  We provide a novel method for constructing asymptotics (to arbitrary
accuracy) for the number of directed graphs that realize a fixed bidegree
sequence $d = a \times b$ with maximum degree $d_{max}=O(S^{\frac{1}{2}-\tau})$
for an arbitrarily small positive number $\tau$, where $S$ is the number edges
specified by $d$. Our approach is based on two key steps, graph partitioning
and degree preserving switches. The former idea allows us to relate enumeration
results for given sequences to those for sequences that are especially easy to
handle, while the latter facilitates expansions based on numbers of shared
neighbors of pairs of nodes. While we focus primarily on directed graphs
allowing loops, our results can be extended to other cases, including bipartite
graphs, as well as directed and undirected graphs without loops. In addition,
we can relax the constraint that $d_{max} = O(S^{\frac{1}{2}-\tau})$ and
replace it with $a_{max} b_{max} = O(S^{1-\tau})$. where $a_{max}$ and
$b_{max}$ are the maximum values for $a$ and $b$ respectively. The previous
best results, from Greenhill et al., only allow for $d_{max} =
o(S^{\frac{1}{3}})$ or alternatively $a_{max} b_{max} = o(S^{\frac{2}{3}})$.
Since in many real world networks, $d_{max}$ scales larger than
$o(S^{\frac{1}{3}})$, we expect that this work will be helpful for various
applications.
