The number of followers on any given day can be calculated from the
number of followers on the previous day, taking into consideration the
respective numbers of pagans and followers as well as the chance that an
encounter between a pagan and a follower will result in conversion (given by
the conversion-rate parameter). Let us calculate the number of followers on a
new dayd+ 1. This will obviously include the number of followersFdon the
previous dayd, plus the number of new converts. How many new converts will
there be on each day? First, conversions only occur when households get into
contact. The number of encounters on every day is given by the total number
of households,N, multiplied by some contact ratecexpressing how many
contacts each household makes during a day. Of all encounters, only encoun-
ters of pagans with followers will result in conversions. To calculate the chance
of an encounter being between a pagan and a follower, we multiply the chance
of a household being followerFd/Nby the chance of a household being pagan
(N-Fd)/N(where we calculated the number of pagan households by deducting
the number of follower households from the total number of households).
Thus the number of encounters between pagans and followers will beN×c×
(Fd/N)×((N-Fd)/N). Finally, we have to multiply the result by the conversion
rate (that is, the probability of such encounters resulting in conversion):r×N×
c×(Fd/N)×((N-Fd)/N). Rearranging this expression, we receivec×r×Fd×
(1 -Fd/N), which has to be added to the number of followers on daydto get
the number of followers on dayd+ 1. Our world includes 51 × 51 = 2601
households. If we start the model with a single follower (Fd= 1), the quantity
Fd/Nwill be very small initially, and consequently the quantityFd×(1-Fd/N)
will be close to the value ofFd. At the“any”setting of the learning parameter,
households check the status of their eight neighbors, that is,c=8.^9 If we
choose to set the conversion rate tor= 0.1, which is the initial setting of the
model, on the second day there will be approximately 1 + 8 × 0.1 × 1 = 1.8
followers. Our expression is the exact definition of thelogistic function, where
our parameterscandrare usually combined into a single parameterR, the
transmission rate.^10
(^9) We can ignore the technical detail that the model only checks the neighborhood of pagan
households, since only they have a chance to be converted, which will be taken into consideration
in the next part of our equation.
(^10) Technically speaking, we constructed adifference equationthat models growth at discrete
time intervals. The logistic function is derived from adifferential equationthat considers change
to be continuous. How can we make sense of the plot of the growth of the movement in the
model (Figure 9.2) in terms of the mathematical expression we just derived? We have seen that
initially the quantityFd/Nwill be very small, and consequently the value of the expressionFd×
(1 -Fd/N) will be close to the value ofFd. On the second day, as we have seen, the value ofFdwill
almost double, but it will be still very small compared toN, giving an almost twofold increase in
the value ofFd×(1-Fd/N). Thus the increase ofFdwill result in a faster growth of the movement
in this phase, resulting in the upward curve on the left-hand side of the plot. When almost all
households are converted, however,Fd/Nwill be close to 1, making the value of (1 -Fd/N) very
Social Networks and Computer Models 197