  The $k$-cap (or $k$-winners-take-all) process on a graph works as follows: in
each iteration, exactly $k$ vertices of the graph are in the cap (i.e.,
winners); the next round winners are the vertices that have the highest total
degree to the current winners, with ties broken randomly. This natural process
is a simple model of firing activity in the brain. We study its convergence on
geometric random graphs, revealing rather surprising behavior.
