Power Law Curve Vs. Logistics Curve

ML Aug 23, 2006

The fact that so many real-world situations follow what looks to be a 1/X long tail power law distribution curve has profound implications. As Chris Anderson says in his book, it may change the shape of business. But a true power law curve is infinite at both ends. As X approaches zero, the head of the curve goes up to infinity, because 1/0 is infinitely high. Few products in the real word, even blockbuster hits, have infinite demand. Even things that can be re-consumed like movies have an ultimate potential consumption of the population of the world watching the same movie over and over for the time period being calculated. It’s a scary picture, but yes, it’s most definitely finite demand.

Similarly with the long tail, which represents the diversity of products, there is a finite supply. For a given time period, it is possible, albeit difficult, to count every single product on the planet–even taking into account things that are not being advertised for sale, but really are.

So you see, even when real-world situations look entirely like a power law curve, in reality, they’re not. As you zoom in on the head, you reach the maximum demand for your top selling product. As you zoon in on the tail, you reach the end of your inventory. This “reaching the end” indicates that there actually was an inflection point in the formula, and regression analysis would reveal that a different formula may be at play, albeit one with some wildly large or small values plugged in to make it look like a 1/X power law curve. Candidate formulas would be the population density curve or the logistics demand curve. They both speak of reaching perfect equilibrium in marketplaces and ecosystems (population and sales drop off as restrained by available resources). And this is today’s main shape of business. Sam Walton’s use of the logistics curve is sometimes credited allowing Wal-Mart to achieve such stellar growth, always having exactly the right inventory on the shelf at the right time and almost never any shortages.

Anyway, a post by Alex Barnett implies that such curves may be at work in keyword traffic distribution, which is of great interest to the HitTail crowd. There are two radically different directions that the ultimate goal of HitTailing could go, based on how HitTailing impacts the shape of the curve over time. If Alex’s observations are true, and HitTailing would enhance and continue the trends he’s observed, then HitTailing will be about changing the shape of the curve into something more like the pure logistics curve. If instead, HitTailing grows the head and lengthens the tail proportionately, then the inflection points are never moving toward the center, and the 1/X power law holds true, and will always be the best formula fit in regression analysis.

I know this post is a bit too techie for the mainstream marketing crowd we’re trying to attract with HitTail. Suffice to say, based on our discoveries, we’re going to adjust the HitTail site to foster the best HitTailing habits. It may be to achieve the purest logistics curve, or it may be to scale up the 1/X curve.

One response
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    Mike Levin

    I’ve only looked at the diagrams so far, but in addition to Alex’s article, this one from Dan Bricklin is worth checking out as well.

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