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 nonnegative integers and of partitioning these numbers into disjoint sets. Indeed, the theoretical material from mathematics of "set" and "function" will underlie the realworld 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 SETUP.
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 EVIDENCEREGION INTERIOR IS SHADED WHITE; CELLS
WITH NUMERALS IN THEM INDICATE NUMBER OF ACCEPTORS IN LOCAL REGION.
aa  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 
1

a  a  a  a  a  a  a  a  
2

a  a 
1

1

a  a  
3

a 
1

a  
4

a 
5

1

a  
5

a  a 
2

a  
6

a 
2


7

a 
1

3

a  
8

a 
1

1

1

a  
9

a  a 
1

a  
10

a  a 
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 
In Figure 2, a map of the same region shows the pattern of acceptors
after two yearsagain, 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 analysisas
well as in planning at municipal and other levels.
aa  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  a  a  a  
2

1

1

a  a  
3

a 
1

1

1

a  
4

a 
6

1

1

1

a  
5

a  a 
2

1

a  
6

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7

a 
1

1

1

3

a  
8

a 
1

1

2

2

a  
9

a  a 
1

1

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a  a 
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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  aa  a 
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.

























RANDOM NUMBERS  a  
a  a  a 
SET 1  SET 2  SET 3 
a  a  a 
6248  4528  8175 
0925  3492  7953 
4997  3616  2222 
9024  3760  2419 
7754  4673  5117 
a  a  a 
7617  3397  1318 
2854  8165  1648 
2077  7015  3423 
9262  8874  2156 
2841  8443  1975 
a  a  a 
9904  7033  3710 
9647  0970  4932 
3432  2967  1450 
3627  0091  4140 
3467  6545  5256 
a  a  a 
3197  7880  4768 
6620  5133  9394 
0149  1828  5483 
4436  5544  8820 
0389  6713  7908 
a  a  a 
0703  5920  2416 
2105  5745  9414 
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 
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 
Construction of the floating gridthe socalled "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:
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, 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
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:
Hagerstrand, Torsten. Innovation Diffusion as a Spatial Process.
Translated by Allan Pred.
University of Chicago Press, 1967.