Recently, the study of systems on networks, general graphs with some combination of nodes and vertices, has become of interest. I will consider a particular network: the small world network. This network combines long-range and short-range interactions, and has become a standard model in the field. I will show that for a wide range of systems the behavior on this network can be described by mean-field critical behavior, and I will analyze the crossover to this behavior.

Finally, I will finish with the example of the Edwards-Wilkinson equation. Recent work of Toroczkai, Korniss, and others, has shown the relevance of this equation to synchronization in parallel processing. I will apply the results above to this example, and find that in some cases the mean-field behavior applies and in some cases it does not. (Some of this is joint work with B. Kozma and G. Korniss.)