Commentary: Inspecting Zipf’s last works

George Kingsley Zipf ¹ is best known for his findings concerning power-laws for word frequency and city-size distributions. The idea of an “isomorphism between linguistic structures and social structures” (Zipf, 1950b) he explored in great detail in his seminal book Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology published in 1949 (Zipf, 2012). Zipf won a Guggenheim fellowship and admission required a physical examination. In the course of this examination, he was diagnosed with cancer (Prün and Zipf, 2002). In 1950, Zipf died at age 48 (Prün and Zipf, 2002).

When a successful scientist dies at a young age, one is tempted to imagine what else they could have found out if had they lived longer. Here, we want to inspect his ideas that might give some hint of how he thought about exploring the context of his possible future work.

Zipf’s Guggenheim fellowship probably best reflects his last scientific undertakings. The respective proposal (Zipf, 1950b) represents a valuable source for what Zipf intended to investigate. Zipf’s work plan consisted of two parts (Zipf, 1950b):

(1) A statistical study of the relationship between the number of different styles of different consumers’ goods, their price, and their rate of style-change in mail order houses.

Zipf’s planned research had two goals. First, he wanted to provide further empirical evidence for his Principle of Least Effort (PLE) (Zipf, 2012). The idea of least effort goes back to Ferrero (1894) and states that people tend to take the easiest route to achieve a goal. Zipf explored the PLE in various contexts, in particular to explain power-law word-frequency distributions and power-law city-size distributions (Zipf, 2012). He theorized that the power-law word-frequency distribution was based on the tendency to communicate efficiently with as little effort as possible. Similarly, he argued that urban systems are organized in such a way that costs of production and consumption are minimized in terms of labor and transport (Rybski and Ciccone, 2023), in short in exploring the competition between cities of different sizes in these terms.

Second, Zipf promised (Zipf, 1950b):

A practical value in disclosing statistical regularities in marketing phenomena that may be helpful in reducing some of the guess-work in production-distribution.

It seems Zipf had shifted his attention towards marketing, business, and more broadly towards economics.

Pre-PLE, Zipf had already analyzed economic data in Zipf (1947). He makes three observations: (i) He finds that the total number of business establishments, manufacturers, and retail stores each scales linearly with the population count of the respective communities, see also Zipf (2012: Fig. 9–3). (ii) Zipf reports that the diversity of these three categories, that is, the number of different business establishments, manufacturers, and retail stores, scales with the community population count to an exponent 1/2. (iii) He observes that the number of employees in an occupation category decreases with an exponent −1/2 with the number of sub-categories. That is, less employees work in more specialized occupations. These findings can be compared with what is studied today under the umbrella of urban scaling. ²

Post-PLE, Zipf analyzed power-law relations for brand-names, manufacturers, factories, and branches (Zipf, 1950c). To some extent, the work was motivated by the U.S. Lanham Act which took effect in 1946/1947 and which regulates the registration and protection of trademarks. For example, Zipf reports a power-law relating the number of products to the number of different manufacturers (many manufacturers make few different products and few manufactures make many different products). Another example is the finding that the number of different factories in cities scales with an exponent 0.9 with their population size. The exponent deviates significantly from his theoretically predicted exponent 1.

These later works indicate Zipf’s ambitions to enter the field of economics, and with the publication of Zipf (1950c) in Econometrica he had a foot in the door. Later, power-law cities-size distribution became mainstream in urban economics (e.g., Ioannides and Overman, 2003; Gabaix, 1999; Krugman, 1996; Singer, 1936) and also more generally power-laws in economics and finance (e.g., Gabaix, 2009; Brock, 1999).

In his Guggenheim proposal (Zipf, 1950b), Zipf also mentions his preliminary studies analyzing registers of manufacturers and mail-order catalogs. Both works were published after proposal submission, the former (Zipf, 1950c) we summarized above and the latter appeared in the Journal of Marketing (Zipf, 1950d). In the spirit of his previous work, in Zipf (1950d) Zipf searches for power-law relations in the occurrence and prices of products in mail-order catalogs. From today’s perspective, both publications (Zipf, 1950c, 1950d) had limited impact with barely any citation.

A similar fate befell another publication. Despite the promising title “The frequency-distribution of wages and the problem of labor unrest” (Zipf, 1950a), Zipf seems to get lost in analyzing wage distributions. Consistent with his previous work, he flips the axes of the frequency distributions, that is, wage on the vertical axis and rank or count on the horizontal axis. In both forms, cumulative and density, he observes relatively flat slopes in double-logarithmic representation, which because of flipped axes correspond to large exponents – entering ranges where the power-laws lose their interesting properties (Newman, 2005).

The last article that we discuss here is little known (we found Zipf (1950e) via Baciu (2020)) and does not even appear in Zipf’s own bibliography (Prün and Zipf, 2002). Maybe, “transformation: systematic empiric social science and the principle of least effort” (Zipf, 1950e) depicts Zipf’s worldview. The essay (Zipf, 1950e) comprises four parts. First, Zipf argues that society consists of groups and each group has its own values and norms. A person from another group is usually not familiar with the values and norms and is thus perceived as “barbarian.” Similarly, Zipf calls “holymen” those individuals representing the authority in regards to the values and norms. They carry the moral responsibility. However, each group considers their own morality absolute. Second, Zipf argues that science undermines the authority of the holymen. Human behavior should be treated as natural phenomenon and studied with objective methods of empirical science. He then discusses four psycho-social approaches, including Gestalt psychology. Following Korzybski’s “The map is not the territory,” he draws the analogy to a person’s “phenomenal world” and “the origin of his sensation on the other” (Zipf, 1950e: p. 16). Third, as quantitative examples of empirical studies, Zipf mentions analysis by Pareto (income distribution), Auerbach (city size distribution), Condon (word frequency distribution), Reilly (retail gravitation), and Stewart (gravitational attraction between cities). He is convinced of the existence of “natural laws governing human behavior” (Zipf, 1950e: p. 17). Finally, Zipf calls for a unifying principle “that breaks down the traditional barriers of departments” (Zipf, 1950e: p. 17). In this context, Zipf also mentions his work around the PLE. Accepting natural laws of human behavior is a prerequisite to systematic empirical social science.

The essay (Zipf, 1950e) and the overall impact of Zipf’s work coincide with the development of Wiener’s Cybernetics and Bertalanffy’s General System Theory (for an introduction, we refer to, e.g., Alterman Blay and Castilho Piqueira (2020, 2024)). Wiener defines Cybernetics as “control and communication in the animal and the machine” (Wiener, 2019). See Gershenson et al. (2021) for a discussion of Cybernetic Cities. With his General System Theory, Bertalanffy aims at defining general principles of dynamic interaction and envisions this to unify science (Bertalanffy, 1950). Whether Zipf, Wiener, and Bertalanffy knew about each other’s work, when publishing, or to what extent they influenced each other, is beyond the scope of this piece. What can be said is that the power-law as signatures of complex systems can be attributed rather to the fractal hype around Mandelbrot’s book (Mandelbrot, 1982).

To summarize, Zipf advocated interdisciplinary research and quantitative social science. While his PLE-book (Zipf, 2012) represents an influential opus with lasting impact, his late work could have benefited from fresh inspiration. Maybe Zipf (2012) was already his publication with the greatest impact (Sinatra et al., 2016). In any case, there is no doubt about Zipf’s achievements and the importance of his life’s work in various disciplines. Zipf might also be seen as an early “power-law enthusiast,” with apparent fascination searching and finding power-laws in diverse fields (see also, e.g., Newman, 2005; Clauset et al., 2009; Mitzenmacher, 2004). Further research is necessary to better understand to what extent Zipf was a forerunner of complex systems science around scaling and respective theories (e.g., Newman, 2011). We welcome speculation about Zipf’s role in developing scaling and power-laws in social and linguistic systems.