It happens occasionally in human history that a completely new idea appears. Not just any idea, but a significant change in the way we humans think about the world and ourselves. A new paradigm becomes apparent.
Sometimes this paradigm shift is heralded by a single book. Perhaps recognized at the time or perhaps later historians pick out a single volume a the point that changed human thoughts.
One such book was, in my opinion, Euclid’s The Elements. What was new about this book was not the mathematics and the results that Euclid presents; most or all of Euclid’s mathematics was already know when the work was written.
What was new—radically so—was Euclid’s presentation of his material. Starting from five “postulates” and five “common notions,” ten axioms in all, Euclid derives his initial propositions, and further and further propositions are derived from the original axioms and previous propositions, creating a tree of knowledge that is always expanding, never closing back on itself.
In short, Euclid introduces deduction to the scientific world, and from him follows what has been the accepted scientific method for over 2,000 years: “divide and conquer”. Divide a complex problem into smaller problems that can either be solved directly or further divided.
This method has been tremendously successful and is the foundation for essentially all human advances in science, engineering, medicine—any form of scientific knowledge. It brought you the printing press and the internet, penicillin and heart transplants, steam engines and space rockets.
It takes some nerve, then, for Albert-László Barabási to challenge this method, but that is precisely what he does in Linked: The New Science of Networks, a book that may mark a turning point in introducing a new scientific paradigm.
His assertion is that many real-life systems are complex because of their connections, not their parts. What this means is that while you can “divide and conquer” the individual constituents in a system and gain a detailed understanding of them and their behavior, that does not let you understand the system as a whole: the “re-assembly” of the system from its parts is too complex for science.
Think about this for a second. Understanding the parts does not always let you understand the whole. What makes some systems interesting is not so much its constituents but their connections. We have of course always know this at the back of our minds, but we have suppressed this, science’s dirty little secret, because the “divide and conquer” approach has been so phenomenally successful.
Now, however, the systems that we wish to study, from the internet to the progress of scientific thought, from corporate governance to global financial crises, from human social networks to international terrorism, are almost all of a type where the traditional scientific approach introduced by Euclid fails us, where the parts are less important than the connections, and where a new scientific methodology is required.
The book is a description of the author’s discovery of the complexity of networks. Personal and anecdotal in nature, it may not be a comprehensive guide to the field, but what it looses in completeness it gains in readability, and for sheer enthusiasm of the subject and for a feeling of the wonder of discovery it is hard to beat.
While Barabási covers human networks, most of the examples are from the physical world. Interested readers may wish to peruse Smart Mobs by Howard Rheingold. Rheingold observes how a combination of mobile communications and computing, high-bandwidth networks and multimedia capabilities, and location information gives rise to new social structures: ad-hoc groups, networking together and sharing/creating/subscribing to a distributed trust system.
A key observation of Rheingold is that the value of a network obviously depends on the number of nodes in the network but the way in which increases depends on the type of network. Sarnoff’s law is valid for linear networks and says that the value of the network grows with the number of nodes.
Sarnoff’s law emerged from the advent of radio and television networks in the early twentieth century in which a central source … broadcasts to a small number of receivers.
In contrast, Metcalfe’s law is good for networks that allow paired connections, and says that the value of the network grows with the number of possible connections, or the square of the number of nodes.
Finally, Reed’s law observes that when a network allows the users to form groups then the value of the network grows exponentially with the number of nodes, taking into account the many group-group connections that are now possible.
Of course these latter networks that allows groups to form, are in many ways the most interesting. Think of human social behavior. But the value lies in the connections and a new scientific method is required to understand these complex systems.