Jan 1, 0001 12:00 AM

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title: “Type-token distributions beyond the Zipf law: a simple model with choice”



location: “XXVII Sitges Conference on Statistical Mechanics, Spain”


street: Piazza Marina, 61

city: Palermo

region: PA

postcode: ‘90133’

country: Italy

summary: “A presentation discussing a statistical model for describing the distributions of movies by popularityn”

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date: “2023-05-29T00:00:00Z”

date_end: “2022-04-27T11:00:00Z”

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publishDate: “2023-04-26T00:00:00Z”


  • Mikhail Tamm
  • Paul Krapivsky
  • Vejune Zemaityte
  • Ksenia Mukhina
  • Maximilian Schich


  • film popularity
  • type-token distributions

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Zipf law is one of the very few regularities believed to permeate quantitative social sciences. It describes distribution of tokens (objects) between a given set of types (classes), and states that if types are order in descending order of popularity (number of tokens in them), then the size of a type is proportional to its rank in some negative power. This law is well established in wealth distribution (Pareto law),1 city science (Zipf law for city sizes),2 linguistics (distribution of word frequencies).3 The same law is believed to hold for the distribution of demand for cultural objects like movies, books, singles, etc., which is supposed to lead to important consequences for marketing.4

Using historical data for film industry we show here that it is not always the case. In fact, the distributions of movies by popularity is (apart from a small number of super-hits) well described by exponential rank-size distribution, i.e., size (popularity) decays exponentially with rank.

In order to describe this behavior we construct a following simple statistical model. Consider a set of $m$ initially empty classes (types) and sequentially add $N$ tokens to these types according to the following procedure: first, choose a subset of a types at random, then determine which of the classes in this subset currently has the largest number of tokens (if there is a draw between two or more leaders, then choose one of them at random) and add an additional token to this class.

In the limit of large number of classes $m → ∞$ and finite number of tokens per class $t = N/m$ the model is described by the infinite set of differential equations $$dC_{k}/dt = C_{k−1}^a − C_k^a,$$ where $C_k$ is the fraction of types with at most $k$ tokens in them. We show that for large $k$ and $t$ the distribution $C_k(t)$ converges to a scaling form $C(x)$ where $x = k/t$, and $C(x)$ satisfies $$(aC^{a−1} − x)C′ = 0$$ with boundary conditions $C(0) = 0, C(∞) = 1$. This equation has a solution $$C(x) = \begin{cases} (x/a)^{1/(a−1)} & \text{for }0 < X < a, \
1 & \text{for } x > a\end{cases}$$ This means that the type-token distribution has a form of a travelling wave with growing average number of tokens per class, the leading class has approximately $at$ tokens in it. Converting this limiting behavior into the form of rank-size distribution we show that for $a ≫ 1$ it converges to the exponential distribution akin to one observed in the data.

Conference website:

  1. W.J. Reed, Physica A, 319, 469 (2003). ↩︎

  2. M. Barthelemy, The Structure and Dynamics of Cities: Urban Data Analysis and Theoretical Modeling, Cambridge University Press (2016). ↩︎

  3. M. Gerlach, E.G. Altmann, New J. of Physics, 16, 113010 (2014). ↩︎

  4. C. Anderson, The long tail: why the future of business Is selling less of more, Hyperion, New York, 2006. ↩︎

Researching screen industries through data analysis and visualisation.