  A book embedding of a complete graph is a spatial embedding whose planar
projection has the vertices located along a circle, consecutive vertices are
connected by arcs of the circle, and the projections of the remaining
"interior" edges in the graph are straight line segments between the points on
the circle representing the appropriate vertices. A random embedding of a
complete graph can be generated by randomly assigning relative heights to these
interior edges. We study a family of two-component links that arise as the
realizations of pairs of disjoint cycles in these random embeddings of graphs.
In particular, we show that the distribution of linking numbers can be
described in terms of Eulerian numbers. Consequently, the mean of the squared
linking number over all random embeddings is $\frac{i}{6}$, where $i$ is the
number of interior edges in the cycles. We also show that the mean of the
squared linking number over all pairs of $n$-cycles in $K_{2n}$ grows linearly
in $n$.
