European Conference on Complex Systems Dresden 1-4 October Scale-free theories in Org Science Pierpaolo Andriani Durham Business School, UK Universita’ di Lecce, Italy Bill McKelvey UCLAAnderson School.
Download ReportTranscript European Conference on Complex Systems Dresden 1-4 October Scale-free theories in Org Science Pierpaolo Andriani Durham Business School, UK Universita’ di Lecce, Italy Bill McKelvey UCLAAnderson School.
European Conference on Complex Systems Dresden 1-4 October Scale-free theories in Org Science Pierpaolo Andriani Durham Business School, UK Universita’ di Lecce, Italy Bill McKelvey UCLAAnderson School of Management, US What a power law is Exponentials vs. Power Laws Exponential y = e –x e = constant Power law y = x - = constant of node linkages of node linkages Number of nodes (log scale) Power-law distribution Typical node No large number Number of links Exponential Network Number of nodes Number of nodes Bell curve distribution Number of links Number of links (log scale) Scale-free Network From Barabasi/Bonabeau, Scientific American, May 2003 Power laws are ubiquitous in natural and social phenomena Some Examples: Rank-Size Rule (Zipf’s law) Krugman on cities: “we are unused to seeing regularities this exact in economics – it is so exact that I find it spooky” (1996) p.40 Simon’s (1955) “lumps and clumps” model 3 rules: 1. Growth 2. Spatially random 3. Growth linearly proportional to size Source: Bak (1996) “How Nature Works” Self-Organized Criticality Self-organised criticality Nc (Earthquakes/Year) Ricther-Gutenberg Law “In the critical state, the sand pile is the functional unit, not the grain of sand” Bak (1997) p.60 Casti _126 Find gutemberg “the system organises itself towards the critical point where single events have the widest possible range of effects” Cilliers (1998) p. 97 Earthquake magnitude (mb ) ~ Log E Scale-free Networks Rules: SEX web scale-free network 1. Growth 2. Preferential attachment (also known as the Matthew effect “for to every one that hath shall be given. . . ” (Matthew 25:29) Nodes: computers, routers Links: physical lines Nodes: people (Females; Males) Links: sexual relationships 4781 Swedes; 18-74; 59% response rate. Liljeros et al. Nature 2001 (Faloutsos, Faloutsos and Faloutsos, 1999) Allometric ¾ mass-metabolism Metabolic ecology: A biological theory of everything? See Whinfield: “In the beat of a heart” West and Brown Life's Universal Scaling Laws PhysicsToday.org 36 Kinds of “Physical” Power Laws Cities Traffic jams Coastlines Brush-fire damage Water levels in the Nile Hurricanes & floods Earthquakes Asteroid hits Sun Spots Galactic structure Sand pile avalanches Brownian motion Music Epidemics/Plagues Genetic circuitry Metabolism of cells Functional networks in brain Tumor growth Biodiversity Circulation in plants and animals Langton’s Game of Life Fractals Punctuated equilibrium Mass extinctions/explosions Brain functioning Predicting premature births Laser technology evolution Fractures of materials Magnitude estimate of sensorial stimuli Willis’ Law: No. v. size of plant genera Fetal lamb breathing Bronchial structure Frequency of DNA base chemicals Protein-protein interaction networks Heart-beats Yeast 38 Kinds of “Social” Power Laws Language—word usage Social networks Blockbuster drugs Sexual networks Distribution of wealth Citations Co-authorships Casualties in war Growth rate of countries’ GDP Delinquency rates Movie profits Actor networks Size of villages Distribution of family names Consumer products Copies of books sold Number of telephone calls and emails Deaths of languages Aggressive behavior among children “No learning” agents (Ormerod) Structure of the Internet equipment Internet links # hits received from website/day Price movements on exchanges Economic fluctuations “Fordist” power structure/effects Salaries Labor strikes Job vacancies Firm size Growth rates of firms Growth rates of internal structure Supply chains Cotton prices Alliance networks among biotech firms Entrepreneurship/Innovation Director interlock structure Italian Industrial Clusters Why do power law matter? Tail of extreme events Evidence from financial markets Rationality, stock market and the butterfly effect Florence 1966 Dresden 2002 New Orleans 2006 Long Tails of Heterogeneous Niches The impact of the Internet on the structure of markets Kevin Laws: the biggest money in the smallest sales Long Tails vs. Averages and Interdependence vs. Independence Which statistics? Do It Yourself (Financial DIY) Download Dow Jones index numbers from: http://www.dowjones.com Take daily variation: take log of each daily index number. Subctract log from following day log Assume variations fit Gaussian and calculate sample variance s2 or s2 = (xav –xi )2 / (n-1) Calculate how typical each crash day is: z = (xi – xav) / s Using z score calculate probability Mandelbrot & Hudson 2004 Probability of financial crushes according to standard financial theory (Mandelbrot, 2004) August 31, 1998 August 1997 July 2002 6.8% Wall Street crush 1 in 20 million 7.7% Dow Jones 1 in 50 billion 3 step falls in 7 days 1 in 4 trillion And finally October 19, 1987 29.2% fall 1 in 10-50 “It is a number outside the scale of nature. You could span the powers of ten from the smallest subatomic particle to the breadth of the measurable universe – and still never meet such a number” Principles Underlying Power Law Statistics 1. Paretian: Mode (most frequent event) < Median (central point) < Mean. (Unstable mean—strongly influenced by large extreme events) 2. Infinite variability/variance 3. Business of extremes. Extreme events are more frequent and disproportionate in size than in a Gaussian dominated world. 4. Scale-free: As with the English coastline, phenomena appear the same no matter what scale the measure 5. Fractal Structure: Self-similarity; fractal statistics. 6. Linear amplification: Fat tails result from amplification of simple causes that may evolve to generate events of any size. Gaussians reflect “quenched” variability; Paretians reflect “amplified” events 7. Cascade dynamics: Generalized self-organized criticality 8. Power Law Distribution : Acts as a universal attractor 9. “Nobody knows anything” principle: Events are probability distribution with infinite variance. Prediction is possible for aggregates only. “In this world nothing is “typical” and every movie is unique” Which approach to statistics? Traditional statistics assume bellshaped distribution, with typical scale (mean) and rapidly decaying tails Neo-classical economics and equilibrium-based management theories assume normal distributions and descriptive/behavioral parameters gathering around means. Extreme events are very rare and therefore negligible Power-law distributions show no mean (scale-free) and exhibit long fat tails (infinite variance). A PL explores the maximum dynamic range of diversity of the variable, limited only by size of network and agent. Extreme events are more frequent and their magnitude is disproportionately bigger than in the bell distribution case. Pluralism in power law causal mechanisms Scale-free Theories Classification Growth-related power laws - ratio imbalances 1 Surface / volume Law Organisms; villages: In organisms, surfaces absorbing energy grow by the square but the organism grows by the volume, resulting in an imbalance (Galileo 1638, Carneiro 1987); fractals emerge to bring surface/volume back into balance West and Brown (1997) show that several phenomena in biology such as metabolic rate, height of trees, life span, etc. are described by allometric power law whose exponent is a multiple of ±¼. The cause is fractal distribution of resources. Allometric power laws hold across 27 orders of magnitude (of mass). 2 Least effort Language; transition: Word frequency is a function of ease of usage by both speaker/writer and listener/reader; this gives rise to Zipf’s (power) Law (1949); now found to apply to language, firms, and economies in transition (Ferrer i Cancho & Solé, 2003; Dahui et al., 2005; Ishikawa, 2005; Podobnik et al., 2006). 3 Hierarchical modularity Growth unit connectivity: As cell fission occurs by the square, connectivity increases by n(n–1)/2, producing an imbalance between the gains from fission vs. the cost of maintaining connectivity; consequently organisms form modules or cells so as to reduce the cost of connectivity; Simon argued that adaptive advantage goes to “nearly decomposable” systems (Simon, 1962; Bykoski, 2003). Complex adaptive systems: Heterogeneous agents seeking out other agents to copy/learn from so as to improve fitness generate networks; there is some probability of positive feedback such that some networks become groups, some groups form larger groups & hierarchies (Kauffman, 1969, 1993; Holland, 1995). Combinations 4 Interactive Breakage theory Wealth; mass extinctions/explosions: A few independent elements having multiplicative effects produce lognormals; if the elements become interactive with positive feedback loops materializing, a power law results; based on Kolmogorov’s “breakage theory” of wealth creation (1941). # of exponentials; complexity: Multiple exponential or lognormal 5 distributions or increased complexity of components (subtasks, Combination processes) sets up, which results in a power law distribution theory (Mandelbrot, 1963; West & Deering, 1995; Newman, 2005). 6 Interacting fractals Food web; firm & industry size, heartbeats: The fractal structure of a species is based on the food web (Pimm, 1982), which is a function of the fractal structure of predators and niche resources (Preston 1950; Halloy, 1998; Solé & Alonso, 1998; Camacho & Solé, 2001; Kostylev & Erlandsson, 2001, West, 2006). Positive feedback loops 7 Preferential attachment Nodes; gravitational attraction: Given newly arriving agents into a system, larger nodes with an enhanced propensity to attract agents will become disproportionately even larger, resulting in the power law signature (Yule, 1925; Young, 1928; Arthur, 1988; Barabási, 2000). 8 Irregularity generated gradients Coral growth; blockages: Starting with a random, insignificant irregularity, coupled with positive feedback, the initial irregularity increases its effect. This explains the growth of coral reefs, blockages changing the course of rivers, (Juarrero, 1999; Turner, 2000; Barabási, 2005). Diffusion limited accretion (DLA). See also “niche constructionism” in biology (Odling-Smee, 2003) Contextual effects 9 Phase transitions Turbulent flows: Exogenous energy impositions cause autocatalytic, interaction effects and percolation transitions at a specific energy level—the 1st critical value—such that new interaction groupings form with a Pareto distribution (Bénard, 1901; Prigogine, 1955; Stauffer, 1985; Newman, 2005). 10 Selforganized criticality Sandpiles; forests; heartbeats: Under constant tension of some kind (gravity, ecological balance, delivery of oxygen), some systems reach a critical state where they maintain stasis by preservative behaviors— such as sand avalanches, forest fires, changing heartbeat rate—which vary in size of effect according to a power law (Bak et al., 1987; Drossel & Schwabl, 1992; Bak, 1996). Markets: When production, distribution, and search become cheap and 11 easily available, markets develop a long tail of proliferating niches Niche containing fewer customers; they become Paretian with mass-market Proliferation products at one end and a long tail of niches at the other (Anderson, 2006). Others – difficult to classify 12 Random walk 13 Event bursts Coin flipping; Gambler’s ruin: Given a stochastic process such as coin flipping and, say, two players with a finite number of pennies to gamble, the probability that eventually one of the players will lose all his/her pennies is 100% (Kraitchik, 1942). Number of tosses required is Pareto distributed (Newman, 2005). Activity prioritization: Individuals show bursts of communication, entertainment, and work activities followed by long delays, as opposed to random (Poisson) distribution (Paxon & Floyd, 1995; Barabási, 2005). Pluralism in the power law world From Gaussian to Paretian Italian Income Distribution Gaussian region Lognormal or multifractal region Power law region From Gallegati The anti-power law ‘camp’ The laggards: Denial The conservatives: No real PL but all lognormal The pragmatists: LN with PL tail – Multimodal distributions with multiple dynamics – LN ~ 98% pdf (Multiplicative independent) and PL ~ 2% (multiplicative interdependent) – The problem: LN and PL are indistinguishable if variance is large and orders of magnitude < 4 – Example: Perline (2005) “Strong, weak and false power laws” Statistical Science, Vol. 20, No. 1, 68–88 From independence to interdependence Gaussian LogNormal Multifractal Independent additive data-point Independent multiplicative data-point Aggregate of clusters of interdependent multiplicative data-point Human height Fractal System of interdependent multiplicative data-point Drunkard walk none global Interdependence The danger of averages What’s Wrong? Where did you say the average was? Readings On Mathematics and power law: – Newman, M.E.J. (2005) ‘Power Laws, Pareto Distributions and Zipf’s Law’, [www document] http://arxiv.org/PS_cache/cond-mat/pdf/0412/0412004.pdf. On Finance, fractal and power law: – – Mandelbrot, B.B. and Hudson, R.L. (2004) The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin and Reward, Profile: London. Sornette, D. (2003) Why Stock Markets Crash? Critical Events in Complex Financial Systems, Princeton University Press: Princeton, NJ. On fractal, phisiology and epistemology – – West, B.J. and Deering, B. (1995) The Lure of Modern Science: Fractal Thinking, World Scientific: Singapore. West, B.J. (2006) Where Medicine Went Wrong, World Scientific, Singapore On Org Science and power law: – – McKelvey B, Andriani P. 2005. Why Gaussian Statistics are Mostly Wrong for Strategic Organization? Strategic Organization 3(2): 219-228 Andriani, P. & McKelvey, B. 2007. Beyond Gaussian averages: redirecting international business and management research toward extreme events and power laws. Journal of International Business Studies (forthcoming) On Market, marketing and power law: – Anderson, C. (2006) The Long Tail: Why the Future of Business is Selling Less of More, Random House Business Books On economics and power law: – Ormerod, P. (2006) Why Most Things Fail, Faber and Faber On Extreme events and power law: – Albeverio, S. and Jentsch, V. (eds) (2005) Esxtreme Events in Nature and Society, Springer Thank you for your patience Any questions?