Ow.

[dmm]

Imagine 256 systems in a square grid. Imagine that each system is connected to its 4 nearest neighbors. Place 16 players uniformly in this grid. Each player will be 4 jumps from each of his neighbors, at the "center" of a 16-system region. I say "center," in quotes, because the true center of each region is between the 4 center systems. So you'd have to pick one of the 4 center systems (say, the lower left) as the universal "center."

OK, now randomize the connections just a bit, so that connections can be either to nearest neighbors or to next-nearest neighbors, with a maximum of 4 connections per system. Also randomize placement of players within the central 4 systems of "their" 16 systems. Looked at this way, it isn't hard to see how you could accidentally end up only two jumps or even one jump away from a neighbor, even though the player distribution was initially uniform. However, you can still count on each quadrant containing 16/4=4 players, so in that sense the distribution is still "uniform."