Of course there are a finite number of farmers, so the exponential growth can not continue forever. Once the innovation starts to become popular, many of the people who one might tell about it are in fact already using it, placing exponential growth for the hazard in tension with Zeno's paradox for the risk pool. Contagious diffusion can only occur when someone who has experienced the innovation encounters someone who has not. Diffusion is slow early on because there are too few adopters who can promote the innovation (a low hazard), and it is slow later on because there are so few potential adopters remaining (a small risk pool), but in the middle lies a "tipping point" of intense diffusion where many people are promoting the innovation to many who have yet to adopt it (a high hazard and large risk pool). The resulting graph is the s-shaped curve shown in the graph and labeled as "endogenous hazard."

Although the example of internal influence described above relies on direct word-of-mouth contagion, the same implications apply to "threshold" or "cascade" models where potential adopters are aware of how many others have adopted the innovation but don't directly communicate with them. For instance, many people who don't make a habit of smashing property and assaulting people on the street will nonetheless join in a sufficiently large riot because safety in numbers means they need be much less afraid of punishment than if they were alone to misbehave. In this model it doesn't matter whether the rioters directly communicate with each other, only that potential rioters have a sense of how large the riot has become. Although in the riot example the potential rioter is directly estimating the size of the mob, this miasmic sort of diffusion is often mediated by things like best- seller lists or website download counts that aggregate and make salient information on popularity. So you may be more likely to buy a book when it becomes a best-seller because the book's popularity gives it more conspicuous placement in bookstores, even if you don't personally know a single individual who has read the book or have even observed strangers reading the book in public.

Thus, we have two distinct patterns for how an innovation might diffuse across a population. In the second style, the proportion of holdouts who adopt in each period is determined by how many actors are already using the innovation. Because the hazard rate is a function of prior adoptions, this is an endogenous pattern or an "internal-influence" cycle. In contrast, in the first style a constant proportion of holdouts adopt in every period. Because a constant proportion cannot be a function of how many people have already adopted, it can be interpreted as reflecting an "external-influence" on the system, or an "exogenous" pattern. Of course these patterns are ideal-typical and real cases can approximate one or the other or even a compromise between them. For instance, the diffusion of tetracycline was mostly exogenous, the diffusion of hybrid corn almost perfectly endogenous, and the diffusion of postwar consumer appliances a compromise between the two patterns. Much of the literature brackets this issue of how different types of innovations spread and instead focus on a single innovation and then ask which actors adopted that innovation particularly early. However, in this book I emphasize the question of the nature of diffusion itself and focus on the question of under what circumstances songs follow the concave curve and under which circumstances they follow the s-curve. This is the type of question that can not be answered by studying a single innovation's diffusion history, but only in comparing those of many innovations, and seeing under what circumstances an innovation's trajectory will follow one path or the other. Such an endeavor requires data on many innovations, and this is a role for which radio singles are well-suited for they occur in such numbers, spread so rapidly, and are so well-documented as to serve the purposes of sociology as admirably as the fruit fly does for those of genetics.

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