Here’s another one of those headlines that captures attention for all the wrong reasons: “Japanese scientists have devised a mathematical formula that can predict the box office performance of a movie based on the level of related activity on social networks and other websites before and during its release.”

I’ve had the opportunity to read the paper, and while the headline isn’t wrong, it’s really just one part of what could be a very important formula for determining and measuring success (or failure) of integrated advertising campaign efforts. Here’s what scientists, led by Professor Akira Ishii from the Department of Applied Mathematics and Physics at Tottori University, really put together:

A mathematical model for the ‘hit’ phenomenon in entertainment within a society is presented as a stochastic process of human dynamics interactions. The model uses only the advertisement budget time distribution as an input, and word-of-mouth (WOM), represented by posts on social network systems, is used as data to make a comparison with the calculated results. The unit of time is days. TheWOM distribution in time is found to be very close to the revenue distribution in time. Calculations for the Japanese motion picture market based on the mathematical model agree well with the actual revenue distribution in time.

First of all, it’s important to understand that the model was based on word-of-mouth represented by posts on social network systems- not social media specifically. I recognize that social media is becoming synonymous with “word-of-mouth,” but, amazingly, people do still talk to one another.

Human interaction in ‘real’ society can be considered using the ‘many-body’ theory. With the popularization of social network systems (SNS) such as blogs, Twitter, Facebook, Google+ and other similar services around the world, interactions between accounts can be stored as digital data. Although the SNS society is not the same as real society, one can assume that communication is very similar. Thus, researchers can use the huge stock of human communication digital data as observation data for real society.

Using this observation, the authors apply statistical mechanics methods to the social sciences. Since word-of-mouth (WOM) is very significant—for example, in marketing science—such an analysis and prediction of the digital WOM in the sense of statistical physics is of importance today.

In their paper, as an applied field of the statistical mechanics of human dynamics, the authors focus their attention on motion picture entertainment, because the logs of communication for each movie in SNS can be distinguished easily, and the market competition between movies can be neglected because of the character of each movie; the markets for the Harry Potter series, the Pirates of the Caribbean series and Avatar can be distinguished, for example. Moreover, traditionally, the motion picture industry in Japan has daily data on revenue for each movie.

The theoretical treatment of the motion picture business has a long history in the social sciences, such as marketing. The traditional method of forecasting motion picture revenue is to assume the following simple model: R = ABCeD, where A, B, C and D represent qualities such as advertisement budget for the movie, strength of WOM, star power, quality of story, quality of music, etc. Then the formula is ‘linearized’ as follows: log R = _1 log A + _2 log B + _3 log B_2D.

Using the huge stock of market data, the coefficients _1, _2, _3 and _4 are determined using accurate statistics. However, before discussing the actual determined coefficients, physicists must question the model of equation itself. The form of equation itself should be considered deeply and should be derived. Moreover, the model of equation has no way to include the dynamics of human interactions because it is too simplified. In an actual society, including the SNS society, communication between humans has some dynamic behaviors, so that a more realistic model to consider the aggregation behavior of communication in society should be used.

Approaches from physicists also exist. These approaches address the statistical law and the dynamics of motion picture popularity at the box office. Sinha and co-workers found the long-tail distribution of the popularity of top movies in theaters and discussed in detail the similarities and differences between two types of hits—blockbusters and sleepers—mainly in the US market. They suggested that popularity may be the outcome of a linear multiplicative stochastic process. They found the lognormal nature of the tail of total income and the bimodal form of the overall gross income distribution. They also discussed the nature of the decay of gross income per theater with time.

Before discussing their work, it is important to note that the Japanese market is very different from the markets of the US and India. Because of the small land size and the high concentration of people in the metropolitan areas, the distribution of Japanese movie theaters is very concentrated in metropolitan areas such as the greater Tokyo area, the greater Osaka area and Nagoya, for example. Moreover, the three major motion picture companies, Toho, Toei and Shotiku, control most theaters in Japan. Because of this concentration of the location of theaters and the control of the major companies, sleeper-type hits never happen in the Japanese motion picture market. All Japanese hit movies are of the blockbuster type. Therefore, Sinha et al’s detailed analysis of sleeper-type hits cannot be applied to the Japanese motion picture entertainment industry.

For blockbusters, Pan and Sinha pointed out that the opening week is the most critical event in the commercial life of a movie. However, from their analysis of their weekly data, they concluded that advertising may not be a decisive factor in the success of a movie at the box office. Asur and Huberman used Twitter logs for movies, focusing their attention on the period comprising one week before the opening and the opening two weeks. However, they did not pay attention to the correlation between daily advertisements and daily weblog (blog) entries. Moreover, they did not pay attention to the daily advertisement budget, but only the total advertisement budget of a certain movie. Ratkiewicz et al discussed online popularity. They proposed a minimal model combining the classic preferential popularity increase mechanism with the occurrence of random popular shifts due to exogenous factors. They analyzed two large-scale networks:

Wikipedia and the Chilean Web. Although their analysis is very interesting and useful, the sudden increase in the popularity of a certain movie has too short a duration for their approach.

The dynamics of the popularity increase have been investigated theoretically by presenting mathematical models to discuss it. The stochastic process has been tried as a way to forecast motion picture revenues, but the approach is still incomplete and could be made more accurate using data from blogs, Twitter or Facebook postings.

A better approach from the point of view of physicists is the so-called Bass model, which was presented as a simple model of aggregation behavior for WOM in 1969. The key concept of the Bass model is a diffusion equation: diffusion ofWOM in society. Many modified Bass models have been presented to analyze WOM for motion pictures. In the Bass model, the authors consider the number of adopters at the time t, R(t). The number of non-adopters is calculated as N–R(t), where N is the number of persons in the market. If advertisements affect the number of people who adopt the products, then the formula for the increasing rate of adoption can be written as: dR(t) / dt = p(N − R(t)), where p is the probability that non-adopters will adopt the product per unit of time due to the advertisement.

People can also be affected by WOM from the adopter. Thus, if considering only the WOM effect, then: dR(t) / dt = q(N − R(t))R(t), where q is the probability that non-adopters will adopt the product per unit time due to WOM from the adopters. Thus, combining both effects, the equation can be written as follows: dR(t) / dt = (N − R(t))(p + q R(t)).

This is the equation of the Bass model. In the Bass model, the advertisement is included only as the factor p. The many modified Bass models include the decrease per time of the advertisement effect using the exponential decay function as follows: dR(t) / dt = (N − R(t)) p e−[alpha](t−t0) + q R(t), where t0 is the time of the release day of the product.

However, the real marketing actions begin several weeks before release. The modified Bass model above does not include such advertisement effects before release. Moreover, the Bass and modified Bass models above do not include rumor effects in real society that are not described using the person-to-person two-body interaction. From the brief review of the previous studies above, the author’s find that the effects of advertisements and WOM are included incompletely and the rumor effect is not included. Therefore, from the point of view of statistical physics, the authors present in their paper a model to include these three effects: the advertisement effect, the WOM effect and the rumor effect. The model presented is applied to the motion picture business in the Japanese market, and they compare their calculation with the reported revenue and observed number of blog postings for each film.

The authors conclude the paper by stating,

“We present the mathematical model of the hit phenomenon as an equation of consumer action where consumer–consumer communication is taken into account. In the communication effect, we include both direct communication and indirect communication. We found the daily number of blog posts to be very similar to the revenue of the corresponding movie. The daily number of blog posts can be used as quasi-revenue. The results calculated with the model can predict the revenue of the corresponding movie very well. We found that indirect communication affects revenue using the calculation and our theory. The conclusion presented in this paper will be applicable to any consumer market.”

I have not done justice to their work. Their paper is both informative and very helpful and well worth downloading.

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