Torsten Hägerstrand, a Swedish geographer at the University of Lund, used the following technique to trace the diffusion of an innovation.

• The thing being diffused (communicated) is an idea;
• the agents of diffusion, or carriers of new information, are human beings;
• the space in which the idea is to be diffused is a region of the world.
  

Hägerstrand traced 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 non-negative 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 years--based 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 related concepts are also enduring ones and they appear over and over again in spatial analysis---as well as in planning at municipal and other levels.
 

Might it have been possible to make an educated guess, from Figure 1 alone, as to how the news of the innovation would spread? Or might one develop Figure 2 from the Figure 1 using some replicable, systematic process?   What are the intellectual and deep challenges to modelling such process? 

• The numerical pattern displayed in the figures is discrete in nature, yet the concept it represents is a continuous one.  Thus, one is faced with the challenge of devising a strategy that satisties both the discrete character of human nature (humans are individual points) and the continous character of a flow (of information or other). 
  
• Most of mathematics, itself, is based on the Law of the Excluded Middle (statements are 'right' or 'wrong'; 'true' or 'false'; 'black' or 'white')--it is a two-valued logic system (Blass and Harary).  There is no 'gray' area.  Yet in much of the natural world there is no 'black' or 'white'--only gray.  Thus, one is also faced with the challenge of aligning mathematical and human process and in circumscribing regions in which the model is valid.

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	0000 to 0095 0096 to 0235 0236 to 0403 0404 to 0543 0544 to 0639 0640 to 0779 0780 to 1080 1081 to 1627 1628 to 1928 1929 to 2068 2069 to 2236 2237 to 2783 2784 to 7214 7215 to 7761 7762 to 7929 7930 to 8069 8070 to 8370 8371 to 8917 8918 to 9218 9219 to 9358 9359 to 9454 9455 to 9594 9595 to 9762 9763 to 9902 9903 to 9999
Figure 3.   The left shows Figure 1 with the grid on the right scaled to have its squares fit the grid on the left.  Animation shows the direction of movement in application of the grid of numbers.  Notice that the grid floats across the distribution from Figure 1 so that it is centered, from left to right, on each non-zero entry.  The middle column shows a set of random numbers taken from a table of random numbers (where one chooses them in the order presented in the table in order to preserve randomization).

Figure 4.  Here the Mean Information Field collects new adopters (red dots) and inserts them in the base configuration from Figure 1.  The transformation described above is animated to illustrate it.

Figure 4 demonstrates the detail of using the Mean Information Field to generate a simulation of the diffusion of an innovation.  Center the floating grid on F2.  The first random number (from Figure 3) is 6248 and it lies in the center square of the MIF. 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 center it on the next cell with a numeral/adopter (as suggested in the animation in Figure 3) and repeat the procedure using the next random number in the sequence.  The animation of Figure 4 shows the use of the entire set of random numbers in Figure 3 and the consequent simulated pattern of new adopters.  Figure 5 shows a static view and illustrates issues associated with edge effect problems.

a a A B C D E F G H I J K L M N  a a a a a a a a a a a a a a a a 1 a a a a a 1 a a a 2 a a 1+1 1 a a 3 a 1+1 a 4 a 1 5+1 1+1 1 a 5 a a 2+1 1 1 6 a 1 2+1 1 7 a 1+1 3+1+3 a 8 a 1+1 1+1+1 1 a 9 a a 1 a 10 a 1 1 a a a a a a a a 11 a a a a a a a a a a 12 a a a a a a a a a a

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 pattern--one might use color, contours, a variety of numerical measures, and so forth.

While the spatial pattern simulated is discrete, it is based on underlying continuous mathematics.  The construction of the MIF assumes, based on empirical evidence, that the frequency of social contact (communication, migration) per square kilometer falls off or decays with distance.   And, this idea is captured easily with a distance decay curve, a continuous curve, with units on the x-axis expressed in terms of kilometers and on the y-axis in terms of the number of migrating households per square kilometer.  More detailed information on the mechanics of construction is available at a variety of locations including at the following linked site. 

Contemporary Vision

Two toolkits that were not available to Hägerstrand for his analysis of the diffusion of an innovation were fractal geometry and Geographic Information Systems (GIS) software.  We consider superimposing these two tools on the classical idea.

Given a view of a gridded locale in the style of Hägerstrand (Figure 6), along with an associated 5-by-5 MIF (Figure 7)  with numerical entries expressed as percentages of the numbers from Figure 3.  Thus, for example, the central cell contains 44.31 percent of the four digit numbers and the likelihood that an individual in that cell will contact another within that cell is 44.31 percent.  Probabilities of the likelihood of contact simulate the spread of knowledge of the innovation over time.  Over time, the information will spread, gaps will fill in and, eventually, the population will become saturated with the information.  Figure 6 shows a very simple pattern of initial adopters, coded as '1' in three cells:  H3, H4, and H5.  The first generation of adopters adds three more adopters, one each in cells H3, G3, and H5, using the first three random numbers from Figure 3:  H3 goes to H3 from 6248; H4 goes to G3 from 0925; and, H5 goes to H5 from 4997.  Because of the weighting of the assignment of probabilities, clearly different initial distributions of adopters will lead to different outcomes of specific locations, even though there may be general pattern similarities.

Figure 6.  Initial adopters are coded 1, in cells H3, H4, and H5.  First generation adopters are coded +1, in cells H3, G3, and H5.

Figure 7.  Mean Informtaion Field from Figure 3 with percentages of four digit numbers entered.  These percentages represent probability of contact.  Thus, when the MIF is centered on adopter, there is a 44.31% chance that his next contact will be in the same cell he is in; there is a 0.96% chance that it will be in two cells away on the diagonal, and so forth in between.

Figure 8a.  Original adopters are in cells H3, H4, and H5.  The zone of interaction is composed of the intersection of any two of the three sets:  it is the blue rectangle.

Figure 8b.  Original adopters are in cells H3, G4, and H5.  The zone of interaction is composed of the intersection of any two of the three sets:  it is 'T'-shaped.

Figure 8c.  Original adopters are in cells H3, F4, and H5.  The zone of interaction is composed of the intersection of any two of the three sets:  it is 'T'-shaped.

Figure 8d.  Original adopters are in cells H3, E4, and H5.  The zone of interaction is composed of the intersection of any two of the three sets:  it is 'T'-shaped.

Figure 8e.  Original adopters are in cells H3, D4, and H5.  The zone of interaction is composed of the intersection of any two of the three sets:  it is 'T'-shaped.

Figure 8f.  Original adopters are in cells H3, C4, and H5.  Here, the zone of interaction is the red core, alone.

As the central adopter cell moves toward the top, note that naturally the associated pattern within the interaction zone preserves the same bilateral symmetry (up to truncation) as the underlying root MIF.  Thus, any configuration characterizing the pattern in this sequence of figures should also exhibit bilateral symmetry.  One such form is a tree.  In fact, if one thinks of each of the three initial adopter cells as a point centered in a cell of 1 unit side, then the three cells in Figure 8a might be represented as a linear pattern, suggested in Figure 9a, top.  In that image, view the tree on vertices H3, H4, and H5 as a generator for a self-similarity sequence (as suggested by  the arms formed in two successive stages in that figure).  In a similar manner, other trees are generated for each movement of the central original adopter, with Figure 9b derived from Figure 8b, and so forth.

Figure 9a., 9b, 9c, 9d, 9e, 9f.  Simple trees based on initial adopter entry positions, from Figures 8a, 8b, 8c, 8d, 8e, and 8f, are used to create self-similar entities as suggested above.  The red dots aid in seeing how angle theta/2 was created in each case, as the inverse tanget of 1, 1/2, 1/3, 1/4, and 1/5 respectively.

Figure 10.  In this general schematic of a tree, the angle φ is created as one base angle of an isosceles triangle whose apex angle is known as θ/2 .  The length of an apical side of the triangle is determined  by  the position of the central adopter in relation to the others.

Table 1.  Calculation of θ for each of the distributions of original adopters in Figure 8.

Certainly when looking at the geometric pattern in Figure 9, one gets an intuitive sense that Figure 8a fills the most space and Figure 8f the least, with tailing off in between.  The challenge is to find a fractal generator that will mimick the pattern, that is derived from the pattern, and will do so within the context of Mandelbrot's formula for fractal dimension, D, as D = (ln N)/(ln (1/r)) where N represents the number of sides in the generator, which in all cases here is the value 2, and where r is some sort of scaling value that remains constant independent of scale (Mandelbrot, 1983).  To find a suitable r, look back to Figure 10, and calculate φ,  the base angle of the isosceles triangle, based on the length of the generator side.  It is constant, independent of scale.  Use the values from Table 1 for the apex angle, subtract that value from 180 and divide the result in half, to get the size of one base angle.  Table 2 shows the results of the calculation.  It also shows the value for fractional dimension using cos φ  as the value for r in Mandelbrot's formula.

Table 2. 

Thus, a measure that reflects the degree of space-filling, based only on the pattern of original adopters,  can follow from use of fractal iteration.  It fulfills the sought purpose of being able to offer a measure of eventual saturation, at the outset of a simulation.  It is not unique and is dependent on generator formulation (as such things must be); but it adds and extra dimension to the previous analysis and it is possible because advances in technology in the latter part of the twentieth century permitted visualization of geometric form previously "seen" only through notation.