The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models
Introduction
Dynamical population models are used to predict average behavior, generate hypotheses or explore mechanisms across many fields of science including ecology [1], [2], [3], epidemiology [4], [5], [6], [7] and immunology [8], to name but a few. Traditionally, epidemiological models focus on the dynamics of “traits” transmitted between individuals, communities, or regions (within specific temporal or spatial scales). Traits may include (i) a communicable disease such as measles [4] or HIV [9]; (ii) a cultural characteristic such as a religious belief, a fad [10], [11], [12], [13], an innovation [14], or fanatic behavior [15]; (iii) an addiction such us drug use [16] or a disorder [17]; or (iv) information spread through, e.g., rumors [18], [19], e-mail messages [20], weblogs [21], or peer-to-peer computer networks [22].
The earliest and by now the most thoroughly studied population models are those used to map disease progression through a human population [23], [24], [25]. These models typically divide a population into classes that reflect the epidemiological status of individuals (e.g. susceptible, exposed, infected, etc.), who in turn transit between classes via mutual contact at given average rates. In this way the models can capture average disease progression by tracking the mean number of people who are infected, who are prone to catch the disease, and who have recovered over time. In addition, these models can be used to identify the role of specific population characteristics such as age, variable infectivity, and variable infectious periods [24]. The division of epidemiological classes according to such characteristics gives rise to more complex models with so-called heterogeneous mixing.
In this paper we apply models similar to those used in epidemiology to the spread of ideas. By the term “idea” we refer generally to any concept that can be transmitted from person to person [26], [27], [28], [29]. It may refer to a technology, which may require effort and apprenticeship to be learned, but it may also be a more fickle piece of information such as a colloquialism or a piece of news. What is important is that it is possible to tell if someone has adopted the idea, understands and remembers it, and is capable of and/or active in spreading it to others.
Pioneering contributions to the modeling of social contagion processes, based on epidemiological models, date back to as early as 1953 [10], [18], [30], [31], [32], [33], [34], [35]. Nearly a decade later, population models were applied to the spread of scientific ideas [36], [37]. Around the same time, a stochastic model for the spread of rumors was proposed and analyzed [19]. In this model, a closed population is divided into three “social” states: ignorant, spreaders, and stiflers. Transitions from the ignorant state to spreaders may result from contacts between the two classes, whereas encounters between individuals who already know the rumor may lead to its cessation. Various recent extensions of this model include a general class of Markov processes for generating time-dependent evolution [38], and studies of the effects of social landscapes on the spread, either through Monte Carlo simulations over small-world [39] and scale-free [40] networks, or by derivation of mean-field equations for a population with heterogeneous ignorant and spreader classes [41]. Other interesting mathematical models, that attempt to capture the capacity for a population or idea to persuade others, or imitate, have also been developed to generate predictions regarding product adoption [42] or public opinion trends [43], [44]. Despite this revival in the modeling of information spread, few of these models have been directly validated by empirical data [36], [37], [42], [44].
Beyond obvious qualitative parallels there are also important differences between the spread of ideas and diseases. The spread of an idea, unlike a disease, is usually an intentional act on the part of the transmitter and/or the adopter. Some ideas that take time to mature, such as those involving apprenticeship or study, require active effort to acquire. There is also no simple automatic mechanism—such as an immune system—by means of which an idea may be cleared from an infected individual. Most importantly, it is usually advantageous to acquire new ideas, whereas this is manifestly not so for diseases. This leads people to adopt different, often opposite, behaviors when interested in learning an idea compared to what they may do during an epidemic outbreak. Thus we should expect important qualitative and quantitative differences between ideas and diseases when using epidemiological models in a sociological context. We explore some of these points below in greater detail, in the context of specific models and data.
In spite of these differences, quantifying how ideas spread is very desirable as a means of testing sociological hypotheses. For example, we can apply dynamical population models to the spread of an idea to validate statements about how effectively it is transmitted, the size of the susceptible population, the speed of its spread, as well as its persistence. Estimating the population numbers and rates is useful in constraining explanatory frameworks. It is also useful for studying how cultural environments may affect adoption, as happens when the same idea is presented to communities in different nations, or conversely when different ideas are presented to the same community.
In this paper, we apply several general models, inspired by epidemiology and informed by our knowledge of the sociology of the spread dynamics, to the diffusion of a specific scientific idea in three different communities. Our test case is the spread of Feynman diagrams, since the late 1940s the principal computational tool of theoretical high-energy physics, and later also used extensively in other areas of many-body theory such as atomic physics and condensed-matter theory. The primary reason to choose this example is that we have detailed historical information about the network of contacts, person by person, by means of which the diagrams spread during the first 6 years after their introduction [45], [46], [47].
This example of the spread of an idea may not transcend automatically to other cases of idea diffusion. Feynman diagrams are primarily a tool for complex calculation. As such their study and assimilation require a period of apprenticeship and familiarization. Transmission of the technique almost invariably proceeded, in the early years, through personal contact, from informal teacher to student and among peer groups of users. In later years the idea became familiar and available in accessible forms so that (in principle) it could more easily have been learned from books and lecture notes. Thus, although our example will clearly not cover every class of ideas it will point, we believe, to features of epidemic models that apply to idea diffusion. It will also reveal features of these models that require modification, thereby producing more realistic candidate models that we expect will prove useful beyond our present analysis.
In Section 2 we give some historical background on the spread of Feynman diagrams in the United States, Japan, and the Soviet Union. We discuss our data sources and the organization of the datasets. Section 3 presents several classes of models of epidemiology (or directly inspired by them), some of their mathematical properties, and the circumstances under which we expect them to apply to the spread of ideas. We apply each model to the historical data in Section 4, and discuss the estimated values for the model parameters in the light of our independent knowledge of how the diagrams spread. Finally, in Section 5 we present our conclusions and give some outlook on the general population modeling of the spread of ideas. Appendix A contains details about our parameter estimation procedure.
Section snippets
Data sources, time series reconstruction, and state determination
Feynman diagrams occupy a central role in modern theoretical physics. Realistic models of high-energy physics, as well as in condensed-matter, atomic, and nuclear physics cannot be solved exactly to generate predictions that can be confronted with experiments. In special circumstances, however, such as when interactions are weak, series expansions in a small parameter permit very good systematic approximations.
In models of particle physics, such as the relativistic quantum theory of
Population models: drawing parallels between epidemics and idea diffusion
Below we shall concentrate on the classical, simplest epidemiological models, based on “homogeneous mixing” in which state variables are only functions of time. In a review of epidemiological models, Hethcote [24] introduced their compartmental characterization (e.g. SIR, SIS, SEIR, etc.) within a global analysis of the field. Such survey also discusses how more complex models can be used to assess the impact of population structure (age, risk, gender, etc.), epidemiological variability (age of
Results and discussion
We now analyze the results of estimating parameters by matching the data on the spread of Feynman diagrams for three distinct countries to several population models discussed above. These models allow us to discuss the effects of the recovered class, of latency, and of competitive idea strands. They also explore several classes of transition mechanisms, both by progression and by contact between population classes.
Table 4 summarizes the results. To gauge the applicability of each model to each
Conclusions
In this paper, we applied several population models, inspired by epidemiology, to the spread of a scientific idea, Feynman diagrams, in three different communities undergoing very different social transformations during the middle years of the 20th century. There is always a tradeoff between the use of models that include more detail (heterogeneous populations) and highly aggregate simple models with a manageable number of parameters. Although a model built under very simplistic assumptions is
Acknowledgements
We thank Gerardo Chowell, Ed MacKerrow, Miriam Nuño, Steve Tenenbaum and Alun Lloyd, for discussions and comments. A. Cintrón-Arias acknowledges financial support from Mathematical and Theoretical Biology Institute and Center for Nonlinear Studies at Los Alamos National Laboratory. Collaboration was greatly facilitated through visits by several of the authors to the Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC, which is funded by NSF under grant
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