NRE 530; Geography: Spatial Analysis, Theory and Practice

Dr. S. Arlinghaus

Measurement of diffusion: a conceptual approach based on the work of Torsten Hagerstrand

Diffusion is a process in which anything that moves, or that can be moved, is spread through a space, from a source, until it is distributed throughout that space. Aerosol sprays smell strongest, immediately, close to the source. Gradually, the odor permeates even the corners of the room, and, once it is uniformly spread throughout the space in the room, it is said to have diffused.

How can diffusion be measured? One way is to trace the positions of things that are being diffused at different times. Torsten Hagerstrand, 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.

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 non-negative integers and of partitioning these numbers into disjoint sets. Indeed, the theoretical material from mathematics of "set" and "function" will underlie the real-world issues of "form" and "process."

To illustrate the mechanics of this process, first work through an example...then more material will follow that suggests the conceptual base on which it rests.

INITIAL SET-UP.

In Figure 1 is a map of an hypothetical region of the world. After one year, a number of individuals accept a particular innovation; their spatial distribution is shown in Figure 1.

MAP BASED ON EMPIRICAL EVIDENCE--REGION INTERIOR IS SHADED WHITE; CELLS WITH NUMERALS IN THEM INDICATE NUMBER OF ACCEPTORS IN LOCAL REGION.
 

In Figure 2, a map of the same region shows the pattern of acceptors after two years--again, based on actual evidence. Notice that the pattern at a later time shows both spatial expansion and infill. These two latter concepts are enduring ones that appear over and over again in spatial analysis---as well as in planning at municipal and other levels.
 

Figure 2. Actual distribution of acceptors after two years.

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 Figure 2 have been generated/predicted from Figure 1? 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.
 

• Construct a "floating" grid (Figure 3) to be placed over the grid on the map of Figure 1, with grid cells scaled suitably so that they match. Center the floating grid on a square in Figure 1 in which there exists an adopter (say 2F)...this is the first cell, working left to right and top to bottom, which contains a numeral. The numbers in the floating grid, used with a set of four digit random numbers, will be used to determine likely location of new adopters. It is assumed that the adopter in F2 (or in any other cell) is more likely to communicate with someone nearby than with someone far away; velocity of diffusion is expressed in terms of probability of contact.
• This assumption regarding distance and probability of contact is reflected in the assignment of numerals within the grid--there 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.

• Given a set (or sets) of four digit random numbers--as below. Center the floating grid on F2. Use the first set of random numbers below. The first number is 6248 and it lies in the center square of the overlay. So in the simulation, the previous Figure 1 acceptor in F2 finds another acceptor nearby in F2. Use the map in Figure 4 to record the simulated distribution (a red entry). In cell F2, enter a red +1 to represent the initial adopter.  Together with the original adopter, there are now two adopters in this cell.

• Pick up the floating grid and center it on G2. The second random number is 0925, located in the first cell northwest of the center cell of the overlay, cell F1. In Figure 4, a 1 in cell G2 to represent the original adopter, and a 1 in cell F1  represents the later (new) adopter.
• Continue this process, shifting the overlay so that it is centered on every cell with at least one early adopter (from Figure 1). Work from left to right, top to bottom. Use a different random number for each early adopter. If there are 3 early adopters in a given cell, use three different random numbers in entering the results in Figure 4. Just use the random numbers (supposedly already randomized) reading down the column.
• How does the simulation (Figures 4) compare to the actual distribution of adopters after two years (Figure 2)?

Construction of the floating grid--the so-called "Mean Information Field" (MIF)

Assumption: the frequency of social contact (migration) per square kilometer falls off (decays) rapidly with distance.

Data from an empirical study:

Units on axes:

x-axis--distance in kilometers

y-axis--number 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, represented by Figure 3.

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

• determine the size of the MIF
• assign particular probabilities.

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 110--an actual number of households.

The total point value of all cells is 248.24--note the symmetry caused by assumptions about ease of movement in all directions outward from the center.

Divide: 4.17/248.24=0.0168---so, 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:

• the surface is uniform in terms of population and transport
• all contacts are equally easy in any direction
• there are an equal number of potential acceptors in each cell

• There is a set of carriers at the start (as in Figure 1)
• information is transmitted at constant intervals
• when carrier meets a new person, acceptance is immediate
• the likelihood of a carrier and another meeting depends on the distance between them.

Hagerstrand, Torsten. Innovation Diffusion as a Spatial Process. Translated by Allan Pred.
University of Chicago Press, 1967.