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Intro To Power Laws (March 2008)


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An introduction to power law distributions, with a focus on branded markets.

Somewhat text-heavy by today's standards, but presentation was created in late 2007.

Published in: Business, Technology
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Intro To Power Laws (March 2008)

  1. 1. Kyle Findlay [email_address] The TNS Customer Equity Company Research & Development March 2008 An Introduction to Power Laws Image: Map of the human genome
  2. 2. How we currently approach the world… <ul><li>It is generally assumed that the normal distribution dominates our world (and our industry) </li></ul><ul><li>However; such an important assumption is definitely worth questioning </li></ul><ul><ul><li>It applies to certain areas such as biometric data (e.g. height, age, weight)… </li></ul></ul><ul><ul><li>… but a few popular science authors have recently cast some doubt on this assumption: </li></ul></ul><ul><li>We have been brainwashed into assuming normal distributions exist everywhere… </li></ul><ul><ul><li>… when the real-world is far more non-linear , complex and systems-based!!!!!!!!!! </li></ul></ul>Philip Ball, Critical Mass (2005) Chris Anderson, The Long Tail (2006) Nassim Taleb, The Black Swan (2007)
  3. 3. What is a power law? <ul><li>Describes certain distributions that are top-heavy and have “ long tails ” </li></ul><ul><ul><li>“A power law applies to a system when large is rare and small is common ”* </li></ul></ul>*Source Date accessed: 25-02-2008 <ul><li>An example power law graph**, being used to demonstrate ranking of popularity </li></ul><ul><ul><li>To the right is the long tail with many small observations … </li></ul></ul><ul><ul><li>… to the left are the few large that dominate </li></ul></ul>**Source Wikipedia http:// Date accessed: 25-02-2008 Long tail (many) Dominant few
  4. 4. Where do we see power laws? <ul><li>Lots of places… </li></ul><ul><ul><li>Customer sales (most sales come from a few customers) </li></ul></ul><ul><ul><li>Market share (a market only has a few market leaders and many smaller brands) </li></ul></ul><ul><ul><li>Popularity of celebrities, musicians… </li></ul></ul><ul><ul><ul><li>… and any other group in an environment that includes social influence </li></ul></ul></ul>% Market share $ $ $ $ $ $ $ $ $ $
  5. 5. Where do we see power laws? <ul><li>More examples… </li></ul><ul><ul><li>Size of cities </li></ul></ul><ul><ul><li>Frequency and magnitude of earthquakes </li></ul></ul><ul><ul><li>Protein families within the human (and other animals’) genome </li></ul></ul><ul><ul><li>Networks : </li></ul></ul><ul><ul><ul><li>The number of connections that individual nodes in a scale-free network have follow a power law e.g. </li></ul></ul></ul>*Source Wikipedia Date accessed: 25-02-2008 *Source Social Network Analysis: Advances and Empirical Applications Forum . Date accessed: 25-02-2008
  6. 6. <ul><li>McPhee’s Double Jeopardy </li></ul><ul><li>Zipf’s Law </li></ul><ul><li>Pareto Principle – seen in many forms: </li></ul><ul><ul><li>The Law of the Vital Few, the Principle of Factor Sparsity </li></ul></ul><ul><ul><li>The 80-20 Rule e.g. “20% of the population controls 80% of the wealth”, “you need money to make money” </li></ul></ul>Power laws in other areas… 20% 80%
  7. 7. <ul><li>Specific example: McPhee’s Double Jeopardy </li></ul><ul><ul><li>The “law” says that the big tend to get bigger and receive more than their fair share </li></ul></ul><ul><ul><li>Contributing factors to a brand’s strength are market factors , advertising , word-of-mouth , customer loyalty , etc. </li></ul></ul><ul><ul><ul><li>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 </li></ul></ul></ul>Power laws in other areas… <ul><li>Size begets size…and ”the rich get richer”… </li></ul>Mmmm, which brand should I buy!?
  8. 8. <ul><li>McPhee’s Double Jeopardy … </li></ul><ul><ul><li>Penetration (usage) vs. frequency of purchase (UK newspaper market) </li></ul></ul>Power laws in other areas… Usage of brand Frequency of purchase <ul><li>So we can see that big brands are used more often… </li></ul><ul><ul><li>… and, to some extent, add to their own “ momentum ” </li></ul></ul>Newspaper 1 Newspaper 2 Newspaper 3 Newspaper 4 Newspaper 5 Newspaper 6
  9. 9. Some Ideas/Brands Have a Kind of Gravity
  10. 10. Some ideas have a kind of gravity… <ul><li>Worldwide percentage of adherents by belief system (mid-2005)*: </li></ul>*Source Encyclopaedia Britannica Online Accessed 25-02-2008 % 0 5 10 15 20 25 30 35 Christians Muslims Hindus Non-religious Chinese Universists Buddhists Ethnoreligionists Atheist Neoreligionists Sikhs Jews Spiritists Baha'is Confucianists Jains Shintoists Taoists Zoroastrians Other religionists
  11. 11. <ul><li>Guild sizes in World of Warcraft* (base = over 10 million subscribers worldwide): </li></ul>Some ideas have a kind of gravity… *Source Life With Alacrity Accessed 26-02-2008
  12. 12. <ul><li>Links to blogs * (based on top 100 most linked to blogs on ) </li></ul>Some ideas have a kind of gravity… *Source Date accessed: 25-02-2008 Curve fit: R 2 = 0.9918
  13. 13. Some ideas have a kind of gravity… <ul><li>Top 25 Global Market Research Organisations (2006)*: </li></ul>Revenue (US$) *Source Marketing News via the AMA website 0 500,000,000 1,000,000,000 1,500,000,000 2,000,000,000 2,500,000,000 3,000,000,000 3,500,000,000 4,000,000,000 The Nielsen Co. IMS Health Inc. Taylor Nelson Sofres plc The Kantar Group GfK AG Ipsos Group SA Synovate IRI Westat Inc. Arbitron Inc. INTAGE Inc. J.D. Power and Associates Harris Internactive Inc. Maritz Research The NPD Group Inc. Video Research Ltd. Opinion Research Corp. IBOPE Group Lieberman Research Worldwide Telephia Inc. comScore Inc. Dentsu Research Inc. Abt Associations Inc. Nikkei Research Inc. Burke Inc.
  14. 14. Power laws apply at all levels… (1) <ul><li>Metabolic rates of mammals and birds*: </li></ul>*Source Dr. Geoffrey West, LANL (KITP Immune System Workshop 11-19-03) Scaling Laws in Biology: Growth, Mortality, Cancer and Sleep
  15. 15. Power laws apply at all levels… (2) <ul><li>Other biological processes closely follow the scaling of body mass: </li></ul>*Source Brown, J.H., et. al. (2002) The Fractal Nature of Nature. Phil. R. Soc. Lond. B 357, 619-626
  16. 16. <ul><li>k = the scaling exponent </li></ul><ul><li>It’s the number we need to identify in order to identify a power law curve </li></ul><ul><li>A few example exponents*: </li></ul><ul><ul><li>2.8 = magnitude of earthquakes </li></ul></ul><ul><ul><li>2.14 = diameter of moon craters </li></ul></ul><ul><ul><li>2.0 = people killed in terrorist attacks </li></ul></ul><ul><ul><li>1.1 = net worth of Americans </li></ul></ul><ul><ul><li>0.8 = intensity of wars </li></ul></ul><ul><li>It is worth noting that these exponents should be taken with a pinch of salt … </li></ul><ul><ul><li>… as settling on a 100% accurate exponent can be challenging </li></ul></ul>The technical bits… f(x) = ax k + o(x k ) *Source The Black Swan (2007) Author: Nassim Nicholas Taleb Published by the Penguin Group
  17. 17. How do we identify power laws? <ul><li>By inspection </li></ul><ul><ul><li>We rank share metrics </li></ul></ul><ul><li>Log-log </li></ul><ul><ul><li>By taking the log-log of the curve… …we get a linear function with slope k </li></ul></ul>log (f(x)) = k log x + log a Slope = k
  18. 18. When do they occur? <ul><li>It’s difficult to say for sure… </li></ul><ul><ul><li>Stable/developed markets? </li></ul></ul><ul><ul><ul><li>Do they occur in markets that have had time to stabilise and for a structure to form ? </li></ul></ul></ul><ul><ul><li>Local minima / maxima? </li></ul></ul><ul><ul><ul><li>Do they occur at a temporarily stable point in the market? </li></ul></ul></ul><ul><ul><li>Before/at/around phase transitions? </li></ul></ul><ul><ul><ul><li>Before, during or after a market structure changes ? </li></ul></ul></ul>
  19. 19. Areas where power laws may be useful… <ul><li>We don’t know where they might be useful yet… but with understanding comes power </li></ul><ul><li>In markets that display a power law, perhaps we can use them to answer: </li></ul><ul><ul><li>What kind of momentum does my brand need to break into a market (or should we be entering it at all)? </li></ul></ul><ul><ul><li>What strength (or lack thereof) does my current brand size lend me? </li></ul></ul><ul><ul><li>What kind of market share can I expect based on my position in the market? </li></ul></ul><ul><li>The reality is that everything is still speculative at this stage… </li></ul><ul><ul><li>… but the new insight such concepts bring is incredibly exciting! </li></ul></ul>
  20. 20. Thank you!