**Guest post by Gabriel Rossman -- **Sociologist at UCLA. His work applies economic sociology to media industries. He blogs at Code and Culture and is the author of Climbing the Charts.

[Since our hostess requested that I talk a bit about my forthcoming book during my guest-blogging stint I'm posting an excerpt describing the two fundamental patterns through which things get popular.]

This book's substantive concern of how songs become hits on the radio is part of a more general class of problems in social science known as the diffusion of innovation. This literature covers a wide variety of substantive areas where actors within a population each decide if and when to adopt an innovation. The seminal studies in this field were about such eclectic phenomena as:

- farmers in Iowa planting a new kind of corn
- firms in heavy industry adopting various new production technologies
- small-town doctors prescribing the antibiotic tetracycline
- postwar households purchasing such appliances as televisions and washing machines

The innovations described in the literature range from drastic changes that reorder the actor's cultural and economic experience to fairly minor variations on incumbent practices for which "innovation" is perhaps too grandiose a term. In current sociology, one of the main applications of diffusion analysis is asking such questions as when firms adopt new business practices or how activists adopt new tactics.

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.

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.

><This article available online at:

http://www.theatlantic.com/business/archive/2012/05/how-things-get-popular/257080/