Posts Tagged ‘power-law’

Last week I introduced the concept of self-similarity and showed its relevance for power law distributions. In this post I discuss the applicability of self-similarity in S-curves.

To recap briefly, self-similarity implies that a structure looks essentially the same at all levels of “magnification” or scale. You can zoom in on any part of a “power curve,” and it will look like… a power curve, with basically the same appearance as at the higher scale.

The same phenomenon can be seen in s-curves, with the difference that the scale invariance is less apparent, at least initially. The following diagram shows how each phase on an s-curve can be broken out into smaller, constituent s-curves at the next lower level. By extension, each of these subordinate s-curves can be parsed in the same, self-similar way. The structure is recursive and nested. If you want to grow, develop, or improve in any way, you must see it as a succession of s-curves at all levels of scale.


This is why I’ve titled this post “Growth is a stairway, not a high jump!”. You make progress in increments, climbing from one step to the next in a succession of achievable bounds. This breaks progress and improvement into (to paraphrase Neil Armstrong) a series of “one small step” moves so you can make “one giant leap” for your bigger purpose… or goals.

This is more manageable from a psychological standpoint as well as logistically. It also makes risk more manageable. As I illustrate in the following diagram, there are risks at each transition to a succeeding s-curve. Risk can arise from making a jump–even a small one–to a higher level of performance and engagement. It can also arise from a drop in performance at this critical juncture. We can seldom know and do everything that is needed at the new level. We need to learn–which is why progress is depicted as an s-curve in the first place. We start out with low performance at the new level and a high potential for mistakes. If we’re focused on learning from our mistakes and on improvement, we get progressively better until we hit the rapid growth stage, and continue up the “learning” curve from there on in. When we hit the inevitable plateau, we must jump–or drop–to the next curve.


The final point is that performance or growth can bog down or slip at any point, for any number of reasons. We can stop or slip back down the curve we’re already on. I call this regression. Even more consequential is when we drop back to a previous curve. I call this retrogression, and I’ve illustrated it in detail in the following diagram. It shows how you can fall from any performance level to any other, usually through neglect, over-confidence, smugness, or simply through inattention to changing conditions in the environment. For instance, new technology, new competitors, changing demographics, all these can make our current success or standing shaky or even irrelevant.


I don’t say this to be overly pessimistic, but rather realistic. Stasis is death. Movement is crucial. Business, life, performance, everything, they are what is called a “red queen” race. You have to work just to stay in place and work even harder to make progress, grow, develop, get better.

We’ll address these issues and many more in my coming posts under the topic of “Ideas,” so stay tuned to this space.

My name is Richard Martin and, as indicated by the title of this blog, I’m an expert on applying readiness principles to position companies and leaders to grow and thrive by shaping and exploiting change and opportunity, instead of just passively succumbing to uncertainty and risk.

© 2016 Alcera Consulting Inc. This article may be used for non-commercial use with proper attribution.

The “Power Law” is one of the most useful concepts for making predictions and decisions in business and management.

The power law shows how two variables–one dependent, the other independent–covary. Mathematically, one varies as a function of the other by being raised to a certain power (exponent).

The following diagram shows this type of relationship. Often these are depicted on log or log-log graphs, but I show the “power curve” as an asymptote on both axes of the graph to highlight the non-linearity of the relationship between the two variables.


A concrete example will help. The great majority of earthquakes are of very low magnitude. High magnitude earthquakes are much rarer than low magnitude earthquakes. In fact, their magnitude varies in inverse exponential proportion to the total number of earthquakes. In practice, this means that there are literally thousands of earthquakes every day around the world, but magnitude 6, 7, and 8 ones are much rarer. The most powerful earthquakes of all, over 9 on the Richter, scale are very rare. They can happen only a few times a century, or even less. This doesn’t mean that the magnitude of any particular earthquake can be predicted. It does however imply that given a sufficiently large sample, we will eventually see a frequency-magnitude distribution that resembles the graph above.

This type of relationship is ubiquitous in nature, and that includes our human and social natures. There was a whole book written on this topic–The Long Tail, by Chris Anderson–with emphasis on the right side of the graph. In his book, Anderson described how the internet has made many businesses or ideas viable which would not previously even have been known. He called this the long tail because there are musicians, artists, artisans, crafts workers, professionals, etc. who can provide their productions and services to people around the world, even though they can’t compete with the more traditional providers who dominate markets by occupying the left side of the power curve. This makes for much more diversity and many more opportunities to get known and appreciated, and to develop a following because it lowers traditional barriers to entry and long-term viability.

This type of relationship is also depicted in the following diagram. I show the relationship between number of clients and the number purchases, interactions, or value of each category of client that characterizes the market and product distributions of most, if not all, companies (including my own clients).


For instance, I’ve been working with a banking client. This graph shows the relationship between number of clients and the number of products/services that each client has with the bank. The total market size for this bank is about 80,000 potential users of its services. Of these potential users of its services, the great majority, about 85 %, have no business relationship with the bank. Of the 13,000 or so that do use the bank’s services, the majority only use less than 3 of over 20 products and services. As we move to the right, there are less clients, but their interactions with the bank are more intensive. In other words, there are are many fewer clients in categories to the right, but they use many more of the bank’s services, which in turn generate much greater value. On the other hand, there are no actual clients who do all of their banking and meet all of their financial needs and objectives, much less use all of the bank’s services. This is why we can depict the lower right part of the curve as an asymptote. You never actually reach complete saturation or use.

We’ve all noticed these types of power-law relationships in our professional and personal lives, our management and business experiences, and even in some natural phenomena. This relationship is often referred to as the 80/20 law, Pareto’s Law, or Zipf’s Law. It shows up in such truisms as: 80 % of my problems are caused by 20 % (rates can vary) of my people; most of my sales and profits come from a small number of sales reps; only a few of my clients provide most of my revenue and profits; this product category accounts for 45 % of my sales, but 70 % of my profits; etc., etc.

The following diagram is a further illustration of the principle. It comes from an online article by Mark McLean of the Toronto Real Estate Board (TREB) and shows an almost perfect example of a power-law distribution in the number of deals done by different categories of real estate agents who are members of TREB.  We can see that only a very small number of agents in TREB can be considered highly successful, prolific even.


Of those agents having completed 6 or less deals in a year, a similar relationship holds, although it’s less stark:


Whatever we wish to call them, power-law distributions and relationships underlie much of the correlations and dynamics that surround us. We can use them in making general predictions and, along with the S-curve phenomenon I described in a previous post, we have two very powerful tools and concepts for understanding the world around us. Moreover, power laws and S-curves are not only ubiquitous, they tend to show what’s called “self-similarity,” or a fractal pattern. I’ll discuss that third powerful concept next week.

© 2016 Alcera Consulting Inc.

This article may be forwarded, reproduced, or otherwise referenced for non-commercial use with proper attribution. All other rights are reserved and explicit permission is required for commercial use.