On "Universality" in Spatial Models

In Cosmos, episode 2, Carl Sagan famously said “Extraordinary claims require extraordinary evidence”.

Simini et al. titled their paper about the radiation model “A universal model for mobility and migration patterns.” That was an extraordinary claim. The model was an intervening-opportunities model with no fitted parameters in its original form. It was not universal in any strong social-scientific sense. It has bugged me for over a decade, but I have a long queue.

The problem is that “Universal” can mean at least four different things.

• The same equation everywhere.

• The same mechanism everywhere.

• The same inputs everywhere.

• The same parameter values everywhere.

The paper suggested all four. The later literature does not support that reading.

Masucci et al. showed that the original radiation model’s thermodynamic-limit assumption causes systematic error in finite systems, especially for large cities. They proposed a finite-size correction. Even then, gravity performed better overall in their England and Wales application. They also asked whether universality means all spatial scales, all times, all places? Their results say no.

Yang et al. introduced an extended radiation model with a scale parameter because the original model did not hold across scales, especially at smaller ones. They also argued that population is not always the right attraction measure. At large scales it may work tolerably, but inside cities, point-of-interest density can do better. Once you need a scale parameter and different destination measures by context, the strong universality claim is over.

Kluge and Levermann argued that the radiation model underestimates long-range migration and contains a conceptual inconsistency with substantial numerical effects. They proposed directional preferences to fix it. Again, the original model did not travel unchanged.

So the history of the model is not: universal law proposed, universal law confirmed. It is: extraordinary claim made, then watered down over time by corrections, extensions, and narrower domains of validity.

That is why the usual horse race between intervening opportunities or radiation and gravity models misses the point. The key issue is not whether gravity or radiation wins on some dataset. The key issue is that the word “universal” was doing work the evidence could not support. A model can be useful, elegant, and influential without being universal. These are not synonymous.

My students reviewing the paper for class back in 2015 noted that the county scale is coarse, that jobs are not proportional to resident population, that travel cost and network structure are ignored, that the behavioural assumptions are thin, and that the universality claim was under-tested across places and scales. They go to the core of the paper.

The paper also overreads what its own commuting data can support. Some of the very long county-to-county “commutes” are prima facie implausible as routine daily out-and-back travel. Census journey-to-work flows are not direct observation of daily mobility as implied in the paper. They are residence-work ties constructed from workers’ reported place of residence and primary place of work during the reference week. The Census residence rules then make the problem sharper: people on a weekly cycle, including commuter workers who live near work part of the week and at a family home the rest, are counted at the residence where they spend most of the week. So these long links do not cleanly measure one behavioural object called daily commuting, instead they mix ordinary daily commuting with multi-residence and weekly-cycle work arrangements. That weakens one of the paper’s claims because a good fit to an administrative residence-work matrix is not the same thing as evidence for a universal behavioural law of daily mobility.

This is not unique to Simini, the same inflation shows up elsewhere in the mobility literature. Noulas et al. write of “universal patterns in human urban mobility” from Foursquare data across sampled cities, Yan et al. offer “universal predictability of mobility patterns in cities,” and later papers escalate further to a “universal model of individual and population mobility on diverse spatial scales” and even “a universal opportunity model for human mobility.” In each case, “universal” does not mean law-like validity across contexts, scales, and measurement regimes. It means something weaker: a common stylized mechanism that performs reasonably well on selected datasets. That is exactly the slippage at issue in Simini as well.

What survived from Simini is Radiation, a model family, which may be a productive starting point for later work. This model family needs finite-size correction, scale parameters, alternative attraction variables, and directional structure, which seems to me that makes it not a universal law. It is another family of approximations.

A social science paper, often written by physicists, borrows a prestige word from physics, makes a broad claim, the editors of Nature get excited, and then the literature quietly retreats to something more modest. Meanwhile the paper collects a large number (over 2000 at this writing) of citations. Fortunately this is a blog post so won’t add to the citations (I hope).

A more accurate title would have been something like: An intervening-opportunities-style model that works ok for some aggregate mobility patterns under some conditions, but that would be less likely to garner over 2000 citations.

References

• Simini, F., González, M. C., Maritan, A., & Barabási, A.-L. (2012). A universal model for mobility and migration patterns. Nature, 484, 96–100.

• Masucci, A. P., Serras, J., Johansson, A., & Batty, M. (2013). Gravity versus radiation models: On the importance of scale and heterogeneity in commuting flows. Physical Review E, 88(2), 022812. Finite-size correction, scale, heterogeneity.

• Yang, Y., Herrera, C., Eagle, N., & González, M. C. (2014). Limits of Predictability in Commuting Flows in the Absence of Data for Calibration. Scientific Reports, 4, 5662. Extended radiation model, scale parameter, limits of the original claim.

• Kluge, L., Levermann, A., & Schewe, J. (2022). Radiation model for migration with directional preferences. Physical Review E, 106(6), 064138.

• Noulas, A., Scellato, S., Lambiotte, R., Pontil, M., & Mascolo, C. (2012). A Tale of Many Cities: Universal Patterns in Human Urban Mobility. PLOS ONE, 7(5), e37027.

• Yan, X.-Y., Zhao, C., Fan, Y., & Di, Z. (2014). Universal predictability of mobility patterns in cities. Journal of The Royal Society Interface, 11(100), 20140834.

• Yan, X.-Y., Wang, W.-X., Gao, Z.-Y., & Lai, Y.-C. (2017). Universal model of individual and population mobility on diverse spatial scales. Nature Communications, 8, 1639.

• Liu, E.-J., & Yan, X.-Y. (2020). A universal opportunity model for human mobility. Scientific Reports, 10, 4657.

Extraordinary claims require extraordinary evidence — Carl Sagan

FIN