When examining the modularity of graphs or networks, concepts like "blocks", "clusters", or "communities" are used, more or less strictly defined as (maximal) subgraphs with some given properties.
A natural generalization of cliques (as maximal complete subgraphs, having diameter 1) would be "diam-k-cliques", i.e. maximal subgraphs with diameter k. Under which name are such graphs filed and can be googled? And which efficient methods to find them are there - or some statistics on them? Is there a Mathematica function for it?
To make "diam-2-cliques" more community-like one might add the simple requirement that the minimal degree of such a subgraph must be 2 (to avoid star graphs as "communities").