Network theory's new math

Experts at Strategy + Business review some of the recent writings on the evolving concept of networks and network standards and how our lives are being shaped by that thinking.

11 min read
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-L?szl? Barab?si, 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.

Barab?si, 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 Barab?si'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. Barab?si, 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 Barab?si. 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, Barab?si 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, Barab?si 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.