For the purposes of a research project, I am wondering if there are any resources on graphs with fractal properties, by which I mean self-similarity in particular. For instance, imagine a graph where nodes could be transformed into subgraphs that were the same as the larger graph, and their nodes could be transformed likewise, etc. I don't mean a graph that is literally exactly like that - but it should have self-similarity at different levels as if it had been created that way, with maybe a little randomness thrown in afterwards.

I was told that Expander graphs were something like what I was looking for, but from what little I understand of their definition, they seem more related to small-world theory than to what I'm looking for.

Edit: I'm because I'm trying ot figure out a way in which representations of social and geographic networks of people could be compressed, probably with significant loss but maintaining basic properties.