Posts Tagged ‘fractals’

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.

In my last two posts under “Ideas” I introduced the concepts of the S-curve and Power Law (a.k.a. Pareto’s Law, Zipf’s Law, or the 80/20 rule).

In this post I discuss the concept of self-similarity. I view it as an adjunct to the S-curve and Power Law that multiplies their effectiveness for anticipating change and other dynamic interactions in society, businesses, and other forms of organization.

According to Wikipedia: “In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.” The term fractal is also frequently used to characterize self-similar structures.

Furthermore, self-similarity is characterized by scale invariance. Again according to Wikipedia: “In physics, mathematics, statistics, and economics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor.

(My emphasis in both quotes.)

In practice, this means that it is difficult or even impossible for an observer to detect the system level depicted just by looking at a picture. As mentioned above, coastlines are the paradigmatic illustration of this phenomenon. You can look at satellite image of a coastline at various scales and, barring the presence of a scale indicator (e.g. a boat or a human on the picture), you can’t determine the scale with any certainty. Moreover, this is also a statistical effect, as the underlying math is the same or very similar at all levels of magnification.

There are many phenomena in nature and society with this characteristic structure. However, for business and strategy, the most crucial realization comes from the self-similar (or fractal) nature of power curves and s-curves. Take any distribution governed by a power law. If you hone in on any particular segment of the distribution, you find that it is also governed by essentially the same power law. In other words, the distribution looks basically the same at all levels of magnification, or scale. The follow diagram shows this effect.


No matter what you’re measuring or tracking–it could be total sales, the performance of your salespeople, the relative impact of your clients–you are likely to notice a power law working at all scales. This was illustrated in my Power Law article last week by the example of real estate agents in the Greater Toronto area. I’ve reproduced those two graphs here, as they show how a power law is in evidence at two different scales.

treb-6-and-under treb-gif

Although not as stark at the level of agents with only 0 to 6 deals, we can see that the two scales are broadly similar. What about practical applications? Well, for one, we can see that the effect is likely to be similar at all levels and in all categories of agents. For instance, if you break out each category (7-12 deals all the way to 201 plus deals), you will probably find the same pattern. A small number of top performers skewing the results of the group upward.

This type of distribution plays havoc with our basic assumptions of normally distributed performance or effects. If we were to assume a normal distribution (Gaussian distribution in technical statistical terms) for real estate agents, we could easily be fooled into thinking that there is an “average” performance, a “typical” real estate agent. But this could not be further from the truth. In a normal distribution, the mode, median, and mean are all very close to having the same value. This means that the arithmetic mean could give a false understanding of the performance distribution for a sales group. In actuality, the mode (most frequent value), median (the middle value), and the actual mean could be different, with the latter possibly heavily skewed in the direction of the highest performing class of sales people. This is what we see with the distributions of real estate agents above.

Would the arithmetic mean of this distribution truly represent the average or typical performance of a real estate agent in the Greater Toronto area? Obviously not. If we look at the numbers of deals, 0-6 is the modal value, and represents about 50% of the total number of agents! This means that the largest number of real estate agents are actually sluggish performers, and even don’t participate in any deals at all! If you’re looking, say, to providing products and services to real estate agents–at least in the Greater Toronto area in 2011–then you’d be better to look at the actual performance distribution at varying scales so you can segment the market properly.

These relationships tend to hold across time and space for any particular phenomenon. We can safely assume that the distribution of real estate agent performance is broadly similar no matter when and where you build your sample. While it’s ultimately a question for empirical investigation, in my experience, self-similar power laws are ubiquitous in market dynamics and human performance. You can apply this insight to all market and performance numbers and you will get similar results. This enables much better analysis, planning, and strategy to gain and sustain a competitive edge.

I’ll explore scale-invariance and self-similarity in s-curves in my next “Ideas” post. In subsequent ones I’ll also look at the broader implications of self-similarity, particular as they relate to hierarchy in organizations, specifically what I call “nested hierarchical planning” and “nested hierarchical vigilance.”

© 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.