techniques or examples of analyzing a series of graphs
I think your question falls in the context of "graph operators". A graph operator $S$ is just a map from the class of graphs to the class of graphs, and you then have a sequence $G$, $S(G)$, $S^2(G)$, etc. 

I do not think that there could possibly be a general way to determine $S^n(G)$ as a function of $n$ and $G$ that could work for all $S$, even for a fixed $S$. For example, if $S(G)$ is the intersection graph of the maximal cliques of $G$ (considered in Harary's book and denoted by $K$), then what you ask is known for only a handful of families, including the complements of $nK_2$, the locally $C_6$ graphs and the so called "clockwork graphs".

Many other graph operators are studied in the book "Graph Dynamics" by Erich Prisner.