Academy of Management
Montreal , 6 August 2010
Empirical Exploration of
Complexity in Human Systems:
Data Collection & Interpretation Techniques
Power law statistics and Pareto Science
Pierpaolo Andriani
Durham Business School, University of Durham, UK
LIKELIHOOD distribution:
sampling & inference
Statistical inference:
Drawing conclusions
about the whole
population on the basis
of a sample
Precondition for
statistical inference:
A sample is randomly
selected from the
population (=probability
sample)
PROB (data given population)
take a
sample
population
mean
variance
etc.
size is
infinite
Representative
agent links
population to
sample level and
allows reduction
of population
complexity to
single agent
complexity
infer a
value
INFERENTIAL distribution:
PROB (population given data)
sample
sample mean
sample
variance
etc.
sample size
From Starbuck: The production of knowledge (2006)
Starbucks: The production of knowledge (2006)
• Consensus favoring use of null-hypothesis significance tests
affords a clear example of paradigm stability. Although
methodologists have been trying to discourage the use of
these tests since the 1950s, the tests have remained very
prevalent, and there is no sign that social scientists are
shifting to other criteria. …. Hubbard and Ryan (2000: 678)
concluded: ‘It seems inconceivable to admit that a
methodology as bereft of value as SST (statistical significance
tests) has survived, as the centerpiece of inductive inference
no less, more than four decades of criticism in the
psychological literature’.
p. 77
Starbucks: The production of knowledge (2006)
• Choosing two variables utterly at random, a researcher has 2-to-1 odds
of finding a significant correlation on the first try, and 24-to-1 odds of
finding a significant correlation within three tries. … the main inference I
drew from these statistics was that the social sciences are drowning in
statistically significant but meaningless noise. Because the differences
and correlations that social scientists test have distributions quite different
from those assumed in hypothesis test, social scientists are using tests that
assign statistical significance to confounding background relationships.
Because social scientists equate statistical significance with meaningful
relationships, they often mistake confounding background relationships for
theoretically important information. One result is that social science
research creates a cloud of statistically significant differences and
correlations that not only have no real meaning but also impede
scientific progress by obscuring the truly meaningful relationships.
p. 49
Starbucks: The production of knowledge (2006)
• I began to think of statistical tests as arcane
rituals that demonstrate membership in an
esoteric subculture
p. 18
Gaussian
A tale of two worlds
World
Gaussian
Paretian
Unit of analysis
‘Things’, entities
Relations
Relations btw UoA
Independence
(or
Paretian
Interdependence
weak interdepencdence)
Variability
Limited
Unbounded
Scaling property
Phenomena are fractal
Property of world
Phenomena have proper
scale
Closure
Statistics
Bell distribution
Pareto
(finite variance distributions)
Scientific ‘approach’
Linear science
Openness
(infinite variance)
Non-linear science
(principle of superposition)
Philosophical origin
Parmenides, Plato, Newton
Eraclitus, Aristotile
Bell curve distribution
Power-law distribution
of node linkages
of node linkages
Number of links
Exponential Network
Number of nodes (log scale)
No large number
Number of nodes
Number of nodes
Typical node
Number of links
Number of links (log scale)
Scale-free Network
From Barabasi/Bonabeau, Scientific American, May 2003
By assuming finite variability and
compressing data around mean/variance,the
Gaussian approach
but also ignores
or downplays
tiny initiating
events on the
left hand side of
the distribution
ignores or
downplays
extreme events
on the right
hand side of the
distribution
http://www.zazzle.com/statisticians_do_it_within_3_standard_deviations_
tshirt-235087605979353103
Rationality, stock market and the
butterfly effect
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 a 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 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).
Gaussian – heights of individuals
Tallest man (Robert Pershing Wadlow) 272 cm
Ratio:
= 3.7
Shortest man (He Pingping) 74 cm
Source : Lada Adamic - http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html
Paretian: city size
Largest city (Mumbai) population 13,922,125
Ratio:
= 605310
Smallest city (Hum, Croatia ) pop. 23
Mumbai, India
Hum, Croatia
Krugman on the Zipf law:
“we are unused to seeing
regularities this exact in
economics – it is so exact
that I find it spooky”
(1996) p.40
Source: Bak (1996) “How Nature Works”
Two tails of a power law
Ricther-Gutenberg Law
Nc (Earthquakes/Year)
Small events tail
Casti _126
Find gutemberg
Earthquake magnitude (mb ) ~ Log E
Extreme events tail
Main properties of Paretian distributions
• Moments:
Pr[X ≥ x] = k*x-α
• Largest value:
• maximum value depends on size of sample
• highly skewed distribution (80/20 Rule)
• Scaling property:
p(bx) = g(b)p(x) for any b
Moments of distributions
 1st: average
– Representative?
– Stable?
number of AOL
visitors to other
websites in
1997*
 2nd: variance
– Finite or unbounded?
– Stable?
• 3rd: Skewness
• 4th: Kurtosis
* Lada Adamic, Zipf, Power-laws, and Pareto - a ranking tutorial, http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html
Largest value
• Financial markets
Central limit theorem doesn’t apply. No convergence to the
mean, no central tendency.
The world shows an unlimited and irreducible stock of
surprises!
Scalability
Scalability
Scalability
Scalability
Scalability
Scalability
Scalability in financial markets
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.
In a Gaussian world:
In a Paretian world:
Challenge: manage the population
Challenge: manage the frontier
– How: reduce population to the
representative agent and define
variance (of population)
– Manage around mean and variance
– Identify outliers and manage the tails
(together with the bulk) of the
distribution
– Manage the tails
Change: gradualism
Change: extreme events
– EEs are exceedingly rare and can be
treated as perturbation (system
restores equilibrium after transient)
– EEs arise in the tails and determine
the industry next structure
Scale-free theories
Scale-free theories
– Don’t exist in Gaussian systems
– The growth of most systems follows a
set of scaling trends that link tiny
initiating events with more significant
or even extreme outcomes.
The danger of averages
Any questions?
Thank you