Diffusion of Innovation
The Anderson School at UCLA
POL 2002-05
Numbers 101: The Diffusion of Innovations
Copyright © 2002 by Richard Rumelt.
This technical note is a quick introduction to the use of diffusion models in forecasting.
We use diffusion models in cases where an innovation diffuses through a population. In this note we focus on the simplest diffusion model: the logistic model. This model produces the familiar
“S” curve in which a period of rapid acceleration is followed by deceleration and, finally, saturation.
The graphs on the right show a logistic growth process. There is some base population that “adopts” the new innovation. The top graph shows the cumulative, or total, number of adoptions rising from 1 to 100 in 20 years.
The inflection point is the time at which the number of adoptions per year peaks (not the growth rate), falling thereafter. On the top graph, the inflection point appears as the time when the curve stops accelerating upward and starts decelerating towards saturation.
The middle graph shows the number of adoptions per year. Technically, it is the derivative of the top graph. There are secondary inflection points on this graph, where the adoption rate changes its curvature. These points correspond to the times of maximum curvature on the top graph.
Finally, the bottom graph shows the growth rate in cumulative adoptions over time. As can be seen, the growth rate falls from a high of about 46% to zero.
Logistic Growth Equations
To fully describe the logistic growth model it is necessary to use symbols and formula. Let the number of people (or companies, or other unit) adopting an innovation at time t be
Logistic Curve --- Cumulative Adoptions
100
80
60
Inflection Point
40
20
0
0
5
10
15
20
15
20
Time
Logistic Curve --- Adoptions
12
10
8
Secondary Inflection
Points
6
4
2
0
0
5
10
Time
Logistic Curve -- Growth in Cumulative Adoptions
0.5
0.4
0.3
0.2
POL 2002-05
Numbers 101: The Diffusion of Innovations
Copyright © 2002 by Richard Rumelt.
This technical note is a quick introduction to the use of diffusion models in forecasting.
We use diffusion models in cases where an innovation diffuses through a population. In this note we focus on the simplest diffusion model: the logistic model. This model produces the familiar
“S” curve in which a period of rapid acceleration is followed by deceleration and, finally, saturation.
The graphs on the right show a logistic growth process. There is some base population that “adopts” the new innovation. The top graph shows the cumulative, or total, number of adoptions rising from 1 to 100 in 20 years.
The inflection point is the time at which the number of adoptions per year peaks (not the growth rate), falling thereafter. On the top graph, the inflection point appears as the time when the curve stops accelerating upward and starts decelerating towards saturation.
The middle graph shows the number of adoptions per year. Technically, it is the derivative of the top graph. There are secondary inflection points on this graph, where the adoption rate changes its curvature. These points correspond to the times of maximum curvature on the top graph.
Finally, the bottom graph shows the growth rate in cumulative adoptions over time. As can be seen, the growth rate falls from a high of about 46% to zero.
Logistic Growth Equations
To fully describe the logistic growth model it is necessary to use symbols and formula. Let the number of people (or companies, or other unit) adopting an innovation at time t be
Logistic Curve --- Cumulative Adoptions
100
80
60
Inflection Point
40
20
0
0
5
10
15
20
15
20
Time
Logistic Curve --- Adoptions
12
10
8
Secondary Inflection
Points
6
4
2
0
0
5
10
Time
Logistic Curve -- Growth in Cumulative Adoptions
0.5
0.4
0.3
0.2