The Tracy-Windom Distribution and the Technology Adoption Lifecycle Phases.

In my recent bookstore romps, the local B&N had a copy of Mircea Pitici’s The Best Writing on Mathematics 2015. I’ve read each year’s book since I discovered them in the public library system in San Antonio years ago. I read what I can. I don’t force myself to read every article. But, what I read I contextualize in terms of what does it mean to me, a product strategist. I’m a long way from finished with the 2016 book. I’m thinking I need to buy them going back as far as I can and read every article. Right now that’s impossible.

I thought I was finally finished with kurtosis, but no I wasn’t thanks to the 2015 book. So what brought kurtosis back to the forefront? Natalie Wolchover’s “At the Far Ends of a Universal Law,” did. The math in that article is about the analytic view of phase transitions or coupled differential equations described by something called the Tracy-Widom distribution. That distribution is asymmetric meaning it has skewness, which in turn means it exhibits kurtosis.

In “Mysterious Statistical Law May Finally Have an Explanation” in the October 2014 edition of Wired magazine, the Tracy-Wisdom distribution is explained. It is linked to distributions of eigenvalues, and phase transitions. The phase transition aspect of the Tracy-Widom distribution caught my attention because Geoffrey Moore’s technology adoption lifecycle is a collection of phase transitions.  The article contained a graph of the Tracy-Widom distribution, which I modified somewhat here.


I annotated the inflection points (IP) because they represent the couplings between the differential equations that comprise the Tracy-Widom distribution. I used thick black and blue lines to represent those differential equations. The Tracy-Wisdom distribution is a third-order differential equation, which is comprised of two second-order differential equations (S-curves), which in turn are comprised of two differential equations each (J-curves).

The cool thing is that we move from a stochastic model to an analytic model.

I removed the core vs tail color coding in the WIRED diagram. In my earlier discussions of kurtosis, the core and tails were defined by the fourth moment, aka the inflection points coupling the first order differential equations. The error persists in this figure because the inflection points were hard to determine by sight. Notice also that the WIRED diagram hints at skewness, but does not indicate how the distribution is leaning. For more on leaning and theta, see Convergence and Divergence—Adoption, De- adoption, Readoption, and More On Skew and Kurtosis. They are taking the Tracy-Widom distribution as a given here, rather than a transformation of the normal. Much about kurtosis is not resolved and settled in the mathematics community at this time.

The dashed vertical line separating the two sides of the distribution intersects the curve at the maxima of the distribution. The maxima would be a mode, rather than a mean. When a normal is skewed, the mean of that normal does not move. The line from the mean on the distribution’s baseline to the maxima slopes meeting the baseline at some θ. Ultimately, the second-order differential equations drive that θ. Given I have no idea where the mean is, I omitted the θ from this diagram.

In the 2015 book, the left side of the distribution is steeper, almost vertical, which generates a curve closer to the base axis, a curve with a tighter curvature, aka a larger value for Κ1 (smaller radius); and the right side is flatter, which generates a looser curvature, aka a smaller value for Κ2 (larger radius)—note that curvature Κ = 1/r.


So both figures can’t be correct. How did that happen? But, for my purposes, this latter one is more interesting because it shows a greater lag when transitioning between phases in the technology adoption lifecycle and in firms themselves, particularly in firms unaware that they are undergoing a phase transition. In Moore’s bowling alley, where the Poisson games occur. The phase transitions are more symmetric and faster. In the transition between the vertical and the IT horizontal, the phase transition can be slower, less symmetric. In the transition between early and late main street, the phase transition is fast. Most firms miss their quarterly guidance here, so they undergo a black swan, which is surprising since a firm should know when they are approaching having sold 50% of their addressable market. A firm should also know when they are approaching having sold 74% of their addressable market, so they wouldn’t hear from the Justice department or the EU. Of course, most firms never get near that 74% number.


Here I aligned a Tracy-Windom distribution with each technology adoption lifecycle phase boundary. I have no idea about the slopes of the s-curves, the second order differential equations. Your company would have its own slopes. Your processes would give rise to those slopes, so collect your data and find out. Knowing your rates would be useful if you were continuously doing discontinuous innovation.

I’ve labeled the phases and events somewhat differently from Moore. TE is the technical enthusiast layer. They don’t disappear at any point in the lifecycle. They are always there. Well, they do lose focus in the software as media model in the vertical phase of the adoption lifecycle.  Likewise in all late phases. BA is the bowling ally. Keeping your six early-adopter (EA) channels of the bowling alley full is key to continuously doing discontinuous innovation. V is the verticals. There would be one vertical for each early adopter. The early adopter is an executive in a vertical market. IT H is the IT horizontal market. Early main street (EM) is another term for the IT horizontal. If we were talking about a technology other than computing, there would still be a horizontal organization servicing  all departments of an enterprise.An enterprise participates in serval horizontal markets. Late main street (LM) also known as the “Consumer Market” where we are today, a market that orthodox business practice evolved to fit, a market where innovation is continuous, managerial, and worse “disruptive in the Christensen way (cash/competition).” The technical enthusiast and bowling alley is wonderfully discontinuous and disruptive positively in the Foster way (economic wealth/beyond the category). L is laggard or device phase. P is phobic or cloud phase. I the phobic phase computing disappears. The technical enthusiasts will have their own Tracy-Windom distributions. Moore’s chasm being one. Another happens when the focus changes from the carried to the carrier  in the vertical phase. And, yet another happens when aggregating the bowling alley applications into a carrier-focused, geek/IT facing product sold in the tornado. Cloud rewrites another. An M&A would cause another as well. That product would sell in the second (merger) tornado (not shown in the figure).

The first second-order differential equation accounts for what it takes to prepare to make the phase transition. The second second-order differential equation accounts for operationalized work during the phase. The diagram is not always accurate in this regard.

More than enough. Enjoy.

Geez another edit, but over packed.


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