Let ${cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The emph{$k$-level} of ${cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${cal L}$ passing below $v$. The complexity (the maximum size) of the $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(ncdot (k+1)^{1/3})$. 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 emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $mathbb{S}^d$ which are orthogonal to a set of random points on $mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $Theta((k+1)^{d-1})$.
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