Torsten Hägerstrand, a Swedish geographer at the University of Lund, used the following technique to trace the diffusion of an innovation.
Hagerstrand traces the diffusion process by imitating it with numbers.
Such imitation, leading to prediction or forecasting of the pattern of
diffusion, is called a simulation of diffusion. To follow the mechanics
of this strategy, it is necessary only to understand the concepts of ordering
the nonnegative integers and of partitioning these numbers into disjoint
sets.
The figure on the left shows the spatial distribution of the number
of individuals accepting a particular innovation after one year of observation
(Hägerstrand, p. 380). The figure on the right shows a map of
the same region and of the pattern of acceptors after two yearsbased
on actual evidence. Notice that the pattern at a later time shows both
spatial expansion and spatial infill (more concentrated use and greater
density per unit of land area). These two latter concepts are enduring
ones that appear over and over again in spatial analysisas well as in
planning at municipal and other levels.
Figure 1.

Figure 2.

Might it have been possible to make an educated guess, from Figure 1
alone, as to how the news of the innovation would spread? Could the righthand
figure above have been generated/predicted from the lefthand figure using
some replicable, systematic process? The steps below will use the grid
in Figure 3 to assign random numbers to the grid in Figure 1, producing
Figure 4 as a simulated distribution, as opposed to the actual distribution
of Figure 2, of acceptors after two years.
6248
0925
4997
9024
7754
7617 2854 2077 9262 2841 9904 9647 3432 3627 3467 3197 6620 0149 4436 0389 0703 2105 

This assumption regarding distance and probability of contact is reflected in the assignment of numerals within the gridthere are the most four digit numbers in the central cell, and the fewest in the corners. The floating grid partitions the set of four digit numbers {0000, 0001, 0002, ..., 9998, 9999} into 25 mutually disjoint subsets.
FIGURE 4. Here, the Mean Information Field collects new adoopters (red dots). The transformation described above is animated to illustrate it.
Given a set (or sets) of four digit random numbersas below. Center the floating grid on F2. The first random number is 6248 and it lies in the center square of the overlay. So in the simulation, the acceptor in F2 finds another acceptor nearby in F2. Record that simulated acceptor as a red dot. Together with the original adopter, there are now two adopters in this cell. Move the MIF over and repeat the procedure using the next random number in the sequence.
FIGURE 5. Simulated distribution of acceptors, using random numbers.
Original acceptors in black; simulated acceptors in red. Consider
what to do with edge effect issues. How does the simulation compare
to the actual distribution of adopters after two years (Figure 2)?
This question leads to a whole set of issues about how to compare patternone
might use color, contours, a variety of numerical measures, and so forth.
Attached is a rough idea using this particular
small example.

Construction of the floating gridthe socalled "Mean Information Field" (MIF) in the original example.
Assumption: the frequency of social contact (migration) per square kilometer falls off (decays) rapidly with distance.
The data are from an empirical study.
Units on axes: xaxisdistance in kilometers; yaxisnumber of migrating
households per square kilometer.
Definition: An area containing probabilities of receiving information from the central point of that region is called a mean information field.
To assign quantitites of four digit numbers to each cell in the MIF, it is necessary to use the curve derived from the empirical study (distance decay curve).
It is used to
The size of the MIF is 25 by 25 kilometers squared. Observation is that the typical household moves no more than 12.5 kilometers. This field is then split into cells 5 by 5 kilometers squared.
Assignment of probabilities
From the graph of distance decay, a point 10 km from the center has a value of .167 associated with it. This is in households per square kilometer; there are 25 km squared in each cell; so the point value of the cell is 25*.0167=4.17. The center cell has a value of 110an actual number of households.
The total point value of all cells is 248.24note the symmetry caused by assumptions about ease of movement in all directions outward from the center.
Divide: 4.17/248.24=0.0168so, assign 168 4 digit numbers to the cells that are 10 km from the center (two to the north, east, south, and west of center).
Thus, the Mean Information Field is constructed.
Some Basic Assumptions of the Simulation Method (Monte Carlo)
Assumptions to create an unbiased gaming table:
Hägerstrand, Torsten. Innovation Diffusion as a Spatial Process.
Translated by Allan Pred. University of Chicago Press, 1967.