Hail The Radiation Model

We have a revolution in how to best predict transport and commuting rates:

The gravity law is the prevailing framework with which to predict population movement, cargo shipping volume, and inter-city phone calls, as well as bilateral trade flows between nations. Despite its widespread use, it relies on adjustable parameters that vary from region to region and suffers from known analytic inconsistencies. Here we introduce a … radiation model [that] predicts mobility patterns in good agreement with mobility and transport patterns observed in a wide range of phenomena. (more)

The gravity law assumes travel is proportional to the product of powers of the to and from populations, divided by some function of the distance between them:

The gravity law assumes that the number of individuals Tij that move between locations i and j per unit time is proportional to some power of the population of the source (mi) and destination (nj) locations, and decays with the distance rij between them as

Tij = mia njb / f(rij)

where a and b are adjustable exponents and the … function f(rij) is chosen to fit the empirical data.

The radiation model fits better by instead looking at how many people live closer than the destination location:

Step one, an individual seeks job offers from all counties, including his/her home county. The number of employment opportunities in each county is proportional to the resident population. … We capture the benefits of [each] potential employment opportunity with a single number, z, [independently and] randomly chosen. … Step two, the individual chooses the closest job to his/her home, whose benefits z are higher than the best offer available in his/her home county. … We denote with sij the total population in the circle of radius rij centred at i (excluding the source and destination population). … The radiation model is

Tij = Ti mi nj / (mi + sij)(mi + nj + sij)

… Ti … is the total number of commuters who start their journey from location i.

Amazingly, this better fitting radiation model only depends on distance indirectly, via population density. It suggests that while distance matters, it is almost never an overwhelming consideration. In the modern world, while political barriers are often insurmountable, distance is detail.

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  • Dimitriy Masterov
  • http://reflexionesfinales.blogspot.com/ russell1200

    That is interesting.

    I think you forgot to give your source.

    I take it that it is : Filippo Simini, Marta C. Gonza´lez, Amos Maritan & Albert-La´szlo´ Baraba´si’s A universal model for mobility and migration
    patterns?

  • Poelmo

    “Amazingly, this better fitting radiation model only depends on distance indirectly, via population density.”

    Right.

    “It suggests that while distance matters, it is almost never an overwhelming consideration.”

    Nope, that depends on f(rij) and the population density. For example if f(r) is the travel time or fuel cost than it is proportional with r while the population density is probably proportional to r^2, then the radiation model depends on r more strongly than the gravity model. Distance can matter, that’s also in the text:

    “Step two, the individual chooses the closest job to his/her home, whose benefits z are higher than the best offer available in his/her home county”

    This suggests that distance is not just a detail in the radiation model: the agents clearly prefer staying closer to home.

    The radiation model makes sense (in theory), but unless I’m missing something, Robin’s conclusion about distance doesn’t. Also, I’m not convinved that such simple models are useful to model a complex economy of emotional human beings. I suspect that in order to fit the empirical data you have to deform any of these models to such levels that not much of the original formula remains and you might as well have started with a random function (I mean it’s always possible to find some polynomial that interpolates the empirical data for a certain time period, but that doesn’t really teach you much about the underlying dynamics and it sure as hell doesn’t allow you to predict the future), but maybe that’s just my distrust of economists speaking.

    • http://hanson.gmu.edu Robin Hanson

      The f() function is only in the gravity model, not the radiation model. And if you’d look at the paper, you’d see that they don’t need to add anything more to the model I described to fit the data well.

      • Poelmo

        “The f() function is only in the gravity model, not the radiation model.”

        Yes, but you (and I) are comparing the dependency on r of the two models, so we have to compare f(r) in the gravity model to the population density in the radiation model.

        “And if you’d look at the paper, you’d see that they don’t need to add anything more to the model I described to fit the data well.”

        I see Dimitri put the link up, and yes, I must say I’m pleasantly surprised with the results in the graphs. Without tweaking the radiation model does seem to fit empirical to a high degree (though the logarithmic graphs could hide smaller discrepancies to the naked eye). Oh well, you learn something new every day…

  • arch1

    The radiation model is very interesting but I don’t understatnd the mi in the numerator.

    Assume that Central Torrance contains 10% of Torrance’s residents.

    Then if I understand it, the radiation model predicts that the number of people who commute from Central Torrance to Pasadena is only 1/100 the number of people who commute from Torrance to Pasadena (since Central Torrance’s Ti and mi are each 1/10 of Torrance’s, while each factor in the denominator is the same for Torrance vs Central Torrance).

    Am I missing something?

  • Dimitriy Masterov

    I would be curious to see what happens if you use travel time rather than straight line distance to do this analysis. The more important issue, however, is that the gravity model ignores the presence of other alternatives (not n_{i} or m_{j}), so in many ways it is a straw man and its poor performance is no surprise. Something like the Huff model would have been a better choice for comparison (http://www.esri.com/library/whitepapers/pdfs/calibrating-huff-model.pdf), especially since that is a probabilistic model. The other issue that I can’t seem to work out is if you can relax the assumption that one of these variables is people. Can we make n_{j} a store, and use square footage instead of population?

  • Doc Merlin

    Brilliant.
    I’ve been saying something similar for years. Laws are the topography of our otherwise flat world.

  • imagination

    Offtopic – have you imagined replacing traders of the City of London with emulations? 🙂

  • richard silliker

    Distance is paramount. It provides latency.

  • Doug S.

    Within a large country, the relevant distances tend to be “practical to drive to” and “impractical to drive to” – if you live in New York State, moving to California and moving to Iowa both involve about the same amount of disruption…

  • Rob

    “Tij = Ti mi nj / (mi + sij)(mi + nj + sij)”

    Why the addition of the Ti term here which wasn’t in the gravity model? That seems unrelated to the gravity v radiation issue.

    Any simple explanation why does the denominator takes the form it does?

    • Smarmet

      The denominator (and the mi*nj in the numerator) come from two probabilities: first, the probability a commuter does not find a good job closer than the target area (mi/[mi+sij]), and second, the probability that they do find a good job in the target area (nj/[mi + nj + sij). The product of those two probabilities gives the probability that a commuter from m will end up in target area n. That probability is multiplied by the number of commuters from county m (Ti) to give the number of commuters expected from m to n (Tij)