Whither? Why?

The contents list gives a fair impression of the coverage attempted. Networks, both deterministic and stochastic, have emerged as objects of intense interest over recent decades. They occur in communication, traffic, computer, manufacturing and operational research contexts, and as models in almost any of the natural, economic and social sciences. Even engineering frame structures can be seen as networks that communicate stress from load to foundation.

We are concerned in this book with the characterisation of networks that are optimal for their purpose. This is a very natural ambition; so many structures in nature have been optimised by long adaptation, a suggestive mechanism in itself. It is an ambition with an inbuilt hurdle, however: one cannot consider optimisation of design without first considering optimisation of function, of the rules by which the network is to be operated. On the other hand, optimisation of function should find a more natural setting if it is coupled with the optimisation of design.

The mention of communication and computer networks raises examples of areas where theory barely keeps breathless pace with advancing technology. That is a degree of topicality we do not attempt, beyond setting up some basic links in the final chapters. It is well recognised that networks of commodity flow, electrical flow, traffic flow and even mechanical frame structures and biological bone structures have unifying features, and Part I is devoted to this important subclass of cases.

In Chapter 1 we consider a rather general model of commodity distribution for which flow is determined by an extremal principle.