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-simila****r** 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.

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.”

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