In community detection, can $k$-cliques overlap?

Question

When finding communities in a network using $k$-cliques, each $k$-clique may considered a community. I have an assignment where there are many $k$-cliques that appear to overlap. Does this mean they are overlapping communities?

Answer 1

Yes, if you define communities as $k$-cliques then any time there are points $A,B,C$ such that

• $A$ and $B$ are in the same $k$-clique

• $C$ is in some $k$-clique which includes $A$

• $B$ is not connected to $C$

the clique containing $B$ and $A$ is an overlapping (but not identical) community with the clique containing $C$ and $A$.