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.

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  • A nice introduction to the topic :) thanks
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  • Critical Mass Philip Ball (born 1962) is an English science writer. He holds a degree in chemistry from Oxford and a doctorate in physics from Bristol University. He was an editor for the journal Nature for over 10 years. Ball's 2004 book Critical Mass: How One Thing Leads To Another examines a wide range of topics including the business cycle random walks, phase transitions, bifurcation theory traffic flow, Zipf's law, Small world phenomenon, catastrophe theory, the Prisoner's dilemma. The overall theme is one of applying modern mathematical models to social and economic phenomena. The book was awarded Aventis Prize for 2005. Long Tail: Why the Future of Business Is Selling Less of More The phrase The Long Tail (as a proper noun with capitalized letters) was first coined by Chris Anderson in an October 2004 Wired magazine article to describe the niche strategy of certain business such as or Netflix. The distribution and inventory costs of those business allow them to realize significant profit out of selling small volumes of hard-to-find items to many customers, instead of only selling large volumes of a reduced number of popular items. The group of persons that buy the hard-to-find or "non-hit" items is the customer demographic called the Long Tail. Given a large enough availability of choice and a large population of customers, and negligible stocking and distribution costs, the selection and buying pattern of the population results in a power law distribution curve, or Pareto distribution, instead of the expected normal distribution curve. This suggests that a market with a high freedom of choice will create a certain degree of inequality by favoring the upper 20% of the items ("hits" or "head") against the other 80% ("non-hits" or "long tail"). [2] The Black Swan In Nassim Nicholas Taleb's definition, a black swan is a large-impact, hard-to-predict, and rare event beyond the realm of normal expectations. Taleb regards many scientific discoveries as black swans—"undirected" and unpredicted. He gives the September 11, 2001 attacks as an example of a Black Swan event. The term black swan comes from the ancient Western conception that 'All swans are white'. In that context, a black swan was a metaphor for something that could not exist. The 18th Century discovery of black swans in Australia metamorphosed the term to connote that the perceived impossibility actually came to pass. Taleb notes that John Stuart Mill first used the black swan narrative to discuss falsification. The high impact of the unexpected Before Taleb, those who dealt with the notion of improbable, like Hume, Mill and Popper, focused on a problem in logic, specifically that of drawing general conclusions from specific observations. Taleb's Black Swan has a central and unique attribute: the high impact. His claim is that almost all consequential events in history come from the unexpected—while humans convince themselves that these events are explainable in hindsight. One problem, labeled the Ludic fallacy by Taleb, is the belief that the unstructured randomness found in life resembles the structured randomness found in games. This stems from the assumption that the unexpected can be predicted by extrapolating from variations in statistics based on past observations, especially when these statistics are assumed to represent samples from a bell curve. Taleb notes that other functions are often more descriptive, such as the fractal, power law, or scalable distributions; awareness of these might help to temper expectations. Beyond this, he emphasizes that many events are simply without precedent, undercutting the basis of this sort of reasoning altogether. Taleb also argues for the use of counterfactual reasoning when considering risk.
  • Complex Networks "A network is named scale-free if its degree distribution, i.e., the probability that a node selected uniformly at random has a certain number of links (degree), follows a particular mathematical function called a power law . The power law implies that the degree distribution of these networks has no characteristic scale. In contrast, network with a single well-defined scale are somewhat similar to a lattice in that every node has (roughly) the same degree. Examples of networks with a single scale include the Erdős–Rényi random graph and hypercubes. In a network with a scale-free degree distribution, some vertices have a degree that is orders of magnitude larger than the average - these vertices are often called " hubs ", although this is a bit misleading as there is no inherent threshold above which a node can be viewed as a hub. If there were, then it wouldn't be a scale-free distribution!"
  • Zipf’s Law Zipf's law, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in physical and social science can be described by a Zipfian distribution, one of a family of related discrete power law probability distributions. The law is named after the linguist George Kingsley Zipf who first proposed it (Zipf 1935, 1949), though J.B. Estoup appears to have noticed the regularity before Zipf. Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, which occurs twice as often as the fourth most frequent word, etc. For example, in the Brown Corpus "the" is the most frequently occurring word, and all by itself accounts for nearly 7% of all word occurrences (69971 out of slightly over 1 million). True to Zipf's Law, the second-place word "of" accounts for slightly over 3.5% of words (36411 occurrences), followed by "and" (28852). Only 135 vocabulary items are needed to account for half the Brown Corpus. Pareto Principle “ The Pareto principle (also known as the 80-20 rule, the law of the vital few and the principle of factor sparsity) states that, for many events, 80% of the effects comes from 20% of the causes. Business management thinker Joseph M. Juran suggested the principle and named it after Italian economist Vilfredo Pareto, who observed that 80% of income in Italy went to 20% of the population. It is a common rule of thumb in business; e.g., "80% of your sales comes from 20% of your clients." It is worthy of note that some applications of the Pareto principle appeal to a pseudo-scientific "law of nature" to bolster non-quantifiable or non-verifiable assertions that are "painted with a broad brush". The fact that hedges such as the 90/10, 70/30, and 95/5 "rules" exist is sufficient evidence of the non-exactness of the Pareto principle. On the other hand, there is adequate evidence that "clumping" of factors does occur in most phenomena. The Pareto principle is only tangentially related to Pareto efficiency, which was also introduced by the same economist, Vilfredo Pareto. Pareto developed both concepts in the context of the distribution of income and wealth among the population.”
  • Origin of McPhee’s Double Jeopardy “ In the 1930s, William McPhee discovered that radio DJs and comic strips that were more popular, had lots more listeners (or readers, for comic books) and that these listeners listened to them for a little longer each day. McPhee thought that it was odd that less popular DJs should 'suffer' in two ways – not only did they have fewer listeners but those listeners didn't listen to them for so long – so he called the pattern 'double jeopardy'.” Reference: Byron Sharp. “The Only Way to Grow Your Brand”. Admap 2003, Issue 438
  • See The Black Swan p. 264 for example exponents
  • 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!