Introduction

What wrote “To be, or not to be: that is the question?” The brain that put words in Hamlet’s mouth consisted of roughly 100 billion neurons, the same number as in your or my brain. But when did we write our last sonnets? This introduction to complex networks is about how the whole sometimes is greater (and sometimes is less) than the sum of its parts. The essence of network science is that the whole is the sum of its parts plus the interactions between them.

This introduction is derived from the first chapters of my Ph.D. thesis, which I defended in September 2006. Read the full story at http://www.diva-portal.org/umu/abstract.xsql?dbid=840.

Why networks?

Complex networks are neither magnetic or charged, nor do they emit light. They are not made of any exotic matter, they are not radioactive and you will never hear: Complex networks offer the world a secure energy supply! So why am I, as a physicist, interested in modeling complex networks?

Networks make it possible to characterize the complex systems of our world, in the same way as a map describes the surrounding landscape. A network is a map of interactions, for example of communication in a social system. As Homo loquens, the speaking man (Fry1977), communication is fundamental in our society. If we want to understand the complexity in different systems in the world, we must first understand their basic interaction patterns. These networks are often neither regular lattices, nor are all units connected randomly—the interaction patterns are complex.

Complex, because a complex system is made up of a large number of components, or agents, interacting in such a way that their collective behavior is not a simple combination of their individual behavior (Newman2002b). Craig Reynolds has expressed it strikingly as “A flock is not a big bird” (Waldrop1992). Moreover, the interactions in a complex system often go beyond the system itself to make it adaptive to an ever changing environment. On the other hand, a simple system is a system with a limited set of interacting units with a behavior that can be described by laws, for example a pendulum or a bouncing ball (Amaral and Ottino2004). Further, a complicated system is a system with a large set of components, each with exact roles that also together act under absolute laws, for example billiards or cars.

When I started by asking what instead of who wrote Hamlet’s famous question, I did not intend to confuse you about who actually wrote Shakespeare’s plays. Instead I wanted to emphasize that he was not isolated. Shakespeare’s network went beyond the 100 billion neurons connected by an even larger number of physical connections inside his brain. Thereby connecting virtually to cultural, historical and social entities like myths, books, and individuals—that in their turn had their connections. Hamlet would probably have used a different vocabulary if Shakespeare’s network had been different.

Cultural, sociological, economic, political, biological, and ecological systems all evolve under the influence of numerous components. The challenge is: how simple can we make the rules of the individuals, constrained by interactions on a network, to reproduce the observed collective behavior in the real world? In particular, how should this network of interactions be constructed?

I now again turn to the analogy with maps and use a passage from Sylvie and Bruno Concluded (Carroll1893).

“What a useful thing a pocket-map is!” I remarked. “That’s another thing we’ve learned from your Nation,” said Mein Herr, “map-making. But we’ve carried it much further than you. What do you consider the largest map that would be really useful?” “About six inches to the mile.” “Only six inches!” exclaimed Mein Herr. “We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!” “Have you used it much?” I enquired. “It has never been spread out, yet,” said Mein Herr: “the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well.”

Specific aims

Obviously, to be useful, a network model must be a simplified version of the pattern of interactions in the complex system it represents, in the same way as a good map is a simplified version of the landscape it represents. The challenge is to create this map of some aspects other than the land that we stand upon. It must be a dynamic map, because neither solar flares nor communicating people are static systems. This calls for a map that is a full model with interacting players over an ever-changing network—a dynamic network model. By characterizing the world in this way we will not only be able to answer specific questions, but also focus on asking general questions. For example, on the way to this dynamic network model, it has been necessary to use, and sometimes develop, structural and statistical measures that are informative not only for a particular network, but for a spectrum of networks from biology to the man-made Internet.

To uncover the interplay between function and structure and the strong connection to the formation process in networks, I have found it necessary to study networks from a perspective of information transfer. In other words, I will treat networks as information machines. With information I here mean any knowledge that has been gathered or received by communication. The connection between information, communication, and networks is central here.

I am in particular interested in how the network structure affects the communication conditions over the network, and vice versa, how the ongoing communication affects the network structure. For example, who communicates with whom and the social structure of a society are strongly integrated. The social network reflects the access to information that different parts of the system experience, and social mobility may be seen as a quest for better information access. Read more about this in my thesis. Here follows an introduction to network science based on the first chapters of the thesis.