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 firms 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 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.