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Kyle Findlay [email protected] The TNS Customer Equity Company Research & Development March 2008 An Introduction to Power Laws Image: Map of the human genome
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Kyle [email protected] The TNS Customer Equity CompanyResearch & DevelopmentMarch 2008
An Introduction to Power Laws
Image: Map of the human genome
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How we currently approach the world…
It is generally assumed that the normal distribution dominates our world (and our industry)
However; such an important assumption is definitely worth questioning
It applies to certain areas such as biometric data (e.g. height, age, weight)…
…but a few popular science authors have recently cast some doubt on this assumption:
Philip Ball, Critical Mass (2005)
Chris Anderson, The Long Tail (2006)Nassim Taleb, The Black Swan (2007)
We have been brainwashed into assuming normal distributions exist everywhere…
…when the real-world is far more non-linear, complexand systems-based!!!!!!!!!!
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What is a power law?
Describes certain distributions that are top-heavy and have “long tails”
“A power law applies to a system when large is rare and small is common”*
*Source kottke.orghttp://www.kottke.org/03/02/weblogs-and-power-laws Date accessed: 25-02-2008
Long tail (many)Dominant few
An example power law graph**, being used to demonstrate ranking
of popularity
To the right is the long tail with many small observations…
…to the left are the few large that dominate
**Source Wikipediahttp://en.wikipedia.org/wiki/Power_lawsDate accessed: 25-02-2008
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Where do we see power laws?
Lots of places…
Customer sales (most sales come from a few customers)
Market share (a market only has a few market leaders and many smaller brands)
Popularity of celebrities, musicians…
…and any other group in an environment that includes social influence
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Where do we see power laws?
More examples…
Size of cities
Frequency and magnitude of earthquakes
Protein families within the human (and other animals’) genome
Networks:
The number of connections that individual nodes in a scale-free network have follow a power law e.g.
*Source Wikipediahttp://en.wikipedia.org/wiki/Complex_network Date accessed: 25-02-2008
*Source Social Network Analysis: Advances and Empirical Applications Forumhttp://www.crim.ox.ac.uk/Social%20Network%20Analysis%20Conference%202005/Conference%20Information.htm.
Date accessed: 25-02-2008
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McPhee’s Double Jeopardy
Zipf’s Law
Pareto Principle – seen in many forms:
The Law of the Vital Few, the Principle of Factor Sparsity
The 80-20 Rule
e.g. “20% of the population controls 80% of the wealth”, “you need money to
make money”
Power laws in other areas…
20%80%
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Specific example: McPhee’s Double Jeopardy
The “law” says that the big tend to get bigger and receive more than their fair share
Contributing factors to a brand’s strength are market factors, advertising, word-of-mouth, customer loyalty, etc.
e.g. a brand with strong distribution channels and high visibility is likely to gain more than its fair share of users simply because it is the most easily available and ‘obvious’ (or only) choice
Mmmm, which brand should I
buy!?
Size begets size…and ”the rich get richer”…
Power laws in other areas…
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McPhee’s Double Jeopardy…
Penetration (usage) vs. frequency of purchase (UK newspaper market)
Power laws in other areas…
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So we can see that big brands are used more often…
…and, to some extent, add to their own “momentum”
Some Ideas/Brands Have a Kind of Gravity
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Some ideas have a kind of gravity…
Worldwide percentage of adherents by belief system (mid-2005)*:
*Source Encyclopaedia Britannica Onlinehttp://www.britannica.com/eb/article-9432620/Worldwide-Adherents-of-All-Religions-Mid-2005Accessed 25-02-2008
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Guild sizes in World of Warcraft* (base = over 10 million subscribers worldwide):
*Source Life With Alacrityhttp://www.lifewithalacrity.com/2005/08/dunbar_world_of.html Accessed 26-02-2008
Some ideas have a kind of gravity…
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Links to blogs* (based on top 100 most linked to blogs on )
*Source kottke.orghttp://www.kottke.org/03/02/weblogs-and-power-laws
Date accessed: 25-02-2008
Curve fit: R2 = 0.9918
Some ideas have a kind of gravity…
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Some ideas have a kind of gravity…
Top 25 Global Market Research Organisations (2006)*:
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*Source Marketing News via the AMA website
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Power laws apply at all levels… (1)
Metabolic rates of mammals and birds*:
*Source Dr. Geoffrey West, LANL (KITP Immune System Workshop 11-19-03) Scaling Laws in Biology: Growth, Mortality, Cancer and Sleep
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Power laws apply at all levels… (2)
Other biological processes closely follow the scaling of body mass:
*Source Brown, J.H., et. al. (2002) The Fractal Nature of Nature. Phil. R. Soc. Lond. B 357, 619-626
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k = the scaling exponent
It’s the number we need to identify in order to identify a power law curve
A few example exponents*:
2.8 = magnitude of earthquakes
2.14 = diameter of moon craters
2.0 = people killed in terrorist attacks
1.1 = net worth of Americans
0.8 = intensity of wars
It is worth noting that these exponents should be taken with a pinch of salt…
…as settling on a 100% accurate exponent can be challenging
The technical bits…
f(x) = axk + o(xk)
*Source The Black Swan (2007)Author: Nassim Nicholas TalebPublished by the Penguin Group
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Log-log
By taking the log-log of the curve…
…we get a linear function with slope k
How do we identify power laws?
By inspection
We rank share metrics
log (f(x)) = k log x + log a
Slope = k
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When do they occur?
It’s difficult to say for sure…
Stable/developed markets?
Do they occur in markets that have had time to stabilise and for a structure to form?
Local minima / maxima?
Do they occur at a temporarily stable point in the market?
Before/at/around phase transitions?
Before, during or after a market structure changes?
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Areas where power laws may be useful…
We don’t know where they might be useful yet… but with understanding comes power
In markets that display a power law, perhaps we can use them to answer:
1. What kind of momentum does my brand need to break into a market (or should we be entering it at all)?
2. What strength (or lack thereof) does my current brand size lend me?
3. What kind of market share can I expect based on my position in the market?
The reality is that everything is still speculative at this stage…
…but the new insight such concepts bring is incredibly exciting!
Thank you!
