Intro To Power Laws (March 2008)

Editor's Notes

• #3: Critical Mass http://en.wikipedia.org/wiki/Philip_Ball 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 http://en.wikipedia.org/wiki/The_Long_Tail 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 Amazon.com 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 http://en.wikipedia.org/wiki/The_Black_Swan_(book) http://en.wikipedia.org/wiki/Black_swan_theory 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.
• #6: Complex Networks http://en.wikipedia.org/wiki/Complex_network "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!"
• #7: Zipf’s Law http://en.wikipedia.org/wiki/Zipfs_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 http://en.wikipedia.org/wiki/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.”
• #8: 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
• #17: See The Black Swan p. 264 for example exponents