1989): the number of times each possible constant-order graph occurs as a subgraph of a Paley graph is (in the limit for large q) the same as for random graphs, and large sets of vertices have approximately the same number of edges as they would in random graphs.
The number of times a specific type of graph appears in a larger graph, particularly when considering constant-order graphs within Paley graphs, approaches that of random graphs for large values of q.  Similarly sized sets of vertices exhibit similar edge counts to those found in random graphs.