Tobler First Law

Extract from Grasland (2009) : SPATIAL ANALYSIS OF SOCIAL FACTS: A tentative theoretical framework derived from Tobler’s first law of geography and Blau’s multilevel structural theory of society

Looking at the contributions published by AAAG on Tobler’s first law of geography, I was very surprised to observe that most contributor’s focused on the famous isolated sentence (“Everything is connected to everything, but near things are more related to each other than distant things”) without exploring its relations with the article where it was published, the article published at the same time period and the following works of Tobler. It is not so surprising if we consider that, out of some very famous paper like “Push Pull migrations lows” or “Geographical filters and their inverse” , many publication of Tobler are difficult to read as they was unpublished preliminary version or purely grey literature.

Tobler’s first law was published in the paper entitled “A computer Movie Simulating Urban Growth in the Detroit Region” in the review of Economic Geography in 1970. In this relatively short paper of 7 pages, the main concern is the elaboration of a model of population forecast by computer simulation, which is a very near from actual cellular automat procedures and other dynamic model like SIMPOP. The first law of geography appears in its complete form only at the third page but the first part (“everything is related to everything”) is formulated much earlier in the second paragraph of the introduction, with a clear position in favor of a positivist and structural approach of social life: “As a premise, I make the assumption that everything is related to everything else. Superficially considered this would suggest a model of infinite complexity; a corollary inference is that social systems are difficult because they contain many variables; numerous people confuse the number of variable with the degree of complexity. Because of closure, however, models with infinite numbers of variable are in fact sometimes more tractable than models with a finite but large number of variables” 93. Tobler clearly consider that a scientific approach implies to write simple models able too capture a large part of the reality, rather than trying to cover every dimension. Moreover, the simplest models are the most general and are therefore likely to be transposed from one field of research to another. About his model, Tobler comments: “The model recognize that people die, are born and migrate. It does not explain why people die, are born and migrate. Some would insist that I should incorporate more behavioral notions (…). My attitude, rather, is that since I have not explained birth, death or migration, the model might apply to any phenomenon which has these characteristics, e.g. people, plants, animal, machines (which are built, moved, destroyed), or ideas. The level of generality seems inversely related to the specificity of the model”.

My own practice of Tobler’s work has led me to the personal feeling that the key of Tobler’s first law can not be found in this paper which is rather a claim in favor of scientific geography but not really a demonstration of the law, at least for the second part (near things are more related to each other than distant things). It is rather in two other papers published at the same period that we can find a clearer demonstration of the interest of a symmetric approach of positions and interactions through distance, which is for me the major contribution of Tobler to theoretical geography. To support my assumption that linking position and process is the core of Tobler’s contribution and the key of interpretation of its first law, it is important to remind also that the paper Geographical Filter and their Inverses was published in 1969, one year before the first law, and focused precisely on the theoretical problem of discovering processes through the analysis of forms: ‘If one assumes that geographical processes operate at various scales, then a filtering by scales could separate processes. The Fourier interpretation of scale is the wavelength, or, equivalently, the form of the spread function. Large-scale processes can thus be separated from small-scale processes as a preliminary step in geographical analyses. Some examples follow. This was really a research program that was defined here and the publications of the following years was therefore a clear attempts to propose empirical validations in various fields.

In the paper published in 1970 and entitled “Geobotanical Distance between New Zealand and Neighboring Islands” 94], Tobler presents for the first time an empirical application of the inversion of gravity model and demonstrate how it is possible to derive positions from interactions and how, more generally, it is possible to link the analysis of flows and dissimilarities. In this work realized with natural scientists, Tobler appears has first author (against alphabetic order) and this fact demonstrates clearly that he realized the core of the demonstration. Briefly said, the paper demonstrate how it is possible to propose a simulation model of diffusion of Botanic species between the islands of new Zealand and to estimate what was the effect of distance on this diffusion. Even if we ignore the initial location of species, it is possible to make assumption on it through an inversion of gravity model. Of course, the model does not fit very well with reality as it is based on Euclidean distance, but further complexity can be added and anyway it is sufficient to demonstrate that near things are more related to each other than distant things. Contrary to the common opinion, Tobler does not believe that the concept of distance can be reduced to its basic geometrical component. But he claims that the simplest model should be firstly applied before to propose more sophistical versions based on residuals. This is very clear in this paper where the authors write: “Model-building is useful not only because it may allow predictions but also because it identifies areas for further research by making assumptions explicit (…) Although the existing model cannot explain all distributions, one-third of the floristic variation between New Zealand and neighboring islands is explained by island location and size alone, demonstrating the importance of spatial arrangement in plant geography”.

One year later, in 1971, Tobler published with S. Wineburg in Nature what we consider as his most fascinating realization. Entitled “A Cappadocian Speculation” [95], this very short paper proceeded to a secondary analysis of archeological data and tried to determine the location and the name of unknown cities mentioned on the cuneiform tablets found by Hrozny near the village of Kültepe in 1925. The author notice immediately in the introduction that “This is theoretical geography in the sense of Bunge, which is conditional on several assumptions.” Contrary to previous researchers that had tried to analyze the relation between cities mentioned on the tablets through historical approach or linguistic approach, Tobler focus immediately on geographical distances that he proposes to estimate through an inversion of the gravity model assumption: ‘On a purely random basis, one would expect the names of large towns to occur more frequently than the names of small towns. The total expectation is thus that the interaction between places depends on the size of the places and the separation between the places. This rather obvious result has been verified in a large number of societies and for many phenomena. Specifically, we expect the interaction to increase as the places get bigger, and to decrease as they are farther apart. Many functions satisfy such a requirement. For social interaction the most common formulation is the so-called gravity model:

Iij = k Pi Pj / d2ij Where Iij is the interaction between places i and j; k is a constant depending on the phenomena; P, is the population of i; P is the population of j; and d is the distance between places i and j ‘[95]. Of course, we ignore what is the size of the cities and the flows between them, but we can propose an estimation of these variables: the size of cities is estimated by the frequency of quotations on the tablets and the flows are derived from the common citation of names. It is now possible to extract the distance by transforming the previous equation in:

dij = (k. Pi Pj / Iij)12

But distance is only an intermediate steps of the research as the real goal is to link the names of the unknown cities mentioned in the tablets with archeological spots that was discovered in the region. To fulfill this objective, Tobler proposed to use a trilateration method in order to derive the positions of cities from their distance and to produce a map of hypothetical position as some name of cities were known and related to precise locations in space. The final result was the famous map presented in Figure 1.

One more time, Tobler use Euclidean distance as basic reference of the analysis but he claims that it is not an obligation and that the model proposed here could be extended to different situations: “Distance may be in hours, dollars, or kilometers; populations may be in income, numbers of people, numbers of telephones and so on; and the interaction may be in numbers of letters exchanged, number of marriages, similarity of artifacts or cultural traits and so on” [95]. In our opinion, the most important discovery of Tobler in 1970-71 is not the efficiency of the gravity model (it was established before), neither the importance of simplicity in scientific model (it was a very common thought in this period). The crucial discovery of Tobler is the fact that positions and interactions can be analyzed in a symmetrical way through the concepts of movements and accessibility. From this point of view, the common distinction proposed by M. Castells 19 and followers between the so-called ‘space of places’ and ‘space of flows’ is purely nonsense. What does really matter is to define what are the relevant geometries for a common description of both positions and interactions. A real criticism of Tobler rely probably more on the choice of his favorite geometry (Euclidean, continuous) than in the ignorance of the importance of flows. But we have seen that Tobler’s always claim that they are no reason to limit the analysis of relations between flows and structure to the case of Euclidean distance. From this point of view, Tobler’s idea are fully compatible with the network society and we have no reason to reject Tobler’s first law on the basis of more and more complex forms of accessibility taking the form of network.

Claude Grasland
Professor of Geography

My main field of research and teaching is spatial analysis of social facts.