Continuum limits [of networks]

For all everyday purposes a region in a network with enough nodes and an appropriate pattern of connections can act just like ordinary continuous space. But at a formal mathematical level this can happen rigorously only in an infinite limit. And in general, there is no reason to expect that all properties of the system (notably for example the existence of particles) will be preserved by taking such a limit. But in understanding the structure of space and comparing to ordinary continuous space it is convenient to imagine taking such a limit. Inevitably there are several scales involved, and one can only expect continuum behavior if one looks at scales intermediate between individual connections in the underlying network and the overall size of the whole network. Yet as I will discuss on pages 534 and 1050 even at such scales it is far from straightforward to see how all the various well-studied properties of ordinary continuous space (as embodied for example in the theory of manifolds) can emerge from discrete underlying networks.