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we will compute the sensitivity of that model to changes in values of some
parameters. Here, we will present only two results of such analyses. To do so, we use
(1) and take the partial derivatives of
p
if
with respect to
p(a1)
and
p(a2)
:
p
∂
(13)
p
c
if
=
(
−
)
.
p
0
∂
i
a
1
p
∂
p
if
c
=
(
−
)(
1
−
)
.
(14)
p
0
∂
i
a
2
Additionally, since
p
i
appears in (13) and (14) and that variable depends on the
number of agents and concepts (
N
,
V
see (2)), we can express (13) and (14) in terms
of
N
and
V
. Moreover, given that for reasonably large values of
N
,
p
i
tends to
1/V
, we
will analyze (13) and (14) taking into consideration that
p
i
≈
1/V.
From (13) and (14) we can see that the sensitivity of
p
if
with respect to
p(a1)
and
p(a2)
is always positive (remember that 0
1), i.e. the larger
p(a1)
and
p(a2)
,
the larger
p
if
. Now, the larger the number of concepts a group has (
V
), the smaller
p
i
will be and the more influential
p(a1)
and
p(a2)
will be on the value that takes
p
if
.
That means that for groups with a large set of related concepts (or many different
versions of the same concept), the probability of true and illusory agreement (
p(a1)
and
p(a2)
) will greatly influence
p
if
. The significance of that influence will also be
determined by the value of
c
0
. Note that for large values of
c
0
,
p(a1)
will have a larger
influence on
p
if
than
p(a2)
and vice-versa. Thus, for groups with many concepts, the
degree of agreement, either true or illusory, and the initial strength of each concept
will dictate whether each concept strengthens or weakens, and eventually disappears.
Several “real world” situations could conform to the dynamics described above. As
an illustration, imagine a social group that has an abstract concept, such as
conservative
. Presumably, people would have many different versions of such
concept (i.e., a small
p
i
), with some people, e.g., considering that conservative is a
view about economics, while others considering that it is a view about values, and so
on. Imagine, furthermore, that this concept's relevance in that society is moderate, in
the sense that it does not persistently determine people's actions (i.e.,
c
0
≠ 1). For
concepts like this, our sensitivity analyses predict that their fate as a cultural
phenomenon will depend mainly on their capacity to generate agreement.
Imagine, furthermore, that
conservative
has
liberal
as a weakly contrasting
alternative concept (liberal is weakly contrasting to conservative because it does not
clearly divide political opinion in two sharply contrasting clusters). Our sensitivity
analyses predict that the fate of this pair will depend on agreement, regardless of
whether it is true (
p(a1)
) or illusory (
p(a2)
). Additionally, as discussed in 4.3 above,
these conditions promote bifurcations akin to social polarization.
Perhaps, an even more interesting situation arises in groups that have a small
number of concepts or versions of them. In that case,
p
i
will be large, and thus
p(a1)
and
p(a2)
will not have a large influence on
p
if
(i.e., the degree of true and illusory
agreement will not have a large influence on the fate of the concepts). Examining
Figure 5, we can see that in the above mentioned situation, the fate of each concept
will be predominantly dictated by its initial strength
c
0
, i.e., an initially rather strong
≤
c
0
,
p
i
≤
