Network theory's new math
Some are born connected, others achieve connection, still others have
connectedness thrust upon them. Everyone is networked. Everyone is either a
node or a hub in someone else's network. Much as the quality of life is
influenced by the quality of our networks, our standard of living is
increasingly determined by network standards. To paraphrase Marshall
McLuhan, we shape our networks and then our networks shape us.
The notion of networks as a dominant organizing principle to explain how
the world really works has attracted enormous interdisciplinary interest.
Physicists are talking to mathematicians who are talking to sociologists
and economists who are talking to physicists. In barely a decade, networks
of researchers have sprung up to research networks. Executives are
beginning to turn to these experts for usable insights into the network
dynamics shaping both threats and opportunities in business.
This is no surprise. Transportation networks have striking similarities to
telecommunications networks. The Internet's technological behaviors map
well onto the ecological behaviors of the biosphere. The complex
interconnections between people in research laboratories around the world
can be cost-effectively etched onto the design of silicon chips. Similarly,
the myriad networks that define corporate connectedness are alike.
Economies aren't merely marketplaces; they're networks. Executives need to
understand network forces, not just market forces.
As Albert-László Barabási, author of "Linked: The New Science of Networks"
(Perseus Publishing, 2002), writes, "The diversity of networks in business
and the economy is mind-boggling. There are policy networks, ownership
networks, collaboration networks, organizational networks, network
marketing--you name it. It would be impossible to integrate these diverse
interactions into a single all-encompassing web. Yet no matter what
organizational level we look at, the same robust and universal laws that
govern nature's webs seem to greet us."
These laws of networks may prove as robust and universal as Newton's laws
of motion. But making network laws, which like Newtonian laws are steeped
in mathematics and metaphor, comprehensible to the layperson is hard work.
The New Yorker's Malcolm Gladwell took a successful first cut with his
best-selling "The Tipping Point: How Little Things Can Make a Big
Difference" (Little, Brown and Company, 2000). Three new books published
this year go far beyond tipping points to present to the conceptually
curious reader important theories that reveal the hidden order of complex
networks.
Barabási, the author of "Linked," is a physicist and leading researcher in
the field who uses the Internet as his dominant research medium for
analyzing the peculiar properties of networks. His book is ideal for those
looking for the perspective of a network researcher and practitioner; it's
even spiced with a few equations. Mark Buchanan's "Nexus: Small Worlds and
the Groundbreaking Science of Networks" (W.W. Norton & Company, 2002) is
the product of a physics Ph.D. who writes for the noted scientific journals
Nature and New Scientist. Although Buchanan draws heavily on Barabási's
work, his intellectual focus is the intriguing so-called small-world
networking theories of mathematicians Duncan Watts and Steve Strogatz.
Small-world theories, which are derived from theoretical mathematics and
practical reality, prove that seemingly distant, disconnected, and
disparate populations, events, or actions can be easily linked to one
another. Like many scientists turned writers, Buchanan is a bit of an
ideologue who seems more comfortable discussing network ecologies than
network economics. Then again, because of the transcendent nature of
networks, the distinctions between ecology and economics aren't that great.
The least scientific but perhaps most stimulating work for business readers
among the three is Howard Rheingold's "Smart Mobs: The Next Social
Revolution" (Perseus Publishing, 2002). Rheingold is neither scientist nor
technologist, but he knows how to talk with those who are and extract the
essence of their thinking and concerns. His previous books on virtual
reality, virtual communities, and the history of digital innovation in
Silicon Valley remain cult classics for the digerati. What makes "Smart
Mobs" so intriguing is not Rheingold's ongoing love affair with the
potential of network technology, but his sure grasp of how people play with
that potential.
Network mathematics
Executives interested in the possible impact of network mathematics on
their businesses and industries have a superb analogy from financial
innovation, the Nobel Prize-winning Black-Merton-Scholes option-pricing
equations. The mathematics was as much a machine tool for creating options
as a diagnostic tool for analyzing them. Clever "quants" could use the
equations to spot "hidden options" in financial instruments and wring
profits from them, or, alternatively, use the equations to customize
innovative financial instruments for their clients. Today, an increasing
number of companies use real options as mathematical tools for pricing the
risks associated with their own business investments.
What Black-Merton-Scholes equations have done for financial innovation and
risk, the new network math discussed in "Linked" and "Nexus" will
ultimately do for network innovation. Scientists and innovators will look
for "hidden networks" within complex systems to figure out whether those
networks are being overly relied upon These analyses will transform how
organizations manage their networks to manage value.
or foolishly underexploited. These analyses will transform how
organizations manage their networks to manage value. Indeed, individuals
and institutions may be able to create just-in-time network activity to
increase reliability and exploit opportunity, much as Black-Merton-Scholes
equations empowered innovative traders to create just-in-time trading of
options and derivatives to better hedge or speculate. In manufacturing,
supply chains represent nothing if not an organizational opportunity to
identify hidden networks of risk and reward.
Consider a speculative example from commercial aviation. For decades,
American Airlines committed itself to a hub-and-spoke network topology
where the vast majority of flights fed into a few key airports. The
economics of this network structure worked for a time, but has fallen prey
to, among other things, ruthless competition from lower-cost competitors
like Southwest Airlines and JetBlue Airways. Southwest dismisses American's
hub-and-spoke network approach in favor of its own point-to-point
structure. And yet, as Southwest continues to expand and sees flight
densities increase at key airports such as San Jose, Oakland and Las Vegas,
isn't it possible that the company will have inadvertently--if not
serendipitously--created "virtual hubs" worthy of profitable exploitation.
Barabási, Buchanan and Rheingold would answer with a resounding yes!
According to small-world theory, networks emerge from links that were never
intended to mesh together. So networks aren't just designed; they evolve.
Order and randomness
As described in "Nexus," the ideas underlying Watts and Strogatz's "small
worlds" are simple, powerful and compelling. In effect, Watts and Strogatz
validated the "six degrees of separation" phenomenon, the belief that any
two people on earth are separated by no more than five people connected to
each other in some meaningful way.
Inspired by earlier research on social networks, the two struggled to find
a coherent mathematical way to describe how these networks were connected.
What Watts and Strogatz found was counterintuitive and profound: By
injecting just a few random connections into a complex network, they could
make that network both more efficient and more effective. The right random
links create small worlds from vast complexities. Randomness can
dramatically improve the performance of a complex system rather than
ruining it.
When Watts and Strogatz published a paper on their small-world theories in
Nature in 1998, it "touched off a storm of further work across many fields
of science," Buchanan writes. "A version of their small-world geometry
appears to lie behind the structure of crucial proteins in our bodies, the
food webs of our ecosystems, and even the grammar and structure of the
language we use. It is the architectural secret of the Internet and despite
its apparent simplicity is in all ways a new geometrical and architectural
idea of immense importance."
This finding on randomness has already had a significant impact on the
design of telecommunications networks and silicon chips. Microprocessor
companies like Intel and Motorola now use elements of small-world theory to
link circuits on their semiconductors to make them run faster and more
efficiently. Engineers are now aggressively exploring the role of
randomness in performance enhancement of their products. Purely rational
design that once treated randomness as the enemy has been transformed;
designers now play with randomness as a tool to create "small worlds" that
exploit this power of serendipitous connection. The result is more robust
networks and ever-faster silicon chips. These innovations wouldn't have
occurred without the proofs outlined by Watts and Strogatz.
It's important to remember--and this theme is stressed in each of the
books--that small-world theory findings are the direct result of
interdisciplinary interaction and observation. Empirical observation is
just as important as clever theory. The beauty of the small-world
hypotheses is that they can be tested in the real world very quickly.
Power laws
Random geometries of small worlds is just one network law that commands
respect. While ambitious managers read Machiavelli to better understand the
laws of social and political power, effective executives need to understand
that mathematical "power laws" profoundly shape laws of personal power.
"If you are not a physicist or mathematician, most likely you have never
heard of power laws," asserts Barabási. In Linked, executives will
recognize their importance, because power laws can reveal as much about
marketing and finance as they do about math and physics.
The "power" in power laws is not a function of Machiavellian manipulation
but the "power" found in exponential functions; numbers squared or cubed or
taken to the 10th power, etc. Power laws strike at the heart of what
businesspeople think they understand about playing the odds and managing
risk. Why? Because power laws are the sworn enemy of a basic statistical
concept: the notion that probabilities present themselves in the average
distribution of bell-shaped curves. In a networked world ruled by power
laws, the bell curve is a dangerous lie.
In fact, power laws describe a radically different kind of distribution.
There are no peaks; no symmetries; no bell curve. Power laws look nothing
like traditional school-taught statistics. Yet they do a far better job of
It's still too early to say how the laws of networks will shape tomorrow's
technologies and sociologies. But it's not too soon to argue that more
individuals and institutions will be more inextricably intertwined with
more networks in the future.
reflecting how much of the real world behaves. The distinguishing feature
of a power law, Barabási writes, is that its distribution is wildly skewed:
Numerous tiny events coexist within the few very large ones that actually
matter.
The distribution of individual wealth in the United States is an excellent
example of a power law; a relatively tiny number of people account for the
overwhelming majority of individual net worth. The distribution of American
and European height, however, is not a power law. There are not a few
hundred giants over 1,000 feet tall and millions of pygmies; there's a more
comforting and symmetrical bell curve distribution. Power laws explain why
computing "the average"--the means, medians and modes--for insight is so
frequently a fool's errand.
Power laws are thus crucial to understand because they force us to look at
those few critical hubs--the O'Hares and Heathrows--that dominate either
the creation of network value or its destruction. "If Watts and Strogatz's
discovery of random connections was a first step into the world of
disorderly and complex networks," Buchanan comments, "then the recognition
of hubs and power law patterns for the distribution of links is second."
But recall the Southwest Airlines network evolution question: Precisely
when does a lowly node evolve into a hub? When should small worlds-oriented
sociologists, economists, or mathematicians declare a cluster of nodes a
hub? How can we be sure a network's links and hubs are distributed by power
laws instead of bell curves? When do a few random connections between
networks create more chaos than cost-effectiveness?
The answers to those questions aren't yet known. Networks have laws, all
right, but even laws are subject to interpretation and experimentation. The
true test of the laws in the context of business and economics will come
from the technologies used and abused by Rheingold's "smart mobs."
Reputation marks the spot
Smart mobs are a sociological phenomenon that Rheingold persuasively argues
will become an everyday reality. These aren't the mobs that storm the
Bastille or riot in the streets (although they could); they're small worlds
of individuals linked and melded by technological networks, especially
through mobile communications. Smart mobs don't just mediate information
and analysis; they mediate passion and behavior.
Where "Linked" and "Nexus" describe how networks behave, "Smart Mobs"
simply yet expansively describes how people behave--and misbehave--within
networks. Rheingold is particularly interested in the just-in-time virtual
marketplaces that networks can create on the basis of trust and reputation.
"A field known as 'experimental economics' has extended game theory into
two specific 'minigames': the 'Ultimatum Game' and the 'Public Goods
Game,'" he writes. "Research using these games as probes indicates the
following:
• People tend to exhibit more generosity than a strategy of self-interest
predicts.
• People will penalize cheaters, even at some expense to themselves.
• These tendencies and the emotions that accompany them influence
individuals to behave in ways that benefit the group.
In other words, e-marketplaces are media as much for social interactions as
they are for financial transactions. That is, who you are and what you're
doing are as important as what you want to buy or what you want to sell.
It's no accident that eBay is still around and making money for both itself
and its, ahem, community of auctioneers. Your reputation on eBay can--and
often does--matter far more than what you are attempting to either buy or
sell.
"Reputation marks the spot where technology and cooperation converge,"
Rheingold writes. "The most long-lasting social effects of technology
always go beyond the quantitative efficiency of doing old things more
quickly or more cheaply. The most profoundly transformative potential of
connecting human social proclivities to the efficiency of information
technologies is the chance to do new things together, the potential for
cooperating on scales and in ways never before possible."
And yet, when novel "networks of scale," as Rheingold describes them,
actually emerge, Barabási and Buchanan insist they will be shaped by the
algorithmic imperatives of small-world theory and power laws. People can't
break these laws of networks any more than they can violate Newton's laws
of motion.
However, mathematical laws can be slavishly obeyed or cleverly exploited.
Indeed, as Newton himself once remarked, "To master nature, one must obey
her." Scientific laws can empower even where they seem limiting.
Entrepreneurs and innovators will figure out how to master networks while
obeying their (apparent) laws.
What is Intel without the ideology of Moore's Law? What is the options and
derivatives marketplace without Black, Merton and Scholes? It's still too
early to say how the laws of networks will shape tomorrow's technologies
and sociologies. But it's not too soon to argue that more individuals and
institutions will be more inextricably intertwined with more networks in
the future. So you shouldn't read these books with the expectation of
rewriting business plans or revising capital expenditures. You should use
them to better understand the networks your business has, and to rethink
what they should be. Perhaps, in the process, you may discover more than
one small world among the disconnected parts of your organization and
marketplace.
To read more articles like this one, visit www.strategy-business.com.
Reprinted with permission from strategy+business, a quarterly management
magazine published by Booz Allen Hamilton.
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