ArticlePDF Available

From Gaussian to Paretian Thinking: Causes and Implications of Power Laws in Organizations

Authors:

Abstract and Figures

Although normal distributions and related current quantitative methods are still relevant for some organizational research, the growing ubiquity of power laws signifies that Pareto rank/frequency distributions, fractals, and underlying scale-free theories are increasingly pervasive and valid characterizations of organizational dynamics. When they apply, researchers ignoring power-law effects risk drawing false conclusions and promulgating useless advice to practitioners. This is because what is important to most managers are the extremes they face, not the averages. We show that power laws are pervasive in the organizational world and present 15 scale-free theories that apply to organizations. Next we discuss research implications embedded in Pareto rank/frequency distributions and draw statistical and methodological implications.
Content may be subject to copyright.
From Gaussian to Paretian Thinking:
Causes and Implications of Power Laws in Organizations
Pierpaolo Andriani
Durham Business School, Mill Hill Lane, Durham, DH1 3LB, United Kingdom
and eBMS-ISUFI, Universita’ del Salento, Italy
Tel. +44 191 334 5385, fax +44 191 334 5201, pierpaolo.andriani@durham.ac.uk
&
Bill McKelvey
UCLAAnderson School of Management, 110 Westwood Plaza, Los Angeles, CA 90095-1481
Phone 310-825-7796; Fax 310-206-2002; mckelvey@anderson.ucla.edu
Copyright. All rights reserved. Not to be quoted, paraphrased, copied, or distributed in any fashion.
April 30, 2009
ii
From Gaussian to Paretian Thinking:
Causes and Implications of Power Laws in Organizations
ABSTRACT
While normal distributions and related current quantitative methods are still relevant for some
organizational research, the growing ubiquity of power laws signifies that Pareto rank/frequency
distributions, fractals, and underlying scale-free theories are increasingly pervasive and valid
characterizations of organizational dynamics. Where true, researchers ignoring power law effects
risk drawing false conclusions and promulgating useless advice to practitioners. This because
what is important to most managers are the extremes they face, not averages. We show that
power laws are pervasive in the organizational world and present fifteen scale-free theories that
apply to organizations. Next, we discuss research implications embedded in Pareto
rank/frequency distributions and draw statistical and methodological implications.
1
While “normal” (bell-shaped) distributions and related quantitative methods are still relevant for a
significant portion of organizational research, the increasing discovery of power laws signifies that Pareto
rank/frequency distributions, fractals, and underlying scale-free (SF) theories are pervasive and valid
characterizations of nonlinear organizational dynamics. Where true, researchers ignoring Pareto
distributions risk drawing false conclusions and promulgating useless advice to practitioners. This is
because, under many circumstances, what is important to most managers are the extremes they face, not
the averages. Given this, we raise the question: How to redirect organization science toward the study of
Pareto distributions in ways that still fall within the bounds of an effective science?
Power laws signify Pareto rank/frequency distributions having “long and fat tails,” potentially infinite
variance, unstable means, and unstable confidence intervals. Pareto distributions are alien to most
quantitative organizational researchers, who are trained via Gaussian statistics to go to great lengths to
configure their data to fit the requirements of linear regression models and related statistical methods. For
example, most of the discussion in econometrics textbooks, such as Greene (2002), aims to accomplish
this. Gaussian distributions have vanishing tails, thereby allowing focus to dwell on limited variance and
stable means. As a result, confidence intervals for statistical significance are clearly defined, stable, and
squeezed in toward the mean, increasingly the likelihood of achieving statistical significance.
The implications for organization science, however, go beyond extreme events. Tools do not exist in a
theoretical vacuum. The adoption of normal distribution statistics carries a heavy burden of assumptions.
Reliance on linearity, randomness, and equilibrium influences how theories are built, how legitimacy is
conferred, and how research questions are formulated. Abbott (2001) says that linear thinking, what he
calls the General Linear Model, defines the philosophical and methodological assumptions upon which
linear science is based. It affects (a) how units of analysis are conceptualized, selected, and operationalized;
(b) how variables are selected; and (c) how the interactions among variables are described by
quantitative/qualitative models. Given these premises, we introduce scalability and the study of SF theories
to begin a reorientation of the organization science paradigm from linear toward a Pareto-based Science
more relevant to nonlinear organizational phenomena.
2
Scalability results from what Mandelbrot (1982) calls “fractal geometry.” A cauliflower is an obvious
example. Cut off a “floret;” cut a smaller floret from the first floret; then a cut piece off the second, and so
on. Now set them in line on a table. Each subcomponent is smaller than the former; each has the same
shape and structure. They are fractal because they all look and behave about the same way. Fractals are
signified by power laws and rank/frequency distributions. Researchers find organization-related power
laws in intrafirm decisions, consumer sales, salaries, size of firms, ecosystems, director interlocks, biotech
networks, and industrial districts, for example. These are all rank/frequency distributions.
Responding to the state of scientific disciplines of many kinds, Gell-Mann emphasized the study of
surface complexity arising out of deep simplicity at the founding of the Santa Fe Institute (1988, p. 3;
his italics). In describing the Santa Fe vision, Brock says the study of complexity “…tries to understand
the forces that underlie the patterns or scaling laws that develop” as newly ordered systems emerge (2000,
p. 30). Many complex systems tend to be “self-similar across levels. That is, the same dynamics drive
order-creation behaviors at multiple levels (West et al. 1997). These processes are called “scaling laws
because they represent dynamics appearing similarly at many orders of magnitude (Zipf 1949). We
present fifteen SF theories, arguing that most apply to organizations. Gell-Mann (2002) argues that in
living systems scalability and scaling laws are as important a means of scientific explanation as is
reductionism and explanation via law-like equations.
We first use findings of ~120 kinds of power laws from natural to social and organizational
phenomena to suggest the pervasiveness and importance of power laws, which typically signify well-
formed rank/frequency Pareto distributions stemming from scalable causes. Next, we classify fifteen SF
theories about scalable causal dynamics that apply to organizations, discussing several in detail. Then we
switch to research implications: How do theory and methods change if we focus on rank-frequency Pareto
distributions rather than squeezing all organization phenomena into normal distributions (or, more
broadly, distributions that rely on finite variance)—as is currently the practice? Finally, we discuss
implications in terms of the basic predictor function, y =
f
(x) +
ε
. How does basic thinking about
3
prediction, data, error term, and statistics, have to change? A conclusion follows.
ENTERING THE THIRD PHASE OF COMPLEXITY SCIENCE
Background
Complexity science has emerged in three phases. Energy: The first phase appeared in Europe, led by
Nobel Laureate Ilya Prigogine (1955, 1997). He built on Henri Bénard’s (1901) study of emergent
structures in fluids. Because these serve to dissipate energy imposing on a system, he labeled them
dissipative structures.” This phase transition, which occurs at the so-called “1
st
critical value” of
imposed energy, defines what we may call “the edge of order.” Schieve and Allen (1981), Haken (1983,
2004), Nicolis and Prigogine (1989), and Mainzer (1994/2007) continue this line of work.
Emergence: This phase was initiated by Nobel Laureates Anderson (1972) and Gell-Mann (1988,
2002) along with Holland (1988, 2002), Kauffman (1993), and Arthur (1994) at the Santa Fe Institute. It
is mostly oriented toward biology and the social sciences—i.e., living systems (Gell-Mann 2002). Focus
is on heterogeneous agents interacting at what was early-on called “the edge of chaos;” this occurs at the
2
nd
critical value” of imposed energy. In between the “edges” of order and chaos is the region of
emergent complexity, what Kauffman terms the “melting” zone (1993, p. 109). Bak (1996) argued that to
survive, organisms have to have a capability of staying within the melting zone, maintaining themselves
in a state of “self-organized criticality,” i.e., adaptive efficacy. Holland (2002) defines emergent
phenomena as multi-level hierarchies, intra- and inter-level causal processes, and nonlinearities.
Nonlinearity incorporates two additional outcomes: the butterfly effect
1
and scalability. Stacey (1992),
Goldstein (1994) and many others apply complexity science to organization studies (Maguire et al. 2006).
Scalability: Though beginning decades ago with Pareto (1897), Auerbach (1913) and Zipf (1949), the
third phase—which includes econophysics (West & Deering 1995, Mantegna & Stanley 2000)—focuses
on power law phenomena (Newman 2005). Econophysics began with Benoit Mandelbrot’s focus on stock
market crashes (1963a). While crashes are negative extreme events, their showing the power-law
signature indicates that the markets were free to go up or down without restraint. Power laws often appear
4
as telltales of self-organization, emergence-in-action, and self-organizing economies (Krugman 1996).
If one plots a well-formed Pareto rank/frequency distribution with both x and y axes as log scales—a
negatively-sloped straight line will appear; this is the inverse power law signature. Power laws often take
the form of rank/frequency expressions such as F ~ N
β
, where F is frequency, N is rank (the variable) and
β, the exponent, is constant.
2
This is in contrast to “exponential” equations stated in terms of the natural
log, e, where the exponent is the variable and N is constant. Power laws show potentially infinite variance
and an unstable or non-existing mean and are frequently “…indicative of correlated, cooperative
phenomena between groups of interacting agents…” (Cook et al. 2004)—but not always, as we will point
out below. Andriani and McKelvey (2007a) present over 80 kinds of power laws (with sources)—16
physical, 24 biological, 21 social, and 23 pertaining to economic, business, and organizational
phenomena. We show an expanded list of 97 social and organizational power laws in Table 1.
>>>Insert Table 1 about here<<<
Power laws indicate long-tailed Paretian rank/frequency rather than “normal” Gaussian distributions—
see Figure 1. The difference lies in assumptions about the correlations among events. In a Gaussian
distribution, data points are assumed independent-additive (hereinafter simply “independent”). These
events generate normal distributions, which sit at the heart of modern statistics. When causal elements are
independent-multiplicative a lognormal distribution results, which turns into a Pareto distribution as the
causal complexity increases (West & Deering 1995)—detailed below. When events are interdependent
and/or interactive, normality in distributions is not the norm. Instead, Pareto distributions dominate
because positive feedback (and other) processes leading to extreme events occur more frequently than
“normal” Gaussian-based statistics lead us to expect. Further, as tension imposed on the data points
increases to the limit, they can shift from independent to interdependent (Boisot & McKelvey 2007).
>>>Insert Figure 1 about here<<<
Phase Three brings a totally new look to organizational applications of complexity science: (1) Power
laws as indicators of effective emergence-in-action; (2) SF theories as explanations of the underlying SF
causal dynamics; and (3) Holland’s “levers” as managerial action tools to foster scalable dynamics.
5
From Reductionism to a New Regularity—Scalability
Brock (2000, p. 29) says,
The study of complexity…is the study of how a very complicated set of equations can generate
some very simple patterns for certain parameter values. Complexity considers whether these
patterns have a property of universality about them. Here we will call these patterns scaling laws.
The increasing discovery of power laws brings scalability and SF theories to prominence (Newman
2005). Many complex systems—resulting from emergent dynamics—tend to be “self-similar” across
levels. That is, the same process drives order-creation behaviors across multiple levels of an emergent
system (Casti 1994, West et al. 1997). These processes are called “scaling laws” because they represent
empirically discovered system attributes applying similarly across many orders of magnitude (Zipf 1949).
Scalability occurs when the appearance of phenomena is independent of the scale used to measure it
(inches, feet, yards, miles), or the same causal dynamic operates at multiples levels.
Gell-Mann (2002) defines “effective complexity” as “regularities” or “schema” found or judged to be
useful. For him, they appear as equations, genotypes, laws and traditions, and business best practices.
What is new is Gell-Mann’s recognition of a new regularity. In doing so, he sets forth two regularities:
Type 1. Reductionist Law-like Regularities: The old simplicity of reductionist causal processes of normal science,
which are predictable and easily represented by equations—the data and information much preferred in classical
physics and neoclassical economics (2002, p. 19). These are the point attractors of chaos theory—defined by
forces, energy conservation, and equilibrium.
Type 2. Multilevel SF Regularities: The new simplicity of insignificant initiating events—what we call “butterfly-
events.” Outcomes over time that result from an accumulation of often random tiny initiating events that have
lasting effects, are compounded by positive feedback effects over time, and become frozen accidents” (2002, p.
20). These are the strange attractors and fractals of chaos theory—never repeating, fostering indeterminacy,
offering a different kind of regularity.
The first process generates regularities characterizing existing empirical organization and management
research. These may be confidently described via Gaussian statistics and allow predictions that become
the basis of schemata and prescriptive solutions. They are the basis of “reductionist” science—using
components to explain a more macro level of behavior. The second focuses on the effects of tiny initiating
butterfly-events.” The butterfly-events of chaotic histories are seldom repeated, are not predictable, and
6
can produce significant nonlinear outcomes that may become extreme events. Consequently, descriptions
of these systems are at best problematic and easily outside the explanatory/scientific traditions of normal
science. Gell-Mann concludes by noting that when butterfly-events spiral up such that their effects appear
at multiple levels and are magnified, we see self-similarity, scalability, and power laws.
Underlying most power laws is a causal dynamic explained via SF theories. Each theory points to a
single pervasive generative cause to explain the dynamics at each of however many levels at which the
scalability effect applies. SF theories yield what Gell-Mann (1988, p. 3) refers to as “deep simplicity.”
Whereas tradition rests on the idea that lower-level dynamics can explain and predict higher-level
phenomena and simplicity comes in the form of (usually) linear mathematical equations—i.e.,
reductionism (Gell-Mann 2002), SF theories point to the same causes operating at multiple levels—the
“simplicity” is one theory explaining dynamics at multiple levels.
SF causes are Holland’s “levers.” “…Almost all
CAS
[complex adaptive systems] exhibit lever point
phenomena, where “inexpensive” inputs cause major directed effects in the
CAS
dynamics” (2002, p. 29).
These “levers” trigger butterfly-effects across multiple levels. A butterfly-effect is a nonlinear outcome
occurring when a single event, out of myriad tiny events, gets amplified to become a stronger effect.
Explaining via SF Theories
Besides those mentioned earlier, researchers find power laws in social networks, industry sectors,
growth rates of firms, bankruptcies, transition economies, drug and movie profits, sales decays, and
economic fluctuations—see Table 1. Power laws are mostly explained by SF theories. We identify 15 SF
theories applying to organizations—see Table 2. We believe that the following logic-chain applies:
1. Successful emergence results in fractals, SF dynamics, and power laws explainable via SF theory;
2. Power laws are far more ubiquitous than heretofore realized and are usually indicators of SF dynamics;
3. Consequently, SF dynamics are also ubiquitous; many SF theories seemingly apply to organizations;
4. If power laws are not obviously evident in organizations then emergence has failed to emerge;
5. Therefore, organization-relevant complexity theory and research have to apply scalability dynamics.
>>>Insert Table 2 about here<<<
Two new complexity thrusts are identifiable. First, roughly one/third of complexity science theory is
7
missing in organizational and managerial applications to date, i.e., the scalability phase—power laws and
the underlying fractals, scalability, and SF theory. Organizations are multilevel phenomena. Almost by
definition then, we can take power law signatures as the best evidence we have that emergence dynamics
are operating at multiple organizational levels. We now know for sure that power laws apply at the overall
industry level (Stanley et al. 1996, Axtell 2001), industry sectors (Aoyama et al. forthcoming), with some
appearing within firms. If power laws are not evident in a particular firm, we can only conclude that
emergence,” if it exists at all, is not multilevel. Building from the interacting food-web literature (Pimm
1982, Solé et al. 2001, Cuddington & Yodzis 2002, Sims et al. 2008), we can also surmise that, absent the
power law signature, a firm’s emergence dynamics are not capable of keeping it competitive with its
changing competitors, suppliers, and customers (McKelvey et al. forthcoming). The bottom line is that
power laws are significant indicators of crucially important managerial and organizational dynamics.
Second, organization change and entrepreneurship researchers should be especially interested in SF
dynamics and related theories. Who more than entrepreneurs wouldn’t like to let loose SF dynamics in
their firms? Think of how many small entrepreneurial ventures stay that way simply because the emergent
growth dynamics they had at the one- or two-level size failed to scale up as levels increased. Think how
many large organizations show failing intrapreneurship for the same reason—the hundreds of “butterfly-
ideas” never become meaningful butterfly-events, never produce butterfly-effects, and never spiral into
multilevel SF causal dynamics producing power law signatures. Jean-Pierre Garnier, CEO of
GlaxoSmithKline says:
…Size is a problem early in the drug-development process. “Drug finders” and innovators may well get
tripped up by bureaucracy and tangled in red tape; good ideas are lost. Even worse, bad ideas may not be
weeded out in time. (Quote is by The Economist 2007, p. 57.)
Complexity theory applied to organizations is silent on the foregoing points. One important move we
recommend now is to learn how to apply SF complexity theories to organization change, OD, and
entrepreneurship/intrapreneurship and strategy. Teaching and preaching complexity theory is out of date
in our organizational world, absent SF theory. These points are further elaborated in Andriani and
8
McKelvey (2007a) and Boisot and McKelvey (2007).
CAUSES OF POWER LAWS IN ORGANIZATIONS
Of the many kinds of power laws we list in Table 1, more than 50 are associated specifically with
firms and organizational processes. Some of the power laws in broader social phenomena also apply to
organizations. These discoveries of organization-relevant power laws offer substantial evidence that well-
formed Pareto distributions are everyday organizational phenomena. Given this, two questions are:
1. What causes power laws in organizations and what theories might describe the causes?
2. To what extent does the existence of power laws undermine prevailing assumptions that organizational
phenomena are linear, equilibrium-seeking, and normally distributed?
In this section, we respond to the first question by arguing that SF theories apply to organizations.
Classifying Scale-Free Theories about Causes of Power Laws
We have assembled enough SF theories that a classification of them seems relevant, as follows:
1. Ratio Imbalances—Theories 1–4: In each, the basic SF cause is some kind of “cost-driven efficiency” requiring
constant or periodic adjustment:
2. Multiple Distributions—Theories 5–7: Here the SF cause of long tails is some kind of combination in the form of
p(y) ~ e
a,b,c,d
⋅⋅⋅⋅
n
, where the distributions underlying variables a, b, c, d
⋅⋅⋅⋅
n are somewhat skewed and tainted with
outliers. The multiplicative effect of the outliers progresses into a long-tailed Pareto distribution (West &
Deering 1995).
3. Positive Feedback—Theories 8–11: In some systems the initial interaction possibilities are such that there is the
possibility, if not probability, of positive-feedback spirals emerging simply as time progresses. The underlying
SF cause is some probability that butterfly events will mutually interact so as to spiral up to produce long-tailed
distributions.
4. Contextual Effects—Theories 12–15: Exogenous effects set SF dynamics in motion. In this set, different kinds of
imposing effects set off SF causal processes. The common effect is context, but in each case the contextual effect
is different and acts to set off a different kind of SF dynamic.
The definition and discipline-base of each theory are given in Table 2.
Ours is the first actual classification of this many theories about SF causes—most publications don’t
mention any SF cause; some mention one. Newman (2005) emphasizes preferential attachment and self-
organized criticality, with minor reference to a few other physical ones (see also Sornette 2000). There are
a few we don’t include. Like the proliferation of power law discoveries, the growing set of SF theories
9
makes it harder and harder not to wonder if they don’t also apply to organizational phenomena.
Do Scale-Free Theories Apply to Organizations?
In Andriani and McKelvey (2007b) we make shorter arguments as to how most of the SF theories in
Table 2 apply to organizations. While each is briefly defined in Table 2, we discuss five of them here in
more depth.
Square-cube/Quarter-power. In biology, many scaling laws take the allometric form Y ~ M
b
, where Y
is some observable and M the mass of the organism. “Allometric” refers to a type of growth in which the
parts of an organism grow at different rates determined by fixed ratios. Among these, West et al. (1997)
cite metabolic rate, height of trees, life span, growth rate, heart rate, DNA nucleotide substitution rate,
lengths of aortas, size of genomes, mass of cerebral grey matter, and density of mitochondria. In
organisms, surfaces absorbing energy grow by the square but the organism grows by the volume,
resulting in an imbalance. By adaptation, fractal structures emerge to keep surface absorption of energy in
balance with the volume’s use of energy. But energy is moved from surface to places in the volume by
capillaries and other tubes in which fluid flows are governed by the quarter-power law. By nature,
organisms adapt in a fashion such that quarter, square, and cube capacities are appropriately balanced.
They are “allometric”
3
scaling laws because they set up rigid relationships.
Allometric growth reflects universal structural constraints in the way organisms use energy. The
emerging field of allometric growth (Whitfield 2006) is redefining biology and more in general the study
about how ecosystems self-organize around fundamental energetic constraints. Insofar organizational
ecosystems use energy and energy-related quantities [money is the equivalent of energy according to
some economists—see for instance Beinhocker (2007)], and conform to general principles of ecosystems
organization, we expect the study of allometric growth to yield compelling inputs to organization science.
Firms operate in ecosystems defined by the need to maximize revenues (exchange area between firm
and customers) and minimize expenses (energy spent for developing, manufacturing and distributing
products). If this revenue-energy constraint can be given a meaningful geometric economic form, we may
discover similar allometric relationships in many organizations. We now present some initial findings.
10
Haire (1959) first applied the square-cube law successfully to four firms. Levy and Donhowe (1962)
confirmed his findings in 62 firms in eight industries. Stephan (1983) applied the square-cube law to
firms in terms of effectiveness. Employees dealing with people outside the firm are “surface”
employees—they bring in the resources from the environment. “Volume” employees are those inside who
produce and coordinate. As firms grow, then, they have to maintain the square-cube ratio by adding more
surface units or making them more efficient.
Carneiro (1987) applies the law to explain the upper bound on the size of villages. The law limits their
size unless they develop what he terms “structural complexity,” where complexity grows at 2/3 power of
a village’s population. Only by doing this do villages avoid splitting in two. Carneiro’s theory is more
general than Stephan’s; Carneiro says social entities can increase in size only by building in structural
complexity. In his data, for example, 100-person villages had 10 “complexity traits” where as 1000-
person villages had four to five times as many. Johnson (1982), studying the governance of primitive
organizations, finds that organizational complexity and leadership diversity scale accordingly with the
allometric principles mentioned above. Much like Carneiro, Johnson finds that the emergence of nested
hierarchical systems seems to be a response to the scalar stress induced by the exponential increase in the
number of communication channels among the parts of the organization. The number of communication
channels scale exponentially with the volume of the organization. A scalar stress increase forces the
organization to elaborate more complex hierarchical systems with the effect of keeping scalar stress under
control. By doing so, the organization changes the surface-volume ratio. From the point of view of the
square-cube law, a decentralized network organization is a way to transform a large portion of the
organizational employees into “surface” employees, thereby correcting scalar stress and bringing the
surface/volume tension in line with allometric growth—but still subject to the ¼ power law.
The ¼ power law applies to the supply-chain materials-flow “tube” that limits both the size of a
supplier and retailer, for example. Zara is a retailer of new high-fashion designs—three weeks from
models in the designer’s mind to new fashions in its stores. How? By bringing manufacturing from China
back to Europe, thereby shorting the “fluid” flow of clothing in transportation corridors and long-distance
11
design communications. The primary factor in the virtual collapse of Citigroup stems from its “silo”
design (Moore 2008). None of its employees having diverse vantage-points of observations of its
activities leading up to the sub-prime meltdown were connected to useful communication flow
channels—no part of the firm could readily learn from any of the other parts.
4
One could call this “a total
¼ power breakdown.”
Combination and Breakage Theories. Kolmogorov (1941) originally applied his “breakage theory” to
coal—when large chunks of coal were smashed so as to be used in furnaces—resulting in small chunks
down to powder, they appeared Pareto distributed. We now see Pareto distributions and power laws in
chromosome breakage (Pevzner & Tesler 2003) and hydrodynamics (Bache 2004). In organizations the
simple breaking up of firms into nearly autonomous modular designs makes breakage theory applicable—
an approach dating back to Simon’s (1962) “nearly decomposable” systems theory.
Oppositely, combination theory holds that the requirement for a power law to emerge is the number of
elements in a complex system and their propensity to interact with one another. West and Deering (1995)
and Newman (2005) both make the case that the combination of exponents results in a power law
distribution—the more of them that are combined the more obvious the power law. Of course it does at
the equation level, but what interacting non-normal phenomena actually occur in organizations?
If several organizational components or behaviors appearing vertically across several levels
individually generate non-normal distributions having somewhat longer tails, and also influence each
other, then combination theory tells us that organizations are inevitably going to contain well-formed
Pareto distributions and show the power law signature. Since they appear across several vertical levels,
there is also a high probability that fractal structures and scalable causes are present—unless, of course,
there are explicit attempts by management to negate them. In combination theory, the occurrence of
interaction is taken as a naturally inherent likelihood as systems become more complex. Since many
larger organizations have many degrees of freedom—and thus are complex by definition—they will show
Pareto distributions, as is preliminarily evident in our Table 1.
As it has been applied to wealth (Montroll & Badger 1974), breakage theory appears as a set of
12
independent-multiplicative elements that are lognormally distributed (West & Deering 1995, p. 152). To
be wealthy, an individual has to have some minimum level of specified kinds of attributes (elements). The
eight elements are: social background, education, personality type, technical ability, communication
skills, motivation, right place-right time, willing to take risks. But, social background, education,
technical ability, communication, and being in the right place at the right time are all potentially
interactive, with an embedded positive feedback effect—e.g., for a family, the more social status,
education, and technical skills, then over time, the more technical skills the more social status, etc. Now,
in the business world, suppose that some set of elements are required in organizations to cause the GEs,
Microsofts, Toyotas, and Wal-Marts of this world—say CEO skills, the right industry, new technology,
“star” employees, special markets, weak competitors, borrowing ability, etc. As these and other elements
become more complexly interactive with positive feedback effects, the distribution of firms changes from
lognormal to Pareto. Axtell’s (2001) research shows that this is, indeed, how U.S. industry appears.
Least effort. Zipf (1949) argued that “least effortexplained his “Zipf’s Law”—a power law of word
usage in English, French, and Spanish. The first question is: What is least-effort? Consider: Why would
you learn words we don’t use? Why would we use words you don’t understand? In both cases it is wasted
effort. Least-effort theory holds that each of us will minimize down to the only words relevant for
meaningful transaction. For example, the 1953 Merriam-Webster’s unabridged dictionary had over
550,000 words; by 1971 it was reduced to ~450,000 words. Merriam-Webster’s collegiate dictionary that
most of us have in our offices is abridged down to around 86,000 words. The Harper Collins Italian
English dictionary contains some 28,000 English words. We get by with fewer and fewer words.
Why is word usage Pareto-distributed? Why are words like the, of, and, and so on, at the top of the
rank/frequency distribution? These words have to fit in with both the “before” words and the “after
words. Adjectives and adverbs, however, only have to fit with “after” words. Some words with narrow
technical means are seldom used—the word “unabridged” is used only once in this paper. High usage
brought more opportunities, historically, for more proposed usages, and more chances for disagreement
on word usage. Higher usage also brings more opportunities for least-effort movements to improve
13
efficiency. The result is increased demand for least-effort and greater payoff. In word usage, then, we
have an interactive market transaction of word usage that slowly works toward increased efficiency. The
basic dynamic is a circular, positive-feedback process where each party moves toward the maximum
efficiency, least-effort attractor basin. It is the opposite of the other positive feedback SF theories.
To make least-effort theory even more compelling and applicable to organizations, we find that it has
now shown to be especially characteristic of changing circumstances. Four recent studies suggest that
Zipf’s Law appears predominantly in the context of change:
1. In testing whether Zipf’s Law applies to Chinese as well as English, Dahui et al. (2005) find that the power law
signature applies only during the period before Emperor Qin Shihuang’s unification (~1720), when Chinese
characters were in flux. They conclude that the law does not apply when the number of characters is stable.
2. Ishikawa (2006) shows that Pareto’s law holds when applied to firms in less populated JSIC
5
2-digit categories
(having fewer firms) where growth rate is high, but a lognormal distribution applies to firms in large categories
(filled with firms) where growth rate is, therefore, low.
3. Dahui et al. (2006) use two computational network models to show that the distribution of firms in growth
markets is a power law but in markets without growth it is Gamma or exponential. They conclude: “…we cannot
get [a] power-law distribution by preferential attachment in a constant market….Economic growth is an
important condition for the power-law distribution of firm size….” (pp. 363–364)
4. Podobnik et al. (2006) find empirically—and test further with a computational model—that time-series indices in
transition economies (i.e., Hungary, Russia, Slovenia, etc.) fit Paretian rather than Gaussian distributions.
Three of the foregoing studies apply to organizations or markets.
Preferential attachment. This positive-feedback process (Barabási 2002) underlies biological and
social networks, going from groups of individuals to groups of organizations. The Internet grows
according to preferential attachment (Dorogovtsev & Mendes 2003). The same happens with cities and
airport hubs—as nodes grow they attract even more people or flights (Barabási 2002). Internet marketing
and sales are very much a positive feedback process (Gladwell 2000). Any time a system grows by adding
nodes to an existing network their growth will amplify historically generated imbalances among the links.
Absent top-down regulation, older or larger nodes will gain more links and generate a Pareto
distribution—as in the biotech industry (Powell et al. 2005, Gay and Dousset 2005). Since organizations
are made of social networks, preferential attachment plays a crucial role in their formation and evolution,
14
thereby providing a solid base for a network-based theory of organizational formation and development.
This “rich get richer” dynamic explains the emergence of central hubs and peripheral groups, which
characterize the geography of most organizations and the inherent concentration (and dispersion) of
decision-making. Other examples are Arthur’s (1994) study of increasing returns—firms making profits
can invest in things that make even more profits. Microsoft is a good modern example, as is Wal-Mart;
the more it lowers prices, the more people come to buy; the more they come to buy, the more Wal-Mart
can lower prices. And then, the more Wal-Mart sells the more pressure it can put on suppliers to lower
price; the more they lower price, the more Wal-Mart can sell.
Self-organized criticality (SOC). This theory is symbolized by Bak’s (1996) and others’ (Frigg 2003)
sandpile experiments. A sandpile subjected to an infinitesimal external perturbation (sequentially adding
single grains of irregularly-edged sand) evolves toward a critical state, characterized by a critical slope,
whereby any additional grain induces a systemic sand movement reaction that can span any order of
magnitude (from one grain to thousands), with a frequency distribution expressed by a power law. This is
counter-intuitive. We generally assume a linear relationship between perturbation size and a system’s
reaction, i.e., small causes yield small effects. This is true before SOC is attained. Thus, before criticality,
each falling grain has a constant probability of displacing an adjacent grain. The probability of an
avalanche therefore scales exponentially with the number of sand grains. This makes large avalanches
highly unlikely. However, at criticality a power law distribution results, given the global connectivity of
the irregular-edged grains making up the sandpile. As Bak (1996, p. 60) writes: “In the critical state, the
sandpile is the functional unit, not the grain of sand.” SOC dynamics arise when an emergent system of
links connects local pockets into a coevolving whole such that small and local fluctuations may be
amplified to achieve systemic effects. More generally, as the tension in the system increases to the SOC
limit (usually as a result of externally imposed tension—in Bak’s SOC this is a function of gravity and
accumulating sand grains—independent data points become interdependent. Mathematically this means
that sandpile behavior obeys a power law of the type: F ~ S
α
, where F represents avalanche frequency
with given size, S.
15
From the dynamics of earthquakes (Gutenberg & Richter 1944), booms and busts in economic cycles
(Sornette 2003, Mandelbrot & Hudson 2004, Malevergne and Sornette 2006), to the dynamics of supply
chains (Scheinkman & Woodford 1994, Wycisk et al. 2008), a common pattern appears across disparate
fields. Many systems exist in the state of criticality—on the critical slope as it were. Bak argues that all
systems in efficaciously adaptive states are in the state of criticality. Needless to say, then, SOC occurs
frequently in markets and organizations (Buchanan 2000). Arguing that individual decisions are sticky
like irregular sand grains, Bak applies SOC to economies. Since the tension between supply and demand
builds and the actions to reduce it are not of equal size or regularity, free-market economies operate at or
near the critical state. Economic fluctuations (business cycles) are SOC (Scheinkman & Woodford 1994).
We see SOC in the price of cotton and financial markets (Mandelbrot & Hudson 2004)—many small
change in the price of stocks and the overall value of the market separated by volatility incidents
averaging one in every four years from 1950 to 1980 and one every two years since 1980.
6
We also see
SOC in consumer product sales (Moss 2002, Sornette et al. 2004), and managerial actions leading to
different sized firms (Stanley et al. 1996, 2000)—all of which show power law signatures.
In the foregoing we detail how five SF theories apply to organizations. Elsewhere, we argue that
almost all apply (Andriani & McKelvey 2007b). We have, thus, potentially fifteen reasons why
organization scientists should take as their New Null Hypothesis (Alderson, 2008) the reality that
organizations and managers very often live in a world of interdependent and not independent events, a
world of Pareto distributions, fractals, and power laws; not Gaussian distributions where stable averages
and finite variances across large samples are what count. Yes, we agree that there are many times and
places where Gaussian statistics apply, but it is simply wrong to assume that they are the rule. If it is
explicitly shown that a normal distribution holds, well then, use Gaussian statistics. But absent this, the
new Null should be presumed to apply.
SOME RESEARCH IMPLICATIONS
All the world believes it [Gaussian distribution] firmly, because the mathematicians imagine that it is a
fact of observation and the observers that it is a theorem of mathematics. (Henry Poincaré 1913)
7
16
We now offer Pareto-driven alternatives, starting from a discussion of the predictor function. Take a
standard predictor function—consisting of a dependent variable y, several independent or explanatory
variables, x
n
, and an error term,
ε
: Thus:
y =
f
(x
1
, x
2
, x
3
,
···
x
n
) +
ε
.
There are two concerns when one shifts from a Gaussian to a Paretian perception of data: (1) what
happens to the predictor function? And (2) what happens to the error term? Organizational researchers
using statistics as their basis of making truth claims—usually translated as findings significant at p < .05
or <.01—generally use statistical methods calling for Gaussian distributions.
For instance, Greene’s textbook, Econometric Analysis, (2002) is in its 5
th
edition and is the standard
for many econometricians and other social science researchers. He begins his ~950 pages of analysis with
linear multiple regression and its five endemic assumptions: (1) independence among data-points; (2)
linear relationships among variables; (3) exogenous independent variables; (4) homoscedasticity and
nonautocorrelation; and (5) normal distribution of error disturbances (p. 10). Mostly, his book focuses on
how to make econometric methods work when one or more of these assumptions are untrue of the data.
Given nonlinearity, for example, Greene says, “by using logarithms, exponentials, reciprocals,
transcendental functions, polynomials, products, ratios, and so on, this ‘linear’ model can be tailored to
any number of situations” (p. 122). Regarding data distributions, he then says:
Large sample results suggest that although the usual t and F statistics are still usable…they are viewed as
approximations whose quality improves as the sample size increases…. As n increases, the
distribution…converges exactly to a normal distribution. (p. 105)
Most standard econometric textbooks, such as Greene (2002) and Kennedy (2003), present methods to
transform datasets into distributions with finite variance. Of these, the normal distribution is by far the
most used due to its stability and conformance to the central limit theorem. However, as Bartels (1977, p.
86) writes:
Economic data are seldom plentiful or accurate enough to distinguish between a hypothesized normal
population and a nonnormal stable one, and since such data are notoriously long-tailed it is difficult to
determine whether the population variance is finite or not.
This is crucial: Mandelbrot (1963) claims that reliance on finite variance is the ‘Achilles’ heel of
17
econometrics. Our Table 1 offers reasonable evidence that increasing n may very well result in Pareto
distributions—the Central Limit Theorem (CLT) doesn’t apply! West (2007) takes it a step further, saying
that Pareto distributions are so ubiquitous that finding a CLT-based average is “exceptional!”
The various robustness
8
tests standard econometric textbooks discuss give evidence that modern-day
researchers have not taken on board Mandelbrot’s 1963 plea:
There is strong pragmatic reason to begin the study of economic distributions and time series by those that
satisfy the law of Pareto. Since this category includes prices, firm sizes, and incomes, the study of Paretian
laws is of fundamental importance in economic statistics (Mandelbrot 1963b, p. 438).
Let’s put this in California earthquake terms—where ~16,000 insignificant quakes occur every year
and a “really big one” once every 150–200 years, with 6- and 7-level quakes occurring within decades. If
one sampled California quakes from 1995 to 2006, all but two would be in the 1–4 range; damage to no
more than a few houses; no one killed. But this would miss the recent 6–7s in urban areas (costing
$billions and over 100 people killed) and the next #8 yet to come. Californians have long concluded that
building codes should be based on the Pareto rather than Gaussian perspective. If California followed
traditional econometric models, above, gathering more data would make quakes appear even more
normally distributed, which is surely not the case.
In effect, application of methods based on Gaussian statistics (or more broadly on finite variance)
models would lead Californians building and living in high-rise buildings to think that using a moving
average of quake variance over the thousands of harmless (average) quakes would lead to effective
building codes. Anyone living through a significant quake in California will tell you this is nonsense. No
amount of so-called “robustness improvements” to the standard linear multiple regression model allow it
to model the effects of extreme quakes on buildings, bridges, lives, and damage costs—i.e., the effects of
fat-tailed Pareto distributions. Robustness “solutions” cannot alter rank/frequency distributions to
conform to Gaussian assumptions.
The Predictor Function—From Gaussian to Pareto Thinking
Consider the typical “linear” prediction:
f
(x
1
, x
2
, x
3
, ··· x
n
)—the predictor function. Suppose we have
18
a simple explanatory theory based on three independent variables: experienced, skilled, and satisfied
employees increase output; thus: y =
f
(x
1
, x
2
, x
3
) +
ε
. In making a prediction like this, we usually think
linearly—x causes y. Furthermore, propositions and even operational hypotheses appear in print with the
expectation of a perfect correlation implicit—minus the effect of the error term. We visualize this as an
upward sloping line in the wished-for plot of each cause of productivity, x
n
(above) against output, y. Of
course, the real world is never like this and so the plot of y by x
1
data-points, for example, appears as
almost a circle at worst (near zero correlation) or a narrow ellipse at best—the thinner the better.
Two essential features, together, are the defining elements of a “normal” Gaussian distribution:
1. The “mean” is stable and meaningful: In the equation: y =
f
(x) +
ε
, define y as weight and x as height. Average
weight of males in the U.S. is 190 lbs; average height is 5' 8.6.” Millions of men are at or very near the mean.
9
2. Variance is finite. Shortest living man is 2' 5"; tallest is 7' 10" (both in China); both are within ½ magnitude.
In Gaussian statistics some variance is essential, but too much is a problem. Worse, if there is too much
variance, confidence intervals widen and getting statistically significant results is more problematic. In
our worker/output example, since human bodies are involved, independence is reasonable: there will be
strong stable means of skill, experience, and satisfaction, with enough variance around the mean to allow
correlations. But, too little variance and there is no meaningful correlation; too much variance and there is
less (or no) chance for significance.
As one moves away from a simple study of bodies, such as our example—which is essentially where
statistics-applied-to-firms started half a century ago—to study firms, distributions appear less obviously
composed of independent data points. As a result, some 70 years of advances in statistics (since the
founding of Econometrica in the early 30s) offer devices econometricians can use to get all the weird
kinds of data in the world of firms redesigned to fit linear regression. Now switch to a Paretian world.
What changes? Consider species and consumer products.
At one end of a Pareto distribution we have a few hundreds of elephants or Wal-Mart.
At the other end we have trillions of mosquitoes or millions of “Ma&Pa” stores (defined as having no paid
employees).
Elephants are huge but mosquitoes are tiny. Elephants eat vegetation, trample the land, and can trash
19
your living room; mosquitoes suck warm blood, fly, and can give you viral diseases. Wal-Mart is huge,
10
enjoys substantial bargaining power over its supply chain,
11
and has powerful lobbying abilities.
12
Tiny
Ma&Pa stores exist in micro niches, buy supplies at the market, and have little if any political power. The
averages of these ends of Pareto long tails are meaningless—Axtell (2008) says a “typical firm” doesn’t
exist! The mean, median, and mode are different. A careful Gaussian study at the average or median may
offer little, if anything, of interest or use about firms at either end. A Gaussian study of Ma&Pa stores
works, but obscures micro-niches and offers little of value to Wal-Mart. And finally, any study close to
the average or median (or elsewhere) ignores scalable dynamics.
What are the implications?
1. When extreme events occur they also alter the value of the mean—pulling it toward the tail where the extreme
event occurred. Hence Pareto means are unstable.
2. Compared to “normal” variance, Paretian variance is potentially infinite. From Ma&Pa stores to Wal-Mart,
profits, assets, and indebtedness range from $1000s to $billions. Profits and assets go from $zero or worse to
$billions—crossing ~11 magnitudes. Mode, median, and mean are not the same. The larger the extremes, the less
frequent or predictable they are. But, when they happen they increase the variance—perhaps more obvious with
things like earthquakes and species abundance, but also apparent in firms, M&A activities, and bankruptcies.
3. Furthermore, since the variance is potentially infinite, the confidence intervals are considerably widened, making
findings less apt to be significant. The CLT is meaningless.
Millions of businesses are single proprietor or Ma&Pa stores with income in the thousands of dollars, and
possibly negative wealth. At the other extreme we see a few giant firms having hundreds of thousands of
employees, billions of dollars in annual income, with assets of hundreds of billions—profits and assets
can range across 11 magnitudes. For much organizational thinking, research, and practical applications,
neither the distributions of the variables nor theorizing about the causal dynamics fits within the Gaussian
assumption set. A different approach is needed, based on scalability.
Suppose we study 4000 people in 500 small stores in small towns. Their owners’ smartness, creativity,
and knowledge of technology, markets, and customers, as well as good or bad attitudes, skills, behavior,
networking, and so on, affect the other store owners and a couple hundred regular customers. This is not a
bad sample, but if we improve it, both sample and error disturbances will become distributed more
20
perfectly “normal.” If we move the same study to 500 rural outlets of Wal-Mart we should end up with
the same high quality “normal” distribution and error term disturbances. So far, so good.
But Wal-Mart is huge, having giant stores, many hierarchical levels, vast profits and assets, and global
reach. So, we expand the sample to 1,000 world-wide. But now, instead of two people at one level in each
small store and 200 small-town customers to deal with, we have employees acting at multiple levels, in
medium to giant stores, who have to deal with many more increasingly diverse customers, subordinates
and superiors, local zoning issues, M&A issues, and so on. They make decisions ranging from local
customer concerns to mid-management store policies affecting $millions in profits to top-management
policies and M&A activities with $billions at stake. No doubt, some aspects of human behavior in the N =
1,000 remains normally distributed. But as we include workers at each higher level of the hierarchy,
things change. As we add levels, the dollar-value of good and bad decisions increases: some effects
increase exponentially; some multiplicatively, and some may show interactivity and positive feedback
effects. Some of these skew distributions may combine to further assure Pareto-distribution effects.
What about timing? In small stores, decisions are pretty much the same from one year to the next. But
at higher levels of Wal-Mart there are “routine” years at all levels and then some years where significant
M&A, supplier re-alignment, or other decisions are made. We could sample across five, even ten years,
and miss the extreme outlier decisions such as buying the UK store chain that appears to be a mistake. If
we study Wal-Mart people at the store-floor level, in one year, N = 1,000 will be normally distributed. If
we study people at all levels across five years we might see a shift from normal to rank/frequency Pareto.
But we may miss key extreme outliers; Wal-Mart doesn’t make the really big decisions on a regular basis.
To the extent our study includes people in larger and larger within-Wal-Mart networks and supply-chain
networks, involves multiple levels, and covers more years, all of the research issues embedded in
rank/frequency research become more likely. It all depends on how much scalability is involved.
The Error Term
As noted above, once we plot Pareto-distributed x and y on log scales, our expectation is a straight
line. In empirical research, Greene (2002, pp. 7–8) observes that the clarity of a predicted relationship is
21
clouded by the normal distribution of error disturbances—mostly, but not always, due to measurement
error. These effects may be due to unknown or uncontrolled variables and/or measurement error. How
does this bear on Pareto-based research?
Two points follow from the foregoing analysis. First, it is clear that in a Pareto rank/frequency world
both the predictor function and error term are influenced by outliers that are fundamentally important to
the validity of the analysis, as opposed to what are typically viewed as “throwaway” outliers in Gaussian
statistics. In the latter, collecting a large sample almost inevitably means that the presumed validity of the
analysis is improved—because the outliers have less effect, with statistical significance more easily
obtained because of the narrower confidence intervals. In a Pareto world this is not true:
1. The analysis is faulty, if not totally meaningless, if the sampling of outliers is insufficient. In a Pareto world,
building up the sample size while ignoring the outliers actually reduces the validity of the statistical analysis —
it’s like designing buildings based on average quakes while ignoring big ones.
2. Even though confidence intervals are widened, the power of the variance is in the long tails—meaning that, if the
outliers are properly sampled, the impact of the increased variance stemming from the tails more than
compensates for the widening of the confidence intervals. For example, a Pareto-distributed independent variable
may be a strong predictor of a Pareto-distributed dependent variable while leaving the error term i.i.d.
(independent, identically distributed, as statisticians prefer); statistical significance is still relevant.
3. Correlations between Pareto and “normally” distributed phenomena are problematic; this needs further study.
Second, the concept of error rests on the “signal plus noise” paradigm, introduced by Wiener (1949) as
part of cybernetics. This long-standing paradigm is based on the assumptions that: (1) a true measure of
the signal exists as a deterministic function; (2) noise is random and its emergence is due to the system-
environment coupling (and also because of measuring errors); and (3) the relationship between noise and
signal is usually additive (Kennedy 2003, p. 8). As noise is assumed to contain no relevant information
about the system, filtering signal from noise is necessary to reconstruct the system’s dynamical response.
In complex systems we have to rethink the signal-noise paradigm. The response of a complex system
is a mix of order and disorder, represented in mathematical terms by deterministic and chaotic functions.
Schroeder (1991) separates nonlinear “noise” into four colors, white (random), pink (deterministic
chaotic), brown and black (Paretian extremes).
13
In a complex system, chaotic fluctuations may reflect the
22
fractal dimension of a system and its scaling properties. Consequently “chaos” can be a fundamental part
of the signal and may convey relevant information about its dynamics.
14
If this is true, then the basis for
the distinction between signal (independent variables that are usually assumed deterministic and
predictable) and noise (chaotic) becomes blurred (West 2006) and, consequently, the separability between
signal and error term is called into question. In other words, if the signal is characterized by chaotic
fluctuations that exhibit long-term correlations (as is usually the case for Paretian functions), and the
separability between signal and noise cannot be based on the presence or absence of noise, then it is to be
expected that the error term shows long-term correlations and “…long-term memory that ties events
together” (West 2006, p. 271) and it, therefore, is not Gaussian. This implies that statistical methods
based on finite variance (i.e., classical regression models) may not be applicable when dealing with
Paretian functions.
SOME METHODOLOGICAL IMPLICATIONS
An SF theory approach in research starts from non-prejudicial views of the environment. Most
conventional research depends on analytical functions and usually assumes linearity. Additional
assumptions, again often implicit, concern evolutionary gradualism and equilibrium, with motion toward
equilibrium considered adaptive in stable niches. Alternatively, research should start with a discussion
whether the phenomena under consideration show weak or strong interdependence among data-points. If
the former, then assume independence and the validity of calculus-based analytical functions. If the latter,
then it is more likely the world is Paretian. As fractals are continuous but infinitely irregular, and
therefore not amenable to differential calculus, the use of analytical functions becomes problematic
(Mandelbrot & Hudson 2004). The little known mathematical fields of fractional calculus and Lévy-based
statistics are more useful (West 2006). Thus new measures are needed. We focus on these next.
Develop appropriate measures of the variables relevant to SF theories; test for SF dynamics:
1. Start with our new null hypothesis. Determine whether a distribution is likely to be subjected to
multiple dynamics, some of which may be Paretian, others Gaussian.
15
Two main questions here:
a) Are the data-points independent or interdependent? and
23
b) Are the data-points additive or multiplicative or interactive and scalable?
2. If the answer is interdependent-multiplicative-interactive, then test whether interdependence increases
going from small to large events. If yes, lognormal distributions likely could show Paretian tails. Then:
a) Don’t exclude outliers. Even Pareto distributions may have inconsistent outliers (due to idiosyncratic causes);
Sornette (2003) calls these “kings” or “black swans.”
b) Look for power law signatures and identify the relevant parameters of the distributions.
c) Is it a rank/frequency Pareto distribution?
3. See if fractal structures exist:
a) Study nestedness and self-similarity so as to establish fractal dimension(s);
b) When looking at spatial, time independent phenomena (or time-dependent phenomena generated by
distributed structures), look at the underlying generating network(s). Identify nodes and links and analyze the
distribution of links to nodes. Calculate power law slopes.
c) Is the rank/frequency Pareto distribution well-formed, as indicated by a power law distribution? It may signal
emergence or not.
d) For time series, determine fractal dimensions D: D = 2 – α, where α is the scaling index of time series; α
indicates the mix of ordered and random dynamics in the series (see Figure 2). α = 0.5 indicates a completely
random time series—there are no underlying patterns of order; it is purely random walk. α = 0 or α = 1 indicate
a completely ordered phenomenon—no randomness anywhere. The interesting case occurs when α 0 or α
1; here the data show a mix of ordered and random dynamics that build from previous fluctuations; the closer
the value is to either end or to the middle, the more dominant the relevant dynamic is: order or randomness.
16
Emergence is most apt to occur as α goes below .25 or above .75. In Stanley et al. (1996) it is α = ~.16.
e) For instance, Stanley et al. (1996) give an example of a hierarchical “Fordist” type organization where the
CEO can order an increase in production, causing a Markov chain along the hierarchical levels—each
subsequent action-step at time t is a replica of action at step t – 1. If it is carried out exactly from top to
bottom of the firm, then the organization is strongly interdependent (α = 0 for total top-down control, where
α is the exponent of the power law describing growth variance), which means that the variance in growth rate
is directly proportional to size. But lower-level managers and employees rarely follow orders exactly. If all
ignore CEO orders, i.e., all act independently, then α = ½. Usually employees follow orders with some
probability and stickiness. Thus, for a 0 < α < ½ or so (based on Stanley et al.), we expect a power law effect
to obtain. Note that 0 < α < ½ could be due to a CEO’s order implemented with some probability or it could
be due to an emergent self-organizing process by employees.
>>>Insert Figure 2 about here<<<
Given measures, then consider the following:
1. Develop theories and hypotheses based on SF theory that are aimed at causes or consequences of
24
extreme events. The tools and measures mentioned above help identify the nature of the phenomenon
and the appropriate SF theory (or mix of). See Table 2 for a list of SF theories.
2. Carry out empirical studies using data at frequently occurring scales—i.e., the hundreds or thousands
of smaller events at lower-level scales comparable to the thousands of smaller quakes. Test whether
these kinds of studies identify causes and consequences of larger extremes at, say, the next higher
scale(s). That is, can we predict emerging fractal structures one level up in scale?
3. Since “extreme” extremes are rare in the real world, take a lesson from the econophysicists and use
computational models to simulate known empirical findings and then test whether they stretch toward
the more infrequent “extreme” extremes in the artificial computational world.
4. Work backwards from existing extreme events described in the organizational or managerial literature.
We have already seen these sorts of studies carried out by official investigations of what led up to the
Challenger and Pioneer disasters, the Bay of Pigs confrontation, Enron, 9/11, and so on. These
findings then can be “reversed” and further tested by tracing backwards from extreme to smaller-scale
employee networks and behaviors via computational modeling.
5.
Use extreme event statistics (Baum & McKelvey 2006) to calculate how extreme a future event might
be. If a power law tail is evident, one can do this simply by looking down the sloping line.
CONCLUSION
We suggest that fractals, rank/frequency Pareto distributions, power laws, and underlying scale-free
theories will help organization scientists deal with Gell-Mann’s “deep simplicity” (1988) and scalability
(2002) explanations of living systems in general, and organizational complexity more specifically. We
demonstrate that power laws are an inextricable aspect of how individuals, organizations, economies and
societies work. To answer the call for causal explanation relevant to organizations, we assemble a list of
fifteen scale-free theories and detail how several apply to organizations and management. These theories
correct two key shortcomings of Gaussian research. First, they signify Pareto distributions and extreme
events as elements of the managerial world that need to be held in account by quantitative researchers,
and second, they put positive-feedback and other scale-free dynamics at the center of analysis.
25
Abbott’s claim that the General Linear Model “subtly shaped sociologists’” thinking (2001, p. 7) (and
the thinking of other disciplines such as economics, management theory, OB, etc) may be at the base of
the growing ineffectiveness between theory and practice. The gap between multiparadigmatic “science”
appearing in journals and practitioner needs (Ghoshal 2005, Van de Ven & Johnson 2006) signifies the
fact that the proliferation of academic disciplines has not produced research useful to practitioners
(McKelvey 2006, McKelvey & Benbya 2007). Several environmental reasons may lie behind this reality:
the ICT revolution, globalization, and radical transformations in Asia, for example, have contributed to
the dazzling acceleration of change. These changes have increased global and local network connectivity
making actors, from individuals to nation states) more interdependent and therefore more exposed to
positive feedback dynamics and consequent rank/frequency distributions (Andriani & McKelvey 2007a).
Unfortunately, theories and methodological tools have not evolved at the same rate and are mostly still
rooted in the time-honored concepts of equilibrium and linearity. In reality, organizational researchers
study an interconnected world full of rank/frequency discontinuities, chaotic dynamics, fractals, Pareto
extremes, and power laws with inappropriate research tools. The consequence is the gap between theory
and practice that some theorists and many practitioners lament. In particular, theories and tools relying on
“averages” and limited variance pledge allegiance to the altar of tradition—they force researchers to
assume homogeneity instead of heterogeneity and averages instead of rank/frequency extremes.
The impact on use of statistics is significant. Researchers should start from the assumption that
phenomena are rooted in interdependent dynamics and that long tails are the effect of scalable causal
dynamics. Means and variance are unstable and cannot be used to represent the phenomenon, unless
independence is demonstrated. We show that predictor and error terms acquire new meaning. We also
show that complexity offers researchers some tools to characterize the mix of order and randomness in the
systems and we give examples about how research could be done in a Paretian world. More specifically:
1. Data about the trillions of mosquitoes or millions of Ma&Pa stores in one tail don’t offer much useful
information about the elephants or Wal-Mart in the opposite tail;
2. Methods that work on the large numbers in the Ma&Pa tail don’t apply to studying extremes like Wal-Mart,
Microsoft, Enron, or the organizational-response behaviors to disasters like Katrina, Pioneer, and Challenger;
26
3. Which is to say, large samples at the mode don’t speak to any other part of the distribution;
4. Studies of normal distributions at the median or mean don’t speak to either tail;
5. Distributions of, and in, firms may not become “normal” just by increasing sample size:
6. Data collection working hard to include all Paretian outliers needs to replace approaches that delete outliers on
the assumption that they are all errors and anomalies;
7. Scalable causes, dynamics, and theories become more important; they are absent from standard econometrics
textbooks and current statistical practices in general;
8. Scalability-relevant methods simply don’t exist in existing research approaches or in management theorizing.
The field of Paretian science, extreme event theory, and complexity is relatively young. From the first
Pareto distribution in Pareto’s 1897 publication, Pareto rank/frequency and then power laws and scale-
free theories have appeared in many instances. However, in comparison with the three centuries of
development of the Newtonian/Gaussian world, Paretian science is far from paradigmatic. There is no
accepted standard for high-quality research; limits of predictability are unknown; tools, frameworks, and
methods are scarcely developed; the “line in the sand” that defines the spheres of influence of Gaussian
and Paretian approaches needs clearer demarcation and new epistemological rules of justification logic.
Scale-free theories offer the promise of explaining extreme events and reducing the fragmenting effect
of social science disciplines on organizational research. Discipline-centric researchers may dislike this
consequence; discipline-neutral researchers will see research advantages and practitioner relevance. But
remember: The average of the rank/frequencies from mosquitoes to elephants, from Ma&Pa to Wal-Mart
retail firms, from small aerospace-oriented foundries to Boeing and Airbus, or small computer repair
stores to Microsoft offers little useful information to any other part of a Pareto distribution. As Brunk says
(2002, p. 36):
Instead of the bulk of the data being produced by one process and the “outliers” by another, all events—both
minuscule and the historically monumental—are produced by the same process in a SOC environment.
While normal distributions call for more standardized management, the long unique tails of rank-
frequency Pareto distributions call for more unique managerial responses. We argue that managers live in
a world of mostly Paretian organizational and economic rank/frequency phenomena, and that the fat/long
tail and chaotic properties of Pareto distributions have to become more evident in organizational empirical
27
research. For some portion of organizational research, the use of the so-called “robustness” enhancement
techniques described in standard econometric textbooks is dysfunctional. Instead of being deleted,
extreme events have to be properly sampled and analyzed. Given current quantitative practices, this is,
indeed, a call for significant change. It is time to change.
ENDNOTES
1
The so-called “butterfly effect” stems Lorenz’s 1972 paper: “Does the flap of a butterfly’s wings in Brazil set off a
tornado in Texas?” These are Holland’s (2002) “tiny initiating events” that scale up to extreme outcomes.
2
Though a power law exponent is constant in a particular function, its exponent may change for different settings,
industries, times, etc. Stanley et al. (1996) find slightly different scaling coefficients across a large sample of firms
for sales, assets, number of employees, etc. Newman (2005) also shows different scaling coefficients.
3
In general the exponent b is a multiple of ±¼.
4
“Besides ensuring that Citigroup has a proper handle on risk, Pandit's other challenge will be to streamline operations.
Over the years, Citigroup has strapped together a vast array of businesses across its five business segments. The bank is
now looking to improve efficiency and reduce overlap.” (quoted from the Morningstar stock analyst report on
Citigroup: http://news.morningstar.com/). In New York, for example, all the acquired business remained in their
original, separate buildings—there was acquisition without integration, i.e., M&A without the “M”.
5
JSIC is the Japanese counterpart to the SIC code in the U.S. Ishikawa studies all 14 Japanese 2-digit industry
classifications, which in the paper he refers to as “job categories.”
6
Just take a look at the market volatility chart in Ghysels, Santa-Clara, and Valkanov (2005) and count the number
of times the red line (volatilities) goes above the black line, which represents the moving average (GARCH) line.
7
Quoted in West and Deering, 1995: 83
8
Other robustness techniques (not based on least square estimation) to deal with datasets that deviate from idealized
assumptions can be found in Rousseeeuw and Leroy (1987). In general, these techniques are not based on normal
distribution and CLT but use instead the t-distribution. They assume finite variance and, like other robustness
techniques, they have developed highly sophisticated tools (trimming, “winsorizing” to deal with outliers, skewness
and long-tailed distributions tend to cut the tails start from trimming, winsorizing, etc.
9
From Wikipedia (accessed Oct. 14, 2007).
10
Wal-Mart is now the largest company in the world; has revenue over $280 billion; serves 138 million shoppers per
year in ~5300 stores (Bianco 2007).
11
“Wal-Mart wields its power for just one purpose: to bring the lowest possible prices to its customers. At Wal-
Mart, that goal is never reached. The retailer has a clear policy for suppliers: On basic products that don't change,
the price Wal-Mart will pay, and will charge shoppers, must drop year after year. But what almost no one outside the
world of Wal-Mart and its 21,000 suppliers knows is the high cost of those low prices. Wal-Mart has the power to
squeeze profit-killing concessions from vendors. To survive in the face of its pricing demands, makers of everything
28
from bras to bicycles to blue jeans have had to lay off employees and close U.S. plants in favor of outsourcing
products from overseas” (Fishman, 2003).
12
Wal-Mart’s lobbying expenses up by 60% in 2007 (See Sarkar 2008).
13
Following West (2006, p. 79) we define chaos as the “kind of randomness…which is generated by the nonlinear
dynamical property of a system.” Chaos can be divided into deterministic chaos, colored and white noise, defined as
follows: white, truly random, is characterized by a power spectrum whose exponent β = 0 (or frequency
independent); colored noise is divided in anti-persistent or mean-reverting [pink; deterministic chaos-based, anti-
persistent; known as 1/f or power spectra with exponent ~f
-1
) and persistent (brown (f
-2
) and black (persistent
reoccurrence of extreme events; f
-β
with β > 2)] (Schroeder 1991). Colored noise and deterministic chaos can be
also characterized by their dimensionality and pattern/path predictability (Dooley & Van de Ven 1999).
14
The origin of chaos in complex systems’ behavior is not always due to the system-environment coupling—
although environmental interactions may contribute to it (Haken 1983), but is often endogenous. Two consequences
follow: First, chaos characterizes consistent dynamics of the system and, therefore, cannot be discarded as noise.
Second, since chaos and noise are both nonlinear, separating them is problematic, though Dooley and Van de Ven
(1999) and Baum and Silverman (2001) start down this path empirically. In Paretian systems a new type of
mathematics and statistics is needed (West & Deering 1995, West 2006).
15
Note that the Gaussian distribution belongs to a broader class of heavy-tailed distributions, the so-called Lévy
stable distributions (West & Deering 1995). Lévy distributions need not be symmetric; they follow a generalized
form of the law of large numbers. Lévy distributions are characterized by four parameters α, β, μ, c, where α (0-2) is
the exponent, β represents the skewness, μ a scaling factor and c a shift factor. For 0<α<2, we get the family of
heavy-tailed distributions which includes the Cauchy, Pareto, etc., most of which show no finite means and variance.
α=2 yields the Gaussian distribution, which is a particular case of a much larger statistical-distribution family. One
referee of this paper notes that since the Gaussian distribution corresponds to a narrow region of the general class,
decisions presuming the generality of Gaussian distributions risk being “brittle.
16
More specifically: 0 < α < 0.5 indicates anti-persistence (see Figure 3). The system “remembers” a fluctuation
and reacts with the opposite. Head is more likely to be followed by tail, a long stride by a shorter one, exploration by
exploitation, or centralization by decentralization, etc.; 0.5 < α < 1 indicates persistence. In finance draw-downs and
draw-ups (Sornette 2003) are repeated sudden changes of stock market values that follow each other. For instance,
the 1987 crash was really three financial crashes repeating (30.7% cumulative loss) in a short time period.
REFERENCES
Abbott, A. 2001. Time Matters: On Theory and Method. University of Chicago Press, Chicago, IL.
Abrams, D. M., S. H. Strogatz. 2003. Modelling the dynamics of language death. Nature 424: 900.
Adamic, L. A., B. A. Huberman. 2000. The nature of markets in the world wide web. Quart. J. Electronic
Commerce 1 5–12.
29
Aiello, W., F. Chung, L. Lu. 2000. A random graph model for massive graphs. Proceedings of the 32nd Annual
ACM Symposium on Theory of Computing, Portland, OR, 171–180.
Alderson, D.L. 2008. Catching the “network science” bug: insight and opportunity for the operations researcher.
Operations Research (forthcoming – acceptance date January 21 2008)
Anderson, C. 2006. The Long Tail. Random House Business Books, London.
Anderson, P.W. 1972. More is different. Science 177 393–396.
Andriani, P. 2003. The emergence of self-organisation in social systems: The case of the geographic industrial
clusters. Unpublished PhD dissertation, University of Durham, Durham, UK.
———, B. McKelvey. 2007a. Beyond Gaussian averages: Redirecting organization science toward extreme events
and power laws. J. Internat. Bus. Studies 38 1212–1230.
———, B. McKelvey. 2007b. Extremes & scale-free dynamics in organization science: Some theory, research,
statistics, and power law implications. Presented at Acad. Mgmt. Conf., Philadelphia, PA, Aug. 6.
Arthur, W.B. 1994. Increasing Returns and Path Dependence in the Economy. U. Michigan Press, Ann Arbor, MI.
Aoyama, H., Y. Fujiwara, W. Souma, Y. Ikeda, H. Iyetomi. Forthcoming. Pareto Corporations: Statistical Life and
Death in Complex Business Networks. Kyoto University, Kyoto, Japan [originally in Japanese].
Auerbach, F. 1913. Das Gesetz der Bevolkerungskoncentration. Petermanns Geographische Mitteilungen 59 74–76.
Axtell, R.L. 2001. Zipf distribution of U.S. firm sizes. Science 293 1818–1820.
———. 2008. Nonexistence of a typical firm in the U.S. economy: Extremely heavy tails in firm size and growth.
Presented at Org. Sci. Winter Conf., The Resort at Squaw Creek, CA, Feb. 8.
Bache, D.H. 2004. From rupture to structure: The physical character of flocs. Water Science and Tech. 50 55–62.
Bak, P. 1996. How Nature Works: The Science of Self-Organized Criticality. Copernicus, New York.
Barabási, A.-L. 2002. Linked: The New Science of Networks. Perseus: Cambridge, MA.
———. 2005. The origin of bursts and heavy tails in human dynamics. Nature 435 207–211.
———, E. Bonabeau. 2003. Scale-free networks. Sci. Amer. 288(May) 60–69.
Bartels, R. 1977. On the use of limit theorem arguments in economic statistics. Amer. Statistician 31 85–87.
Baskin, K. 2005. Complexity, stories and knowing. Emergence: Complexity & Organ. 7 32–40.
Battiston, S., M. Catanzaro. 2004. Statistical properties of corporate board and director networks. Euro. Phys. J. B –
Cond. Matt. 38 345–352.
Batty, M., R. Carvalho, A. Hudson-Smith, R. Milton, D. Smith, P. Steadman 2008. Scaling and allometry in the
building geometries of Greater London. European Physical Journal B 63 303–314.
Baum, J.A.C., B.S. Silverman. 1999. Complexity in the dynamics of organizational founding and failure. M.R.
Lissack, H.P. Gunz, eds. Managing Complexity in Organizations. Quorum Books, Westport, CT, 292–312.
———, B. McKelvey. 2006. Analysis of extremes in management studies. D.J. Ketchen, D.D. Bergh, eds., Res.
Method. in Strat. and Mgmt., Vol. 3. Elsevier, Oxford, UK, 123–197.
Beinhocker, E. 2007. The Origin of Wealth: Evolution, Complexity, and the Radical Remaking of Economics.
Harvard Business School Press, Cambridge, MA
Bénard, H. 1901. Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en
30
régime permanent. Ann. Chim. Phys. 23 62144.
Bentley, R.A., M.E. Madsen, P. Ormerod 2008. Physical space and long-tail markets. Physica A 388 691–696.
Bianco, A. 2007. Wal-Mart: The Bully of Bentonville. Doubleday Business, New York.
Bettencourt, L.M.A., J. Lobo, D. Strumsky. 2007. Invention in the city: Increasing returns to scale in metropolitan
patenting. Research Policy. 36 107–120
Boisot, M., B. McKelvey. 2007. Extreme events, power laws, and adaptation: Towards an econophysics of
organization. Best Paper Award. Acad. Mgmt. Conf., Philadelphia, PA, Aug. 7.
Braha, D., Y. Bar-Yam. 2007. The statistical mechanics of complex product development: Empirical and analytical
results. Mgmt. Sci. 53 1127–1145.
Brock, W.A. 2000. Some Santa Fe scenery. D. Colander, ed. The Complexity Vision and the Teaching of Economics.
Edward Elgar, Cheltenham, UK, 29–49.
Brunk, G.G. 2002. Why are so many important events unpredictable? Self-organized criticality as the “Engine of
History.” Japanese J. Poli. Sci. 3 25–44.
Buchanan, M. 2000. Ubiquity: The Science of History…or Why the World Is Simpler Than We Think. Weidenfeld &
Nicolson, London.
———. 2004. Power laws & the new science of complexity management. Strategy + Business 34 1–8.
Buldyrev, S.V, N.V. Dokholyan, S. Erramilli, M. Hong, J.Y. Kim, G. Malescio, H.E. Stanley. 2003 Hierarchy in
social organization. Physica A 330 653–659.
Bykoski, V. 2003. Was Zipf law—Now: Life is about hierarchies
http://necsi.org:8100/Lists/complex-science/Message/5478.html
Carneiro, R.L. 1987. The evolution of complexity in human societies and its mathematical expression. Internat. J.
Comparative Soc. 28 111–128.
Castellano, C., M. Marsili, A. Vespignani. 2000. Nonequilibrium phase transition in a model for social influence,
Phys. Rev Lett. 85 3536–3539
Casti, J.L. 1994. Complexification. Abacus, London.
Cederman, L.E. 2003. Modeling the size of wars: From billiard balls to sandpiles. Amer. Poli. Sci. Rev. 97 135–150.
Chmiel, A.M., J. Sienkiewicz, K. Suchecki, J.A. Holyst 2007. Networks of companies and branches in Poland.
Physica A 383 134–138.
Clauset, A., C.R. Shalizi, M.E.J. Newman. 2007. Power-law distributions in empirical data.
http://arxiv.org/abs/0706.1062
Cook, W., P. Ormerod, E. Cooper. 2004. Scaling behaviour in the number of criminal acts committed by
individuals. J. Stat. Mech.: Theory & Exper. http://www.iop.org/EJ/abstract/1742-5468/2004/07/P07003/JSTAT
Cuddington, K., P. Yodzis 2002. Predator-prey dynamics and movement in fractal environments. Amer. Naturalist
160 119–134.
Dahui, W., L. Menghui, D. Zengru. 2005. True reason for Zipf’s law in language. Phys. A 358 545–550.
———, Z. Li, D. Zengru. 2006. Bipartite produce-consumer networks and the size distribution of firms. Phys. A 363
359–366.
31
Delaney, P.F., L.M. Reder, J.J. Staszewski, F.E. Ritter. 1998 . The strategy-specific nature of improvement: the
power law applies by strategy within task. Psych. Sci. 9 1–7.
DeSolla Price, D.J. 1965. Networks of scientific papers. Science 149 510–515.
De Vany, A. 2003. Hollywood Economics. Routledge, New York.
Dezsı, Z., E. Almaas, A. Lukács, B. Rácz, I. Szakadát, A.-L. Barabási. 2005. Fifteen minutes of fame: The
dynamics of information access on the web. http://mokk.bme.hu/archive/15fame_pr2005
Diatlov, V. 2005. Information technology in practice of organising: Meeting fragmentation and interdependence by
the incremental political delivery of information systems in financial services. Unpublished PhD dissertation,
School of Management, University of Southampton, Southampton, UK.
Dodds, P.S., D.J. Watts, C.F. Sabel. 2003. Information exchange and the robustness of organizational networks.
Proc. Nat. Acad. of Sci. 100 12516–12521.
Dooley, K.J., A.H. Van de Ven. 1999. Explaining complex organizational dynamics. Organ. Sci. 10 358–372.
Dorogovtsev, S.N., J.F.F. Mendes. 2003. Evolution of Networks. Oxford University Press, Oxford, UK.
Dumé, B. 2005. Global terrorism follows a power law. Physicsweb, Feb. 1–2.
http://physicsweb.org/articles/news/9/2/6/1
Fishman, C. 2003. The Wal-Mart you don’t know. FastCompany, 73:
http://www.fastcompany.com/magazine/77/walmart.html
Frigg, R. 2003. Self-organised criticality—what it is and what it isn’t. Stud. History Philo. Sci. 34 613–632.
Fujiwara, Y., 2004. Zipf law in firms bankruptcy. Physica A 337 219–230.
Gaffeo, E., M. Gallegati, A. Palestrini. 2003. On the size distribution of firms: Additional evidence from the G7
countries. Physica A 324 117–123.
Gay, B., B. Dousset 2005. Innovation and network structural dynamics: ?Study of the alliance network of a major
sector of the biotechnology industry. Research Policy 34 1457–1475.
Gell-Mann, M. 1988. The concept of the Institute. D. Pines, ed. Emerging Synthesis in Science. Addison-Wesley,
Boston, MA, 1–15.
———. 2002. What is complexity? A.Q. Curzio, M. Fortis, eds. Complexity and Industrial Clusters. Physica-
Verlag, Heidelberg, Germany, 13–24.
Ghoshal, S. 2005. Bad management theories are destroying good management practices. Acad. Mgmt. Learn & Ed. 4
75–91.
Ghysels, E., P. Santa-Clara, R. Valkanov. 2005. There is a risk-return trade-off after all. J. Financial Econ. 76 509–548.
Gladwell, M. 2000. The Tipping Point: How Little Things Can Make a Big Difference. Little, Brown, Boston, MA.
———. 2006. Million-dollar Murray. The New Yorker, Feb. 13, 82(1) 96.
Glaser, P. 2009. Fitness and inequality in an increasing returns world. Working paper, UCLAAnderson School of
Management, Los Angeles, CA.
Goldstein, J.A. 1994. The Unshackled Organization. Productivity Press, Portland, OR.
Greene, W.H. 2002. Econometric Analysis, 5th ed. Prentice-Hall, Englewood Cliffs, NJ.
Gunz, H., B.B. Lichtenstein, R. Long. 2001. Self-organization in career systems: A view from complexity science.
M@n@gement 5(1) 63–88.
32
Gutenberg, B., R.F. Richter. 1944. Frequency of earthquakes in California. Bull. Seismo. Soc. of America 34 185–188.
Hackett, A.P. 1967. 70 Years of Best Sellers, 1895–1965. Bowker, New York.
Haire, M. 1959. Biological models and empirical histories of the growth of organizations. M. Haire, ed. Modern
Organization Theory. Wiley, New York, 272–306.
Haken, H. 1983. Synergetics, An Introduction, 3rd ed. Springer-Verlag, Berlin.
———. 2004. Synergetics: Introduction and Advanced Topics. Springer-Verlag, Berlin.
Hamilton, M.J., B.T. Milne, R.S. Walker, O. Burger, J.H. Brown 2007. The complex structure of hunter-gatherer
social networks. Proc. Royal Soc. London B 274 2195–2202.
Hegyi, G., Z. Néda, M.A. Santos 2007. Wealth distribution and Pareto’s law in the Hungarian medieval society.
Physica A 380 271–277.
Huesch, M.D. 2009. Power-law distributions in cardiac surgeon average volume? Working paper, Fuqua School of
Business, Duke University, SC.
Holland, J.H. 1988. The global economy as an adaptive system. P.W. Anderson, K.J. Arrow, D. Pines, eds. The
Economy as an Evolving Complex System, Vol. 5. Addison-Wesley, Reading, MA, 117–124.
———. 1995. Hidden Order. Addison Wesley: Reading, MA.
———. 2002. Complex adaptive systems and spontaneous emergence. A.Q. Curzio, M. Fortis, eds. Complexity and
industrial clusters. Physica-Verlag, Heidelberg, Germany, 24–34.
Iansiti, M., R. Levien. 2004. Strategy as ecology. Harvard Bus. Rev. 82(3) 68–78.
Ishikawa, A. 2006. Pareto index induced from the scale of companies. Physica A 363 367–376.
Johnson, G.A. 1982. Organizational structure and scalar stress. C. Renfrew, M. Rowlands, B.A. Segraves-Whallon,
eds. Theory and explanation in archaeology, Academic Press, New York.
Jones, C. 2005. The shape of production functions and the direction of technical change. Quart. J. of Economics
120 517–549.
Kauffman, S.A. 1993. The Origins of Order. Oxford University Press, Oxford, UK.
Kennedy, P. 2003. A guide to econometrics (5th ed.). MIT Press, Boston, MA
Kolmogorov, Á.N. 1941, Uher das logarithmisch Normale Verteilungsgesetz der Dimensionen der Teilchen hei
Zerstuckelung. Commits Rendus (DoLlady) de L'Academie des Sciences de l'URSS 31 99–101.
Kraitchik, M. 1942. Mathematical Recreations. Norton, New York.
Krugman, P. 1996. The Self-Organizing Economy. Blackwell, Malden, MA.
Lee, Y., L.A.N. Amaral, D. Canning, M. Meyer, H.E. Stanley. 1998. Universal features in the growth dynamics of
complex organizations. Physical Rev. Lett. 81 3275–3278.
Legara, E.F., C. Monterola, D.C. Juanico, M. Litong-Palima, C. Saloma 2008. Earning potential in multilevel
marketing enterprises. Physiza A 387 4889–4895.
Levy, S., G. Donhowe. 1962. Explanations of a biological model of industrial organization. J. Business 35 335–342.
Levy, M., S. Solomon. 1997. New evidence for the power-law distribution of wealth. Physica A 242 90–94.
Lewin, R. 1992. Complexity: Life at the Edge of Chaos. University of Chicago Press, Chicago, IL.
Liljeros, F., C.R. Edling, L.A.N. Amaral, H.E. Stanley, Y. Aberg. 2001. The web of human sexual contacts. Nature
411 907–908.
33
Lorenz, E.N. 1972. Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? Paper
presented at the 1972 meeting of the American Association for the Advancement of Science. Washington, DC.
Lotka, A.J. 1926. The frequency distribution of scientific productivity. J. Washington Acad. Sci. 16 317–323.
Maguire, S., B. McKelvey, L. Mirabeau, N. Oztas. 2006. Complexity science and organizational studies. S.R. Clegg,
C. Hardy, T. Lawrence, W. Nord, eds. Handbook of Organizational Studies, 2nd ed. Sage, Thousand Oaks, CA,
165–214.
Mainzer, K. 1994. Thinking in Complexity. Springer-Verlag, New York. [5th ed. published in 2007.]
Malevergne, Y., D. Sornette 2006. Extreme Financial Risks: From Dependence to Risk Management. Springer-
Verlag, Berlin.
Mandelbrot, B.B. 1963a. The variation of certain speculative prices. J. Bus. 36 394–419.
———. 1963b. New methods in statistical economics. J. Political Econ. 71 421–440.
———. 1982. The Fractal Geometry of Nature. Freeman, New York.
———, R.L. Hudson. 2004. The (Mis)Behavior of Markets. Profile, London.
Mantegna, R.N., H.E. Stanley. 2000. An Introduction to Econophysics. Cambridge Univ. Press, Cambridge, UK.
McKelvey, B. 2006. Comment on Van De Ven and Johnson’s “engaged scholarship”: Nice try, but…. Acad. Mgmt.
Rev. 31 822–829.
———, H. Benbya. 2007. Using Pareto-based science to enhance knowledge for practical relevance. Paper
presented at the Third Annual Organization Studies Summer Workshop, Rethymno, Crete, Greece, June 79.
———, B.B. Lichtenstein, P. Andriani. Forthcoming. When organizations and ecosystems interact: Toward a law of
requisite fractality in firms. M.J.L Moreno, ed. Chaos and Complexity in Organizations and Society. UNESA,
Madrid, Spain.
Montroll, E.W., W.W. Badger. 1974. Introduction to Quantitative Aspects of Social Phenomena. Gordon and
Breach, New York.
Moore, H.N. 2008. Vikram Pandit, Citigroup’s chess master. Online Financial News, 8(May).
http://www.efinancialnews.com/homepage/content/2450595389
Moss, S. 2002. Policy analysis from first principles. Proc. Nat. Acad. Sci. 99 (Suppl. 3) 7267–7274.
Nagel, K., M. Paczuski. 1995. Emergent traffic jams. Physics Rev. E 51 2909–2918
Newell, A., P.S. Rosenbloom. 1981. Mechanisms of skill acquisition and the law of practice. J.R. Anderson, ed.
Cognitive skills and their acquisition. Erlbaum: Hillsdale, NJ, 1–55.
Newman, M.E.J. 2001. The structure of scientific collaboration networks. Proc. Nat. Acad. Sciences 98 404–409.
———. 2005. Power laws, Pareto distributions and Zipf’s law. Contem. Phys. 46 323–351.
Nicolis, G., I. Prigogine. 1989. Exploring Complexity: An Introduction. Freeman, New York.
Odling-Smee, F.J., K.N. Laland, M.W. Feldman. 2003. Niche Construction. Princeton U. Press: Princeton, NJ.
Ormerod, P., C. Mounfield. 2001. Power law distribution of the duration and magnitude of recessions in capitalist
economies breakdown of scaling. Physica A 293 573–582.
———., B. Rosewell, R. Colbaugh, K. Glass. 2005. Knowledge and cognition in the social sciences: Modeling with
low cognition agents. Presented at the Conf. on Human Complex Systems, May 19, Lake Arrowhead, CA.
34
Pareto, V. 1897. Cours d'Economie Politique. Rouge & Cie, Paris.
Pevzner, P., G. Tesler. 2003. Transforming men into mice. W. Miller et al., eds. Proceedings of the 7
th
Annual
International conference on Research in Computational Molecular Biology ACM Press, New York, 247–256.
Park, J. 2008. Riding the wave: A multi-level model of mergers and acquisitions wave activity. Working paper. Katz
Graduate School of Business, University of Pittsburgh, Pittsburgh, PA.
Patriarca, M., A. Chakraborti, G. Germano 2006. Influence of saving propensity on the power-law tail of the wealth
distribution. Physica A 369 723–736.
Pimm, S.L. 1982. Food Webs. University of Chicago Press, Chicago, IL. [2nd ed. 2002.]
Plerou, B., L.S.N Amaral, P. Gopikrishnan, M. Meyer, H.E. Stanley. 1999. Similarities between the growth
dynamics of university research and of competitive economic activities. Nature 400 433–437.
Podobnik, B., D. Fu, T. Jagric, I. Grosse, H.E. Stanley. 2006. Fractionally integrated process for transition
economics. Physica A 362 465–470.
Poole, M.S., A.H. Van de Ven, K. Dooley, M.E. Holmes. 2000. Organizational Change and Innovation Processes:
Theory and Methods for Research. Oxford University Press, New York.
Powell, W.W., D.R. White, K.W. Koput, J. Owen-Smith. 2005. Network dynamics and field evolution. Amer. J.
Sociology 110 1132–1205
Preston, F.W. 1948. The commonness, and rarity, of species. Ecology 29 254–283.
Prigogine, I. 1955. An Introduction to Thermodynamics of Irreversible Processes. Thomas, Springfield, IL.
———, (with I. Stengers). 1997. The End of Certainty. Free Press, New York.
Redelico, F.O., A.N. Proto, M. Ausloos 2008. Power law for the duration of recession and prosperity in Latin
American countries. Physica A 397 6330–6336.
Rousseeuw, P.J., A.M. Leroy. 1987. Robust Regression and Outlier Detection. Wiley, New York
Saito, Y.U., T. Watanabe, M. Iwamura 2007. Do larger firms have more interfirm relationships? Physica A 383 158–
163.
Sarkar, D. 2008. Wal-Mart lobbying up 60 percent in 2007. Associated Press, March 7
th
.
Scheinkman, J., M. Woodford. 1994. Self-organized criticality and economic fluctuations. AER 84 417–421.
Schieve, W., Allen, P.M. 1981. Self-organization and Dissipative Structures. U. Texas Press, Austin, TX.
Schmid, L. 2004. Are living beings in a state of self-organized criticality? A new interpretation of data on work
incapacity due to low back pain. Medical Hypotheses 62 458–461.
Schroeder, M. 1991. Fractals, Chaos, Power Laws. Freeman, New York.
Simon, H.A. 1962. The architecture of complexity. Proc. Amer. Philos. Soc. 106 467–482.
Sims, D.W. et al. 2008).Scaling laws of marine predator search behaviour. Nature 451 1098–1103.
Solé, R.V., D. Alonso, J. Bascompte, S.C. Manrubia. 2001. On the fractal nature of ecological and
macroevolutionary dynamics. Fractals 9 1–16.
Solomon, S., P. Richmond. 2001. Power laws of wealth, market order volumes and market returns. Physica A 299
188–197.
Song, D.-M., Z.-Q. Jiang, W.-X. Zhou 2009. Statistical properties of world investment networks. Physica A 388
2450–2460.
35
Sornette, D. 2000. Critical Phenomena in Natural Sciences. Springer-Verlag, Heidelberg, Germany.
———.2003. Why Stock Markets Crash? Princeton University Press: Princeton, NJ.
———, F. Deschâtres, T. Gilbert, Y. Ageon. 2004. Endogenous versus exogenous shocks in complex networks.
Phys. Rev. Lett. 93 228701-1–4.
Souma, W., H. Aoyama, Y. Fujiwara, Y., Ikeda, H. Iyetomi, T. Kaizoji 2006. Correlation in business networks.
Physica A 370 151–155.
Stacey, R.D. 1992. Managing Chaos. Kogan Page, London.
Stanley, M.H.R., L.A.N. Amaral, S.V. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M.A. Salinger, H.E. Stanley.
1996. Scaling behaviour in the growth of companies. Nature 379 804–806.
Stanley, M.H.R., L.A.N. Amaral, P. Gopikrishnan, V. Plerou 2000. Scale invariance and universality of economic
fluctuations. Physica A 283 31–41.
Stephan, G.E. 1983. A research note on deriving the square-cube law of formal organizations from the theory of
time-minimization. Social Forces 61 847–854.
The Economist 2007. The nimble sumo. August 4, p. 57.
Turner, J.S. 2000. The Extended Organism. Harvard University Press, Cambridge, MA.
Van de Ven, A.H., P.E. Johnson. 2006. Knowledge for theory and practice. Acad. Mgmt. Rev. 31 809–821.
Watts, D. 2003. Six Degrees: The Science of a Connected Age. Norton, New York.
Warren, K., G. Craciun, D. Anderson-Butcher. 2005. Everybody else is: Networks, power laws and peer contagion
in the aggressive recess behavior of elementary school boys. Nonlinear Dyn., Psych. Life Sci. 9 155–173.
West, B.J. 2006. Where Medicine Went Wrong. World Scientific, Singapore.
———. 2007. The average person is truly exceptional: Where medicine went wrong. 17
th
Annual Conference for the
Society for Chaos Theory in Psychology and Life Sciences, Chapman University, Anaheim, CA, August 28.
———, B. Deering. 1995. The Lure of Modern Science: Fractal Thinking. World Scientific, Singapore.
West, G.B., J.H. Brown, B.J. Enquist. 1997. A general model for the origin of allometric scaling laws in biology.
Science 276(4) 122–126.
———, J.H. Brown. 2004. Life's Universal Scaling Laws. Phys. Today 57(9): 36–42.
Wiener, N. 1949. Time Series. MIT Press, Cambridge, MA.
Whitfield, J. 2006. In the Beat of a Heart: Life, Energy and the Unity of Nature. National Acad. Press, New York.
Wycisk, C., B. McKelvey, M. Hülsmann 2008. “Smart parts” logistics systems as complex adaptive
systems. Int. J. Phys. Distrib. Logistics Mgmt. 38 102–125.
Zanette, D.H., S.C. Manrubia. 2001. Vertical transmission of culture and the distribution of family names. Physica A
295 1–8.
Zanini, M. 2008. Using ‘power curves’ to assess industry dynamics. The McKinsey Quarterly, November.
Zhang, J., Q. Chen, Y. Wang 2009. Zipf distribution in top Chinese firms and an economic explanation. Physica A
388 2020–2024.
Zipf, G.K. 1949. Human Behavior and the Principle of Least Effort. Hafner, New York
36
Figure 1: Gaussian vs. Pareto Distributions
Bell Curve
Power Law
Linear axes Log axes
Power Law
Bell
Curve
Figure 2: Persistence and Anti-persistence Behavior in Time Series
Order
Randomness
Order
0
0.5
1
Anti-persistence region
(like followed with increasing
probability by unlike)
0 = oscillatory pattern
Persistence region
(like followed with increasing
probability by like)
1 = ballistic trajectory)
37
Table 1: Some Examples of Social and Organizational Power Law Phenomena
1. Size of nations by population (Buldyrev et al. 2003)
2. Fractal structure of hunter/gatherer social networks
(Hamilton et al. 2007).
3. Hierarchy of social group size (Zhou et al. 2005)
4. Economic fluctuations (Scheinkman & Woodford 1994)
5. Growth rate of countries GDP (Lee et al. 1998)
6. Duration of recessions (Ormerod & Mounfield 2001)
7. Recessions & prosperity in Latin America (Redelico et al.
2009)
8. Transition economies (Podobnik et al. 2006)
9. Distribution of wealth (Pareto 1897; Levy & Solomon 1997)
10. Financial crashes (Sornette 2003)
11. Casualties in war (Cederman 2003)
12. Political complexity in communities (Carneiro 1987)
13. Size of cities (Zipf 1949)
14. Area, height, volume, size of buildings (Batty et al. 2008)
15. Costs of homeless in cities (Gladwell 2006)
16. Number of religious adherents (Clauset et al. 2007)
17. Price movements on exchanges (Mandelbrot & Hudson ’04)
18. Scientific discoveries (Plerou et al. 1999)
19. Copies of books sold (Hackett 1967)
20. Cascades in book sales (Sornette et. al 2004)
21. Sales of fast moving consumer goods (Moss 2002)
22. Movie profits (De Vany 2003)
23. Market share distribution of UK retail outlets (Moss 2002)
24. Cotton prices (Mandelbrot 1963)
25. Blockbuster drugs (Buchanan 2004)
26. Wealth dist. of investors (Solomon & Richmond 2001)
27. Saving effects on wealth distribution (Patriarca et al. 2006)
28. Medieval wealth distribution (Hegyi et al. 2007)
29. Job vacancies (Gunz et al. 2001)
30. Changing language (Dahui et al. 2005)
31. Deaths of languages (Abrams & Strogatz 2003)
32. Social networks (Watts 2003)
33. Sexual networks (Liljeros et al. 2001)
34. Social influence (Castellano et. al. 2000)
35. Co-authorships (Newman 2001)
36. Publications and citations (Lotka 1926; deSolla Price 1965)
37. Actor networks (Barabási & Bonabeau 2003)
38. Scale-free business networks (Souma et al. 2006)
39. Number of inventions in cities (Bettencourt et al. 2005)
40. Traffic jams (Nagel & Paczuski 1995)
41. Frequency of family names (Zanette & Manrubia 2001)
39. Global terrorism events (Dumé 2005)
40. Revenues of top 500 Chinese firms (Zhang et al. 2009)
41. Learning rates in heart surgery (Huesch 2008)
42. Firm size (Axtell 2001)
43. Firm size, interfirm relationships (Saito et al. 2007)
44. Growth rate: Japanese SIC industries (Ishikawa 2006)
45. Growth rates by sales; internal structure of firms (Stanley et
al. 1996)
46. Growth rates: universities, countries (Stanley et al. (2000)
47. Economic effects of zero-rational agents (Ormerod et al. ‘05)
48. Delinquency rates (Cook et al. 2004)
49. Aggressive behavior among boys (Warren et al. 2005)
50. Supply chains (Scheinkman & Woodford 1994)
51. Complex product development (Braha & Bar-yam 2007)
52. Bankruptcy of firms (Fujiwara 2004)
53. Robustness in org. networks (Dodds et al. 2003)
54. Learning strategy (Delaney et al. 1998)
55. Cognitive skills: “power law of practice” (Newell &
Rosenbloom 1981)
56. # of phone calls,emails (Aiello et al. 2000; Ebel et al. 2002)
57. Website hits per day (Adamic & Huberman 2000)
58. News website visitation decay (Dezsı et al. 2005)
59. “Fordist” power (Diatlov 2005)
60. Alliance networks among biotech firms (Barabási &
Bonabeau 2003, p. 207)
61. Branch networks of Polish firms (Chmiel et al. 2007)
62. Worldwide investment networks (Song et al. 2009)
63. Antibody alliances in biotech (Gay & Dousset 2005)
64. “Power curves” in U.S. industries (Zanini 2008)
65. Entrepreneurship/innovation (Poole et al. 2000)
66. Italian industrial districts (Andriani 2003a)
67. M&A waves (Park 2008)
68. Director interlock structure (Battiston & Catanzaro 2004)
69. Microsoft’s ecosystem (Data from Iansiti & Levien 2004;
analysis by Colon Drayton)
70. Market capitalization in industries (Glaser 2009)
71. Earnings, multilevel marketing by firms (Legara et al. 2008)
72. Biotech networks (Powell et al. 2005)
73. Growth of firms (Lee et al. 1998)
74. Productivity of innovation (Jones 2005)
75. Work incapacity from back pain (Schmid 2004)
76. Intra-firm decision events (Diatlov 2005)
77. Type of political officers, size of community (Johnson 1982)
78. Decision-making and queuing (Barabási 2005)
79. Physical space, long-tail analysis (Bentley et al. 2008)
80. Japanese (J.) income*
81. J. income tax 1887-2003
82. J. firms’ sales
83. J. firms’ profit
84. J. company income
85. J. iron/steel fabrication sector
86. J. electrical machinery sector
87. J. wholesale sector
88. J. steel, other metals sector
89. J. general machinery sector
90. J. chemical, petroleum prod. sector
91. J. retail trade sector
92. France: size by total assets
93. France: size by sales in France
94. UK: Size by total assets
95. UK: size by number of employees
96. Italy: size by total assets
97. Italy: size by sales in Italy
98. Italy: size by number of employees
* The source of the power law distributions from 64–82 is H. Aoyama et al. (Forthcoming). Similar distributions have been found in
many other industrialized countries—see for instance Gaffeo, Gallegati, Palestrini (2003).
38
Table 2: Empirical Basis of Scale-free Causes of Power Laws*
Rules
Explanation
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 (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-Random walk
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).
3-Hierarchical
modularity
Growth unit connectivity: As cell fission occurs by the square, connectivity increases by nn–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).
4-Event bursts
Activity prioritization: Individuals show bursts of communication, entertainment, and work activities
followed by long delays, as opposed to random Poisson) distribution (Barabási 2005).
5-Combination
theory
# of exponentials; complexity: Multiple exponential or lognormal distributions or increased complexity of
components subtasks, processes) sets up, which results in a power law distribution (Mandelbrot 1963, West
& Deering 1995, Newman 2005).
6-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).
7-Interacting
fractals
Food web; firm & industry size: The fractal structure of a species is based on the food web S. Pimm quoted
in Lewin 1992, p. 121), which is a function of the fractal structure of predators and niche resources (Preston
1948, Pimm 1982, Solé et al. 2001, West 2006).
8-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; now found to apply to language, firms, and economies
in transition (Zipf 1949, Ishikawa 2006, Podobnik et al. 2006).
9-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
(Barabási 2002, Newman 2005).
10-Spontaneous
order creation
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 1993, Holland 1995).
11-Irregularity
generated
gradients
Coral growth; blockages: Starting with a random, insignificant irregularity, coupled with positive feedback,
the initial irregularity starts an autocatalytic process driven by emergent energy gradients, which results in
the emergence of a niche. This explains the growth of coral reefs, innovation systems (Turner 2000, Odling-
Smee et al. 2003).
12-Phase
transition
Turbulent flows: Exogenous energy impositions cause autocatalytic, interaction effects and percolation
transitions at a specific energy level—the 1
st
critical value—such that new interaction groupings form with a
Pareto distribution (Prigogine 1955, Nicolis & Prigogine 1989).
13-Contagion
bursts
Epidemics; idea contagion: Often, viruses are spread exponentially—each person coughs upon two others
and the network expands geometrically. But, changing rates of contagious flow of viruses, , stories, and
metaphors, because of changing settings such as almost empty or very crowded rooms and airplanes, result in
bursts of contagion or spreading via increased interactions; these avalanches result in the power-law
signature (Watts 2003, Baskin 2005) due to the small-world structures of the transient underlying networks.
14-Self-
organized
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 adaptive stasis by preservative behaviors—
such as sand avalanches, forest fires, changing heartbeat rate, species adaptation—which vary in size of
effect according to a power law (Bak 1996).
15-Niche
proliferation
Markets; when production, distribution, and search become cheap and easily available, markets develop a
long tail of proliferating niches containing fewer and fewer customers; they become Paretian with mass-
market products at one end and a long tail of niches of the other (Anderson 2006).
* Additional power law causes are mentioned in West and Deering (1995), Sornette (2000), and (Newman 2005).
... Como resultado, eles se tornam não somente mais ricos em recursos e detêm mais prestígio, como também eles ocupam posições mais centrais nas redes (Barabási, 2016). Essa dinâmica "do rico se tornando cada vez mais rico" (Merton, 1968, p. 610) explica a origem dos hubs e dos atores periféricos nas redes (Andriani & McKelvey, 2009;Kurt & Kurt, 2020). ...
... Poucos atores atraem mais relacionamentos, resultando em redes que se distanciam da distribuição gaussiana, mas se aproximam da distribuição de lei de potência. Essa distribuição possui médias instáveis, variância infinita, e caudas gordas devido aos hubs, ou seja, atores altamente conectados (Andriani & McKelvey, 2009). Essa rede é também conhecida como Centro-Periferia (Sun & Liu, 2016;Kurt & Kurt, 2020). ...
... Isso porque a evolução da rede não se adapta à distribuição Poisson, mas à distribuição de lei de potência (Clauset et al., 2009). Poucos atores situados na cauda da distribuição possuem uma maior probabilidade de afetar a evolução da rede (Andriani & McKelvey, 2009). ...
Article
Full-text available
RESUMO Este artigo analisa a evolução na rede na internacionalização da firma enfatizando o mecanismo generativo ligação preferencial. Ligação preferencial significa que os atores mais conectados tendem a formar um maior número de relacionamentos. O artigo utiliza dados quantitativos longitudinais de um caso único, representado pela internacionalização de uma divisão operacional de uma firma multinacional norte-americana no mercado brasileiro. A análise de dados é baseada no algoritmo computacional de Clauset et al. (2009) e no PAFit, um novo método estatístico. O objetivo é identificar até que ponto a evolução da rede segue uma distribuição de lei de potência e o grau em que a ligação preferencial afeta a evolução da rede. Os resultados indicam que a evolução da rede na internacionalização da firma segue uma distribuição de lei de potência. Tal evolução é afetada pela forma sublinear de ligação preferencial. Poucos atores acumulam desproporcionalmente um elevado número de relacionamentos. A ligação preferencial não se manifesta homogeneamente na evolução da rede. Ela tem um efeito mais forte no início da rede. Este artigo contribui ao avançar uma abordagem relacional e baseada em processo da internacionalização da firma. Ele mostra que a evolução da rede na internacionalização da firma indica uma rede maior e mais esparsa ao longo do tempo. Os atores mais conectados formam hubs, sinalizando mais status, poder e recursos.
... As a result, they become not only more resourceful and prestigious, but they also occupy central positions in the networks (Barabási, 2016). This "rich getting forever rich" dynamic (Merton, 1968, p. 610) explains the origins of hubs and peripheral actors in networks (Andriani & McKelvey, 2009;Kurt & Kurt, 2020). ...
... Few actors attract more relationships, resulting in networks that depart from the Gaussian distribution but approximate the power-law distribution. This distribution has unstable means, infinite variance, and displays fat tails caused by hubs, e.g., highly connected actors (Andriani & McKelvey, 2009). This network has also been called Core-Periphery (Sun & Liu, 2016;Kurt & Kurt, 2020). ...
... That is why the network evolution does not fit a Poisson distribution but a power-law distribution (Clauset et al., 2009). Few actors situated in the tail of the distribution have a higher likelihood of affecting the network evolution (Andriani & McKelvey, 2009). ...
Article
Full-text available
This paper analyses the network evolution of the internationalizing firm, focusing on a generative mechanism called preferential attachment. Preferential attachment means that more connected actors are likely to form more relationships. This paper uses longitudinal quantitative data of a single case of the internationalization of an American multinational firm's operational division in the Brazilian market. The data analysis is based on Clauset et al.'s (2009) computational algorithm and PAFit, a new statistical method. The aim is to identify the extent to which the network evolution follows a power-law distribution and the degree to which preferential attachment affects the network evolution. It finds that the network evolution of the internationalizing firm follows a power-law distribution. It is affected by a sub-linear form of preferential attachment. Few actors accumulate a disproportionally high number of relationships. The preferential attachment does not homogeneously manifest itself in the network evolution. It has a strong effect on the network onset. This paper contributes by advancing a relational, process-based approach to the internationalization of the firm. It shows that the network evolution of the internationalizing firm grows over time and becomes sparser. More connected actors form hubs, meaning increased status, more power and resources.
... As a key cross-level characteristic of the relationship between a unit and the broader organization to which it belongs, this choice is strategically important because it reflects the nature and intensity of coupling within an organization (Weick, 1976). Organizational autonomy is a scale-free concept (Andriani and McKelvey, 2009): a unit-parent relationship can exist at the multiple levels of analysis that define a unit and include it within organizational boundaries at the next level up (i.e., parent); examples include teams within a functional department, functional departments within a division, and divisions or subsidiaries within a firm. Yet as Wiedner and Mantere (2019: 3) summarized, despite being a foundational construct, "organizational autonomy is rarely explicitly defined in management and organization theory." ...
Article
Through a 21-year longitudinal study of the relationship between Italian supercar manufacturer Automobili Lamborghini and its parent, German carmaker Audi AG, we examine how a unit’s degree of organizational autonomy is renegotiated over long periods of time. Using detailed empirical data, we develop a process model of the dynamics of organizational autonomy in a unit‒parent relationship. This process model shows an ongoing dialectical tension between parent managers’ autonomy-reduction efforts and unit managers’ autonomy-extension efforts, and it reveals oscillations in the unit managers’ discretion over resource-orchestration decisions. Driving this dialectic are parent managers’ appraisal respect for the unit, their search for firm-wide strategic integration, and unit managers’ organizational identity and concern for distinctiveness. Our process model captures concurrent feedback loops that endogenously produce these oscillations between lower and higher autonomy. We then conceptualize a harmonic domain in the unit–parent relationship, in which these oscillations persist without deviating toward amalgamation or separation. Finally, we develop a theory of change in autonomy by identifying a theoretical link between resource orchestration and specific dimensions of organizational identity. Our study highlights the dialectical, dynamic, and ongoing nature of organizational autonomy.
... While the power law has been considered a suitable distribution in many fields including online communities (Johnson, Faraj & Kudaravalli, 2014) and organizations (Andriani & McKelvey, 2009), recent studies in statistics challenge its apparent pervasiveness (Clauset, Shalizi & Newman, 2009;Broido & Clauset, 2019). According to these studies, power law distributions are complicated to detect because fluctuations occur in the tail of the distribution, and because of the difficulty of identifying the range over which power law behavior holds. ...
Article
Full-text available
Peer production online communities are groups of people that collaboratively engage in the building of common resources such as wikis and open source projects. In such communities, participation is highly unequal: few people concentrate the majority of the workload, while the rest provide irregular and sporadic contributions. The distribution of participation is typically characterized as a power law distribution. However, recent statistical studies on empirical data have challenged the power law dominance in other domains. This work critically examines the assumption that the distribution of participation in wikis follows such distribution. We use statistical tools to analyse over 6,000 wikis from Wikia/Fandom, the largest wiki repository. We study the empirical distribution of each wiki comparing it with different well-known skewed distributions. The results show that the power law performs poorly, surpassed by three others with a more moderated heavy-tail behavior. In particular, the truncated power law is superior to all competing distributions, or superior to some and as good as the rest, in 99.3% of the cases. These findings have implications that can inform a better modeling of participation in peer production, and help to produce more accurate predictions of the tail behavior, which represents the activity and frequency of the core contributors. Thus, we propose to consider the truncated power law as the distribution to characterize participation distribution in wiki communities. Furthermore, the truncated power law parameters provide a meaningful interpretation to characterize the community in terms of the frequency of participation of occasional contributors and how unequal are the group of core contributors. Finally, we found a relationship between the parameters and the productivity of the community and its size. These results open research venues for the characterization of communities in wikis and in online peer production.
Article
Rare events and other nonerror outliers (such as the COVID‐19 pandemic) are important phenomena in global strategy contexts. Despite their salience, however, they have hardly been studied systematically in our field (or organizational research at large). We suggest that this is due to a dominance of the Gaussian paradigm, which (often unrealistically) assumes linearity and independence of observations. Moreover, case‐based qualitative studies which offer contextualization have been underrepresented. We thus call on researchers to abolish the practice of habitually discarding outliers, reflect on nonnormal distributions, and pursue more qualitative studies. Journal editors and reviewers should widen their assumptions regarding “acceptable” papers and reflect on the requirement of contributing to big “T” theories. Finally, PhD training should juxtapose fundamental paradigms and associated implications for epistemological choices. Extreme occurrences, such as organizational crises, recessions, or pandemics, are challenges most practitioners deal with and worry about. Understanding their determinants, characteristics, and dynamics allows for heightened vigilance, preparedness, and ultimately performance. Yet, much of global strategy research (and organizational research at large) has focused on “average” phenomena, based on methodologies that assume bell‐shaped distributions and independent observations. In this note, we argue that this is not a realistic way to think about most social phenomena. In fact, most are characterized by their high degree of interdependence among elements, as well as a relative commonness of “rare” events and outliers. As a result of embracing the reality of nonnormality, scholars will be able to offer more relevant guidance to practitioners.
Article
While industrial marketing often comprises a process that, at least in principle, mirrors Bayesian reasoning, the notion of Bayesian inference has predominantly been utilized in the marketing field as a methodological tool. This article suggests that the practice of industrial marketing itself should be (re)conceptualized as a Bayesian process of belief-updating that entails a continuous cognitive cycle of formulation of hypotheses (i.e., beliefs about the market) and the subsequent updating of those hypotheses through exposure to market evidence (e.g., data from the market). A Bayesian perspective on industrial marketing enables a synthesis of a broad body of extant research as well as a focus on the interconnection between executives' market beliefs (theories-in-use) and belief-updating (assessing the validity of those beliefs in view of market evidence). A view of industrial marketing as a Bayesian process not only enhances our understanding in general but also fosters insights into market learning in uncertain and volatile situations. A Bayesian conceptualization suggests a new understanding of industrial marketing that also informs a typology of marketing approaches. We outline opportunities for developing a better understanding of the Bayesian foundation of industrial marketing.
Chapter
Colombia’s national science, technology, and innovation (STI) system planned resources to include more people in higher education systems (HES) to create the human capital needed to foster socio-economic development. Two of the main concerns and central indicators in HES are the inclusion of undergraduate students and PhDs. However, in several countries, the supply of PhDs seems to exceed the demand in universities despite institutional pressures exerted by accreditations and rankings to increase the number of PhDs into their faculties. Additionally, in Colombia, the number of freshmen has decreased in recent years. This imbalance leads to a governance problem in adjusting the inclusion of human capital in HES, which is further compounded by regional inequalities. This chapter proposes a model to classify and forecast the inclusion of students and PhDs within a contingency theory framework to contribute to resource planning according to environmental conditions and to allow better adjustments of governance efforts in its STI system. Using a sample of 257 Colombian universities, the model measures the changes in the probability distribution curve between PhDs and students adjusted according to their territorial conditions. The results show that changes in inclusion depend on the development level of the regions. The main conclusion is that there exists a group of regional universities with greater capacity for growth in the inclusion of students and PhDs that can contribute more to the development in their territories.
Chapter
Full-text available
Two Caribbean Hope Spots were declared as bottom-up protection scheme based on communities governance to improve coral reef conservation effectiveness in Colombia. Data acquisition through a citizen science program with the participation of scuba diving operators, fishers and other volunteer observers since 2008 received 1958 geo-referenced reports of observations and semi-structured interviews with videos and photos. Applying the transdisciplinary conceptual dimensions of the Sustainable Development Goal 14—Protection of Life below Water—was measure the performance of ecological, social and economic indicators illustrated with three cases studies: (1) Spatial Collaborative Multicriteria Decision Support System to evaluate the National Park Corales del Rosario per dive sites Varadero Reef had higher performance indicators under all the perspectives. (2) Opportunity Cost of Dredging vs. Protecting Varadero Coral Reef; the minimal benefit of protection was US$244 million/Km² with an expected Annual Revenue>10%. (3) Change in Fisheries Productivity in Hope Spot Capurganá Reefs shows a decline with industrial fisheries and after the appearance of the lionfish Pterois volitans of commercial important native stocks up to 2013; afterwards a gradual replenishment was due to exclusive artisanal fisheries and biological control of the invasive species. Compared with the Park, coral reefs in the Hope Spots have higher healthy coral cover>70% and more frequent sightings of vulnerable species listed on the IUCN Red List (7–10 per). They also have higher fishers’ community stewardship, effective bottom-up schemes of protection and less visitors in contrast with the Park. More ecological complex food webs of resilient coral reefs in the Hope Spots with open seawater currents at the bottom, mixed at the surface during the rainy season with run-off producing a natural shadow that avoids impact from climate change, and low sedimentation rates make them potential climate change refugia. These results point out the urgency to connect the case studies within an effective protection framework, increasing governance to guarantee ecological connectivity.
Article
The COVID-19 pandemic has highlighted the salience of different supply networks merging together to provide critical products in short supply. The automotive supply network and the medical device supply network, for instance, became intertwined to meet rising demands for ventilators. As such, coming to the fore is a search mechanism, based on recombination of multiple components, conceptualized as jury-rigging behavior. Designing and changing the interdependencies across firms/supply chains become relevant. Drawing on complex adaptive systems and Ashby's law of requisite variety, we use a computational model to examine these mechanisms’ isolated and combined effect on supply chain network resiliency. Our results show statistically significant difference in the adaptiveness and mortality rate between a jury-rigged supply chain network and a non-jury-rigged one. In particular, the effect of jury-rigging is diminished by the level of coupling among the supply chains.
Article
Full-text available
The boards of directors of the largest corporations of a country together with the directors form a dense bipartite network. The board network consists of boards connected through common directors. The director network is obtained taking the directors as nodes, and a membership in the same board as a link. These networks are involved in the decision making processes relevant to the macro-economy of a country. We present an extensive and comparative analysis of the statistical properties of the board network and the director network for the first 1000 US corporations ranked by revenue (Fortune 1000) in the year 1999 and for the corporations of the Italian Stock Market. We find several common statistical properties across the data sets, despite the fact that they refer to different years and countries. This suggests an underlying universal formation mechanism which is not captured in a satisfactory way by the existent network models. In particular we find that all the considered networks are Small Worlds, assortative, highly clustered and dominated by a giant component. Several other properties are examined. The presence of a lobby in a board, a feature relevant to decision making dynamics, turns out to be a macroscopic phenomenon in all the data sets.
Chapter
In many disciplines of science we deal with systems composed of many subsystems. A few examples, mainly taken from topics in this book, are listed in Fig. 1. Very often the properties of the large system can not be explained by a mear random superposition of actions of the subsystems. Quite on the contrary the subsystems behave in a well organized manner, so that the total system is in an ordered state or shows actions which one might even call purposeful. Furthermore one often observes more or less abrupt changes between disorder and order or transitions between different states of order. Thus the question arises, who are the mysterious demons who tell the subsystems in which way to behave so to create order, or, in a more scientific language, which are the principles by which order is created.
Chapter
This paper introduces concepts associated with complex adaptive systems (cas), linking those concepts at some points to economic planning. The paper begins (section 1) with an informal description of the notion of a cas and then (section 2) discusses the critical role of "building blocks" in understanding cas. Using these ideas, the paper goes on (section 3) to discuss the phenomenon of "emergence", wherein the whole of the system's behaviour goes beyond the simple sum of the behaviours of its parts. The body of the paper (sections 4 through 6) looks at the role of modelling in predicting the behaviour of cas, examining the kinds of model that will serve this purpose. The paper concludes (section 7) with a brief discussion of the relevance of these ideas to economic planning.