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probably the coefficients will converge to one. This coincides with the region of
combinations for p(a1) and p(a2) , shown on Figure 4, where the coefficients converge
to one. To see that, we can rewrite (1), replacing c 0 = 0.5:
p
p
2
p
p
if
i
(4)
+
=
.
p
a
1
a
2
1
i
If we replace in (4) p i for its value 0.1919 and, for example, set p if = 0.6, we obtain
p(a1) + p(a2) = 1.248. On the other hand, if the sum of p(a1) and p(a2) is smaller
than 1.0, we also obtain a parallel line to (3) but located below (3). In this case, p if will
be smaller than 0.5, and thus we will get a convergence to zero of the coefficients. For
example, if we put p if = 0.4 in (4), we get the line p(a1) + p(a2) = 0.753. That line is
located within the region of combinations of p(a1) and p(a2) shown in Figure 4,
where we obtain that dynamic.
However, from Figure 4, we can also see that on the upper left corner of the graph,
the line p(a1) + p(a2) = 1.0 does not represent all the combinations of p(a1) and
p(a2) where the ABM exhibits the bifurcation. Thus, the simple probability model
only partially explains the empirical results.
5.2
Markov Chain Model
Since the model in 5.1 calculates p if only for the initial state of a simulation run, it
cannot fully capture the dynamical nature of the ABM. Remember that the concepts'
coefficients change during a run, as well as the interaction probabilities among agents.
In the ABM, that means that c 0 and p i will change as the simulation run advances.
Thus, we need to build a model that captures that dynamical aspect of the ABM. To
do so, we use a simple Markov chain, with four states, as described in Table 1.
Table 1. State transition probability matrix of the Markov chain
S t+1 (j = 0)
W t+1 (j = 1)
S t (i = 0)
p if +
1 - p if +
W t (i = 1)
p if -
1- p if -
Table 1 indicates that if a concept strengthens (state S t (i = 0) ), then the
probability that it will increase in the next step (state S t+1 (j = 0) ) is p if + , and that it
will weaken is 1 - p if + (state W t+1 (j = 1) ). On the other hand, if a concept weakens
(state W t (i = 1) ), then the probability that it will strengthen in the next step (state S t+1
(j = 0) ) is p if - and that it will weaken (state W t+1 (j = 1) ) is 1- p if - . In the ABM, each of
those p if has a meaning. From expression (1), we know that p if depends on c and p i ,
which change during a simulation run. The description of the ABM states that if a
concept strengthens, then its coefficient will increase by a certain
Δ
c , and the same
will happen with p i , which will increase by
Δ
p i . If the concept weakens, then the
coefficient c will decrease by
c , but p i will remain the same. Therefore, using those
facts, we can write the following expressions for p if + and p if - :
Δ
 
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