You're not going to be able to "effectively" propagate that transaction in 1 second to 7000 nodes with 3 neighbors. You would never have distributed consensus among those nodes. If user's can't get distributed consensus, then they will gravitate toward the subset of node(s) which has(have) the closest thing to it.
Why 1 sec, not 1 min?
Consensus among nodes is not required, it's perpendicular to consensus about ledger state.
I can't comment on the gravitation thing without knowing what reference strategy is being used in your scenario, I know at least one strategy that doesn't lead to fragmentation.