City Travel Scaling

Here’s a fascinating factoid I found in Geoffrey West’s new book Scale:

An extremely simple but very powerful mathematical result for the movement of people in cities. .. Consider any location in a city. .. predicts how many people visit this location from any distance away and how often they do it. .. It states that the number of visitors should scale inversely as the square of both the distance traveled and the frequency of visitation. ..

Suppose that on average 1600 people visit the area around Park Street, Boston from four kilometers away once a month. .. only 400 people visit Park street from 8 kilometers away once a month. .. how many people visit Park street from four kilometers away but now with a greater frequency of twice a month. .. also … 400 people. (pp.347-9)

As cities are basically two-dimensional in space and one-dimensional in time, this implies that most visits to a place are by people who live nearby (not so surprising), and also by people who visit very infrequently (quite surprising). I’d love to see an urban econ model embodying this pattern. Alas West cites “Markus Schlapfer and Michael Szell”, but no publication, nor could I find one online.

The book Scale is on an important yet neglected topic: basic patterns in large systems such as organisms, ecosystems, cities, and firms. Alas West rambles, in part to avoid talking math directly, so you have to skim past many words to get to the key patterns; I bet I could have described them all in ten pages. But they are indeed important patterns.

I found myself distrusting West’s theories for explaining these patterns. He talks as if most of the patterns he discusses are well explained, mostly by papers he’s written, and he doesn’t engage or mention competing theories. But I’ve heard that many disagree with his theories. In particular, though West claims that a 3/4 power law of organism metabolism versus mass is explained by piping constraints (West offers different theories for trees and for animals with pumped blood), while researching Age of Em I learned:

Does our ability to cool cities fall inversely with city scale? Actually, no. We have good fractal pipe designs to efficiently import fluids like air or water from outside a city to near every point in that city, and to then export hot fluids from near every point to outside the city. These fractal designs require cost overheads that are only logarithmic in the total size of the city.

So if this metabolism pattern is due to piping constraints, it is because evolution never managed to find the more efficient piping designs that we humans now know.

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  • gwern

    This ‘donut hole’ of visitors/distance reminds me of living on LI. We would take in exchange students from, say, Sweden or Vietnam or Italy; naturally, while they were here, we or the nonprofit would take them to see American sites like lighthouses or the Statue of Liberty or Broadway plays or the Cloisters or delis or the Bronx Zoo, since they ought to see more than just suburbs and the inside of the school while they were here. It’s easy for us to drive to the LIRR station in Ronkonkoma, hop on the train in to Penn, and then take the subway whereever you need to go, and then come back on the last 11PM train. (We also had a number of other reasons to go in: a family friend was a student in Manhattan so that led to a lot of back and forth, another friend had relatives in Queens, siblings would go through NYC en route to upstate NY schools…) Similarly, if their families visited the USA or some of our own relatives came to visit, they often would go in to NYC too. And naturally, when I talk to people who live in NYC or at least the boroughs, many of them have also visited such places as well – the Cloisters is a great place for a picnic lunch, the Zoo is obligatory for kids, Broadway plays are pricy but worthwhile (assuming it’s not ‘Hamilton’), etc. So, who do you think, of the kind of people I talked to, was the least likely to have gone to the Statue of Liberty etc?

    Our *neighbors*! Sure, we went into NYC fairly regularly for a variety of reasons, but they hardly ever did. It was no more difficult for them than it was for us, they could do the same thing, and it’s an easy enough trip that people do it as a working commute (one reason we preferred weekends, the LIRR parking lots were much easier to navigate). So why didn’t they? Asking them, the impression I always got was that, well, they could do it next month, or maybe next year; they didn’t have any particular reason to do it *now*, and today turns into tomorrow and into the next day and one still hasn’t gone to see _Lion King_ and maybe next year… If they flew into New Orleans or Orlando, they would make sure they visited the major destinations or at least made a conscious decision to drop destinations from the list, and if they lived in NYC they would have eventually gone to this or that on a whim or because they were passing by, but they apparently lived just far enough away to eliminate any sense of urgency but not close enough to make it easy enough to do ordinarily. Thus paradoxically, LIers often know less about NYC than a visiting foreigner.

    • RobinHanson

      Yes, thats also a familiar pattern here near Washington DC. Not sure how well it fits the above empirical distribution though.

  • Gunnar Zarncke

    A very comprehensive treatment of structures in social ecosystems across a wide range of scales (from countries to parts of rooms) can be found in Christopher Alexanders A Pattern Language: Towns, Buildings, Construction (I can highly recommend the book; disclaimer: I own two of the quite expensive volumes of the series).
    In particular he explicitly refers to the distance frequency relationship and draws consequences for architecture from it here:

  • Lord

    In that case, one must ask why evolution hasn’t. Circulatory systems are fractal within material limits. Life doesn’t operate at superconducting/superfluid temperatures though, and pumping occurs periodically rather than continuously, or are there other constraints that make what we ‘know’ not possible? Many a slip..

    • RobinHanson

      Not all organisms use periodic pumps. The available better designs don’t require new materials or temperatures.

      • Lord

        Or energy availability/efficiency? Or adaptability/durability/life span? Or is this something evolution simply hasn’t had time for yet?

  • James Cambias

    For a readable but very thorough investigation of circulatory systems, seek out Steven Vogel’s _Vital Circuits_.

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  • Andrew Swift

    A more effective way to do the simple but powerful calculation would be to use the time it takes for a trip rather than the distance.

    In Toulouse, where I live, it takes a minimum of 15 minutes for me to get downtown, about 5 km. in Hossegor, where I go on vacation, I can go about five times the distance (get several towns away) in the same amount of time — no traffic and no stoplights.

  • Silent Cal

    The pipes post says that with good fractal pipe designs,

    the fraction of city volume devoted to cooling pipes goes as the logarithm of the city’s volume

    (emphasis mine)

    If this is written correctly, there is a certain size at which these fractal piping designs will require all of the city’s volume. (Source is paywalled and the abstract doesn’t tell us what we need).

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