Before revision:  We investigate the complexity of k-levels in random line and hyperplane arrangements. When the arrangement is obtained from any fixed projective line arrangement of n lines by choosing a random cellto contain the south-pole, we prove an upper bound of O((k+1)^2) on the expected complexity of the k-level .
After revision:  Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the k-level denotes the vertices at distance at most k to a marked cell, the prove an upper bound of O((k+1)^2) on the expected complexity of the k-level .