  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 $O(n\cdot (k+1)^{1/3})$. 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 $\Theta((k+1)^{d-1})$.
