At the most basic level, one can study diffusion simply by drawing a graph and looking at its shape to see whether it is more concave or more s-shaped. The graph shows typical curves of each ideal type. The shape of the graph is informative because different processes create differently shaped graphs; thus, seeing the shape of the graph gives very strong clues as to the process that created it. In a diffusion graph the x-axis is time, which can be denominated in whatever unit is appropriate. Many of the canonical studies measure time in years, but tetracycline spread in a matter of months, and pop songs usually spread even faster. The y-axis is how popular the innovation is at a particular time. Usually the y-axis is cumulative, showing how many actors have adopted the innovation to date, though sometimes they are plotted as instantaneous, showing how many actors are adopting in each period.
This implies that diffusion is about seeing how many actors adopt the innovation in each period, and it is, but this can be misleading. The reason is that it's quite a different thing for a hundred out of a thousand to adopt than for a hundred out of a hundred. The number of actors who have yet to adopt as of a time is the "risk pool," and the proportion of the risk pool who adopt in a time interval is the "hazard" rate. For a given hazard, the raw number of adoptions decreases as the risk pool shrinks. This is a case of Zeno's paradox, in which fleet- footed Achilles races a tortoise but allows the reptile a head start. If in each minute he closes half the remaining distance, then after the first minute he will have closed 1/2 the distance, after the second minute, 3/4 of the initial gap, then 7/8, 15/16, 31/32, etc. Returning to diffusion, imagine that a thousand doctors have a hazard rate of 10 percent for adopting tetracycline. In the first month 100 doctors (a tenth of 1,000) will write their first prescriptions for tetracycline; in the second month 90 will adopt, for a total of 190 doctors prescribing it; in the third month 81 will adopt, for a total of 271, and so on. In this example the hazard remains constant at one-tenth per month. Therefore, the proportion of the risk pool converted in each period is the same, but the raw volume decreases rapidly. This results in the concave-shaped curve labeled constant hazard" in the graph, which shows rapid growth initially and asymptotically limited growth thereafter.
So far we have assumed that the hazard is constant. This may be warranted if we imagine that there is some constant force acting in the population and encouraging actors to adopt the innovation, such as a marketing campaign with a fixed budget. For this reason these curves are often known as "external influence" in that the innovation is being spread by something outside of the population adopting it. However, imagine that the innovation is spread as an endogenous process within the population, perhaps by word of mouth. This might be because there is no marketing budget or because the actors simply don't trust advertisements or salesmen to provide impartial advice. For instance, imagine that farmers are deciding to plant a new type of maize that presents higher risk but offers higher reward. Most farmers are hesitant to make so radical a change, but one farmer is willing to experiment with the seed and, on seeing his higher crop yields, he tells two neighbors about his satisfactory experience and they try it. After their own satisfactory experiences they in turn each tell two others. If each person using the corn tells two new neighbors about it, then one farmer will plant it in the first year, three in the second, nine in the fourth, twenty-seven in the fifth, eighty-one in the sixth, and so on. This pattern shows slow diffusion at first, but follows exponential growth so that once the innovation reaches a critical mass of the population, it diffuses rapidly.