What do road and railway systems, electrical circuits, mingling at parties, ancient mazes, and the Internet have in common? The answer is that they all involve interconnections – they are what mathematicians call networks. Peter M. Higgins is the author of Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections, which examines the hidden mathematical structures that underpin many real world phenomena. Below is an excerpt from the first chapter of the book, which reveals just how important networks are.
Who discovered networks? The question is almost like asking who discovered drawing—the urge to start doodling pictures of networks is almost overwhelming as soon as we begin thinking about a situation in which there is a multitude of connections. The advantage of the picture is that it allows you to see all the connections at once and we can remind ourselves of any one of them simply by flitting our eyes around the diagram.
Perhaps networks have been underestimated because they are so common, yet at the same time they seem to lack any structure. The mathematical topics that have been studied extensively for thousands of years are numbers and geometry. Numbers are pervasive, they allow us to tally and compare, and have an undeniable natural order. Geometric objects are pretty and visual, providing all manner of symmetries that can strike you before a word is said, so the attraction of geometry is very powerful and immediate. Networks on the other hand are none of these things. Networks are not numbers of any kind, nor are they truly geometrical even though we can draw pictures of them. They represent quite a different realm of mathematics. And not only of mathematics, for everyone appreciates the importance of networking—the real measure of our comprehension of the world is our understanding of how all the various parts come together and affect one another.
Moreover, the use of the word ‘network’ in this context is more than just a metaphor. Some of the most difficult and technically demanding research in the social and political sciences centres on studying the nature of networks of international organizations of all kinds, whether they be legal, cultural, and diplomatic, or scientific, commercial, and sporting. Relatively small nations and organizations can have profound influence on world affairs. Sometimes this can be tracked to their strategic or cultural importance or to dominant individuals. However, substantial and sometimes less visible influence often stems from the way they are placed within relevant networks and how they draw from and feed into these webs.
It is fair to say that the first genuine problem in networks dates to the eighteenth century when the famous Swiss mathematician, Leonhard Euler, showed how to solve the now celebrated riddle of the Bridges of Königsberg by finding a simple general principle that dealt with any question of that kind. But more of that later. This does alert us however to the fact that networks have been studied from the mathematical viewpoint for centuries. None the less, it is striking how their relative importance keeps growing and growing. In part this is due to examples of networks springing up in modern life—we need look no further than the internet to find a massive and important instance of a network that has come into being almost spontaneously. This network pervades most aspects of the modern world and has taken on a life of its own. The internet acts as a vehicle for another network, the World Wide Web. These networks differ in two ways, one physical and the other mathematical. The Web is visible but intangible and floats on top of the internet, which is a physical array consisting of routers and their connections. The Web is also a directed network for there are links directed from one page to another, but not necessarily in the reverse direction. This gives it a very different character from networks in which all connections are mutual and two-way.
It all goes much deeper than that however. Professional mathematicians have tended to have a similar reaction to that of the general public to the underlying idea. The notion of a network of connections is so simple and natural that there looks to be not much to it. To be sure, even in the eighteenth century Euler showed that even a simple example can yield an interesting problem. All the same, it was felt that the depth and interest of the mathematics involved could hardly be on a level comparable with really serious science, such as that which explains how the Earth and the Heavens move. Since the time of Isaac Newton, calculus, the mathematics of change and movement, has been a well-spring of scientific inspiration and was seen as the heir to classical Greek geometry, representing the pinnacle of mathematical practice and sophistication. Indeed Leonhard Euler himself perhaps did more than anyone who has ever lived to develop the methods of Newton, the so called differential and integral calculus. By comparison, problems about networks were regarded as a poor relation, little more than recreational puzzles, fit only for those who could not contribute to the really tough stuff.
Networks, however, spring many surprises. And they truly are surprises because no one would expect objects with virtually no mathematical structure to yield anything of interest. After all, a network is any array of points on a page with lines drawn between some of them in any fashion at all. The idea would seem to be far too general to yield anything that went much beyond the obvious. However, there is a whole world to be explored by those prepared to search and the results have consequences for real networks of people and telephone lines. For instance, at any party that ever there was, or ever will be, or ever could be, there will be two people with the same number of friends at the gathering—this, and many results like this, are unavoidable consequences of the nature of networks, as we shall soon witness.
Part of the trouble has been that mathematics itself has been slow to wake up to what was happening. Problems about networks keep arising irresistibly, even when you are not looking for them. I myself spend a lot of time on my own speciality that is a certain area of algebra. What has happened in my own field has been mirrored elsewhere. Certain intractable problems have arisen and, in the end, progress is only made when they are represented in terms of networks whereupon it transpires that what is holding you up is a question about whether or not certain patterns can or cannot arise in a network. No use sneering—it turns out that nets were really what you have been studying all along.