You don't need a PhD in mathematics. Firstly, they explain the (already existing) concept of a closure (think for example transitive closure, which seems to be

You don't need a PhD in mathematics. Firstly, they explain the (already existing) concept of a closure (think for example transitive closure, which seems to be exactly what they're going for). Y.ϕ is a notation that means we apply operator ϕ on Y. It seems like they work with a closure operator that goes akin to "if we have a graph G where A relates to B and B relates to C, then G.ϕ additionally has A relating to C". Then they look at how a social network evolves under the (formalized) assumption that no two individuals become one individual. Then they prove, under this condition, that, as the network evolves, some separate sets remain separate. That is, certain structures in the graph remain stable. That's a broad overview, I think. I only spent like 15 minutes reading and writing this, so I might be mistaken.