Fields and Flows on Random Graphs

Random graphs have become objects of increasing interest over the last thirty years. Two sets of physical models which have particularly stimulated this development are those of polymerisation and of percolation. The early polymerisation work is associated especially with the names of Flory, Stockmayer, Gordon and Good. An approach in which one sets up a reversible Markov model of association/dissociation and analyses its equilibrium properties is set out systematically in Whittle (1986). Percolation theory is firmly associated with the name of Hammersley, who initiated and so greatly developed the subject. For reviews see Hammersley and Welsh (1980) and Kesten (1982). Finally, largely in isolation from either of the above movements, pure mathematicians developed an interest in the subject, beginning with Erdős and co-workers, and continued notably by Stepanov and Bollobás. The work of this school is systematically presented in Bollobás (1985). However, there are now new applications developing, which will require fundamental theoretical advances. I think especially of the study of neural networks. These are viewed as random graphs, partly because their size and complexity makes the statistical approach inevitable, and partly also because (just as in communication theory), the further one penetrates into the subject, the more one realises that the statistical approach is ‘right’. The interest of a neural network is that impulses and activity of some kind are propagated around it. One is then motivated to a study which has received only sporadic attention hitherto and of which this article can only be a token: of directed dynamics on a random graph. The view of a neural net as a random graph is explicit in the papers of Kauffman (1969), Little and Shaw (1978) and Hopfield (1982), for example. More recent papers are those by Derrida and co-workers and by the author, listed in the references, particular aspects of which we shall refer to later.