Geoscience Reference
In-Depth Information
This will converge under certain circumstances to
=
∑
−α β
−
1
x
(
1
)
x P
,
(2.4)
k
j k
j
which is similar to the case where the reaction is incorporated into the diffusion - where α
= 0 and
β = 1 with
∑∑
xt
(
+=
1
)
β
xtP
()
x tP
() .
(2.5)
k
j
jk
j
jk
j
j
In this form, Equation 2.5 is a first-order Markov chain or process that under certain conditions of
strong connectivity of the transition matrix [
P
jk
] converges to the steady-state
x
=
∑
where
x
k
is a stable probability vector that results when the diffusion has completely worked itself out. In
this case, the final state of the system [
x
k
] is independent of the initial conditions [
x
k
(
t
= 1)]. A variety
of models ranging from spatial interaction to social opinion pooling have been based on these kinds
of structure (Batty, 2013). The most well known is the
PageRank
algorithm that attributes the suc-
cess of a web page to the process of dynamics implied by the reaction-diffusion equation where the
reaction is constant for any page, that is, from the process where
x P
k
j
jk
j
1
∑
xt
(
+= +
1
)
α
(
1
− α
)
xtP
() .
(2.6)
k
j
jk
N
j
Note that this process assigns the importance of the event (page) or state
x
k
(
t +
1) to the random
chance α/
N
plus the assignment of importance of all other events to the event in question given by
(
j
∑
This in fact is the original variant of the algorithm used to rank web pages in
the Google search engine (Brin and Page, 1998).
The CA model can be written in terms of the diffusion model without any reaction and external
drivers. Then the equation becomes
1−α
)
xtP
j
() .
jk
+=
∑
xt
(
1
)
xtP
() ,
(2.7)
k
j
jk
j
where
the vector
x
k
(
t +
1) is the pattern of cells which are switched on at time
t
+ 1
x
k
(
t
) is the initial or predecessor pattern of cells at time
t
the matrix
P
jk
is a translation of the set of cellular rules into a form that maps
x
k
(
t
) into
x
k
(
t +
1)
It is in fact rare for this structure to be used to represent CA models largely because convergence is
not strictly a property of such models for these tend to be models in which the occupancy of cells
increases through time, enlarging a space or lattice rather than being distributed across a fixed space.
However, what is significant in terms of this structure is the fact that we can code the rules in the
matrix [
P
jk
] as pertaining to any size of neighbourhood, enlarging the neighbourhood to one which
covers the entire system if that is the way the system works. In this sense, the generic model not
only includes strict CA but also CS models which tend to be the norm when it comes to applica-
tions. If we add the reaction component, then all this does is to ensure that, as well as the cellular
rules applying to the existing configuration
x
k
(
t
), the existing configuration is preserved as well (or
at least a fraction α of this is). In short, this can be used to ensure that once a cell is switched on, it
remains on. If we then add back the external driver, then this lets us override the cellular rules and
the reaction if necessary, and this is equivalent to introducing exogenous change.
To illustrate how we can simulate a simple CA using the generic structure in Equation 2.7, in
Figure 2.3, we take the tree-like generator from Figure 2.1 and we display this in a 5 × 5 system.
