Archive for February, 2015

Limit Cycles

February 21, 2015

I’m reading Colin Adams’ Zombie & Calculus. Zombies are not an interest of mine. There are no Zombie Rescue stickers on my car. I’ve never seen a Zombie movie and won’t. All right, I’ve escaped the Zombies. My differential game.

In the book the protagonists, those who have not yet been caught by the Zombies, and the antagonists, the Zombies engage in differential games of capture and pursuit. A differential game is a game played between two or more players where each player has their own differential equation. A differential equation expresses a rate like iterations per year. Some design considerations might involve rates like how fast will we drive away our geeks fast will we attract consumers. Or, bringing it back to capture and pursuit, what can we do about that fast follower?

In the book, the Zombies, playing these games, eventually caught almost everyone. Well, eventually caught up with almost everyone is more like it. Almost? Well, the author lived to tell the tail. Meanwhile back in the games, some of the protagonists were able to see each individual’s attempt to stay away from the Zombies. They saw every experiment. In one of the experiments, the protagonist took a circular path. The Zombies kept their eyes on the runner. The Zombies ran in a circle, so they could keep their eyes on the runner. The Zombies kept moving, but could never get closer to the runner. The runner was faster; the Zombies, slower. The ratio of their speeds determined how far the runner’s circle was from the Zombies’ circle. The runner circle is the limit cycle. Cycles like in trigonometry class where we spent time going round and round. Not cycles like those in The Lincoln Lawyer. Nobody was spiraling. Nope, just circles. So, lets look at a limit cycle.

Limit Cycle 01

The faster person, the red dot, runs on the black circle, the outer circle, the limit cycle. The slower person, the blue dot, runs on the gray circle, the inner circle. The blue arrow illustrates the speed of the slower person. The blue arrow is a vector. Enough of that. The slower person keeps the faster person in sight along the thin blue line, the tangent line. Stop that. Oh, another differential game. Colin Adams, the author didn’t throw down a theorem to define the limit cycle. No, he evaded, the proof-based mathematics, he had his own differential game going. He was pursuing an audience that wouldn’t stand for that.

In the next figure we’ll illustrate what happens when the slower person speeds up or slows down.

Limit Cycle 02

The faster person, the red dot is still on the black circle. At the top, the second person has sped up, which made their circle bigger and closer to the limit cycle. In the middle, the second person has slowed down, which made their circle smaller and more distant from the limit cycle. The blue circle is just there to remind us that there are larger scopes even for the larger player. Ultimately, a limit cycle comes with a category. The market leader as determined by market power sells more, has more customers, more money, and more people than anyone else. If you compete with this company you won’t exceed their limit cycle until they bail out of the category. That’s something they wouldn’t do if they saw the point in staying behind.

And, yes, nobody competes for market power these days. Here in the late main street/consumer phase of the technology adoption lifecycle, its all about promo spend and the myth that you can beat the limit cycle. Las Vegas will take that bet.

But, just for grins, lets say your pursuer finally managed to match your speed, what happens next?

Limit Cycle 03

Once you’re pursuer is as fast as you are, they won’t be inside the circle looking at you. They will realize that they still can’t catch you. They’ll stumble on the Wayne Gretzky quote about being where the puck is going, instead of where it was just now. Well, your staff saw the day when a chunk of their offer that that this pursuer competes on would commoditize, and they made sure that they had some discontinuous technological innovation that everyone could ride for the next twenty years. They put a real option on the circumference of the current limit cycle, an unweighted control point ready to bare some weight.

Limit Cycle 04

So your pursuer was surprised–“Oops,” but they keep trying to catch you. Unfortunately a miracle happens.

Limit Cycle 05

They end up inside your new limit cycle. Your pursuer can’t be faster than you, so being slower, the parallelogram flips rotating them inside your new limit cycle. An odd thing happens though. The center of their cycle is not concurrent, is not the same, as the center of your cycle. The more knowledge it took to create your new offering, the more learning they will have to do. They will be elliptical, not shown, until their process maturity has them following you again without additional learning. Mean while, they dream of the day they own the limit cycle, aka they day you cash out of the category. The elliptical brings with it some messier math.

So as product managers, we compete about and in many limit cycles.

Limit Cycle 07

Each of our features are either about a limit cycle, like the red dot on black circles, or in a limit cycle, like the red dot on the thin blue circle. Each red line in the factor analysis on the right corresponds to the radius of a limit cycle on the left, except inside the blue circle. The red dot is you. Inside the blue circle the factor line is attached to your red dot. All the limit circles are inside a hyperbolic surface. We could just say F2 is a belief function under distribution. Or, we could learn to draw the hyperbolic surface first and fit the circles inside it later.

Realize that you are a fast follower of your whole product components’ providers. You are using someone’s APIs, you follow them, you chase them. It doesn’t feel like pursuit and capture, because they keep their third-party developers informed. Or, do they? Value chains involve limit cycles as well. And, after doing the math, it might be quicker just to buy their company and integrate their functionality–the pursed captures their pursuer, not a good thing when Twitter or Microsoft did it.

As for the Zombies, the last page still eludes me.


Spatio-Temporal Maps

February 16, 2015

Saturday, I looked some of the pages linked to a visualization site I came across over the past week or so. A visualization of the trains out of London and how the trains changed travel times.

Spatio-Temporal Mapping

Spatio-temporal maps have been a topic of mine for decades. Each of the circles represents a half an hour. Here things are fairly straightforward. But, the realities on the ground are quite different. Yes, the train takes you there quickly. Yet, step off the train, and suddenly it still takes five minutes to cross the street in front of the train station. Things slow down. The 3.5 hours it took to get to Neiuweschans did not deform the map so the physical and temporal, the spatio-temporality, stayed aligned. If you map by time, distance gets deformed. Well, not in this viz. But, try flying to China and then taking the train to your home, several days away via that train. That map would get bent up quite a bit.

Since San Antonio has been my hometown if I ever had one, and having worked in Austin at too very different times, I travelled back and forth a lot. At times I took the Greyhound. Once, on Thanksgivings Day, I blew a hose, popped the hood, opened my trunk, someone pulled over, took a look, dug around in the camper topped-pickup truck found a hose, put it on, filled my car with water, and sent me on my way gratis. Dad didn’t get to tell me off. Thanks. And, I didn’t have to walk 45 minutes to get to a phone in what was at that time ag wilderness. Now it’s convenance stores, burbs, same distance, different mindset. Same spatial, different temporal. Well,  tore up the concrete since then and changed the physical as well. Roads get wider, flatter, and straighter over time.

So here would be something to map and look at the math behind intrinsic curvature. Well, try to if you will. So I pulled the northbound distance and time data for the cities between San Antonio and Austin. I threw away the first attempt. I modified the second. I waffled on how to represent the gaps and lags. Then, I realized that the physical, the miles were continuous, as were the times. I used horizontal bars for each quantity. To get both bars ends to line up meant bending the bars, aka introducing curvature. Except for one thing, I’m working in MS Paint. I know all the tricks. Actually, I learned a few Saturday.


There were four segments. Two of the towns were obvious. The third was just pulled out of necessity. The Greyhound used to pull into Kyle, so I know it’s there, along with DQ and a Wachenhut staffed county jail on the frontage road.

I did not pull the southbound data.

So here’s my spatio-temporal map.

Spatio-Temporal Map 01

The graphic at the bottom uses spheres to represent each maximum quantity. The minimum quantity is a smaller circle on the maximum quantities sphere. I put the two larger spheres out front and the smaller ones behind them, so both the spatial quantities and the temporal quantities could maintain their continuity. The line charting had gaps. That just isn’t real. The world is seemless, so the spheres let me present that continuity.

The leftmost sphere represents the minutes traveled with the larger circle. The smaller circle represents the miles travelled. The leftmost sphere is blue because travel time trumps distance travelled. That making it a slow segment of the trip. The travel represented by the second sphere is black, because the miles travelled trumps the time travelled. Again, this sphere has a smaller circle on it representing the shorter quantity, time. The two spheres contact each other miles to miles. That would be the smaller circle on the first, and the outer circumference of the sphere on the second. The third sphere is like the second, but they contact each other times to times, aka the small circle on the second sphere to the outer circumference of the third. The fourth sphere contacts the third, miles to miles.

Too complicated, I know. With a better 3D package, it could be clearer.

Each sphere’s rotational axis is shown. The system of spheres are organized along a curve, rather than a plane.

I went on to draw a curvature graph based on the same data, but the formulas didn’t give the correct results. The arc length of the arcs are too long. The red line edits a spike out of the graph, because it too isn’t real. Again, the road is seamless.

Spatio-Temporal Map 02

I continued my quest for a curvature view with this, the last one.

Spatio-Temporal Map 03

No explanation for this one. I did manage to get a nice arc, but instead of covering the first three points, it went all the way to the fifth point, to Austin, fitting well, but missing the point.

In these diagrams, I used hard data. Well, hard for the moment. I did go back to get the southbound data and found it at odds with the northbound data. Construction can account for why the trip south took more time. Traffic could do same. I did get to that data later in the day. Escaping the big cities eats up the time on this trip. Still, we can consider the hard data. But, there is a soft component. Back during the dot-com one days, the trip was lit up all night. The convenience stores/gas stations stayed open all night. This shortened the psychological distance. These days the speed limits have been reduced for revenue generation purposes–new speed traps. But, I still remember making the trip from south Austin to north San Antonio in 45 minutes. Forget that. But, yes, the speed limits affect the psychological distance as well.

So tying this to product management? Two things:

  • The original train visualization showed a process over geography. Sprints are such a process. And, I’ve talked about product manager geography previously in this blog. For me a roadmap is a map, not a list. Populations are like lakes. Come up with your own analogies. Make your map a real map.
  • Consider my maps to be maps of a user experience. When I reach New Braunfels, please don’t make me open a Wal-Mart popup window. I don’t have time to stop. Your features are organized like trips. Different outcomes, different trips. Done again and again, it becomes familiar and routine. The user knows their way until Agile changes something and DevOps thought nothing of injecting a bug into the user’s UX. Click here, then click there is a geography, a series of histograms/long tails, and flow or psychological time.

Enjoy. Comments.


February 7, 2015

When you continuously look over time at your frequency of use long tails for the functionality in your product, you end up with a surface. I’ve not converted my histograms into nice neat equations, but this YouTube brought surfaces to mind. Similarly, you content marketing, financial results, and progress across the technology adoption lifecycle, if you do such things can be surfaces as well.

Requirement fitness can likewise be modeled. This time, another YouTube brought this to mind. Take the original curve to be the customer’s curve, so the second curve represents our having met that customer’s requirements. Notice that the second curve is the content layer of our software as media, and that our underlying technology is not represented at all. Products foster adoption. Products in the B2B early adopter projects are about the client’s content, not the underlying technology whose adoption is being fostered. The shapes of these surfaces will persist in a given population. In the aggregate late main street phase, the B2B early adopters’ surfaces can be brought back as mass customizations, or simpler templates.

Overall, the technology adoption lifecycle looks like a NURBS curve. It shares characteristics of a uniform B-spline as the curves and distributions are different between the bookends of startup and bankruptcy. Given the uniform B-spline works, we can move to NURBS curves to represent the changes in distribution/curves moving from the Poisson game to the vertical’s normal, the horizontal’s normal, and the late main street’s/consumer’s normal. Laggard/device/cloud is probably a normal as well. The Telco normal was said to be 10x that of the first dot com that lived in the early main street phase of the technology adoption lifecycle. Such weight shifts would show up in the number of standard deviations of each of the normals, and in the weights of the NURB curves. A uniform B-spline repeats the k+1 curve until you reach the n-k curve where the curves are symmetric to the first k curves. This repetition illustrates the purpose of management in the sense of not declining on the decline side of the technology adoption lifecycle. Not declining will still look like a decline since the area under the curve still adds up to one no matter how big in terms of standard deviations the normal is or becomes. The probabilities get thinner, the margins decline.

Discontinuous innovation gives you new distributions, new populations of prospects, new revenue sources. Continuous innovation stretches out your current distribution which in turn thins out your probabilities, and rapidly regresses to the mean. Rapidly might mean five years or so, but the typical CEOs tenure is two years, so even a continuous innovation will make its sponsoring CEO into a business press hero.