Multi-Agent Systems, AAMAS
’14, Paris, France, May 5-9, 2014, pages 277–284.
Watts, D. J. (1999). Networks, dynamics, and the small-
world phenomenon. American Journal of Sociology,
105(2).
Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics
of small-world networks. Nature, (393):440–442.
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