  We study the expected adjacency matrix of a uniformly random multigraph with
fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a
variety of analyses of networked data sets, including modularity-maximization
and mean-field theories of spreading processes. Its structure is
well-understood for large, sparse, simple graphs: the expected number of edges
between nodes $i$ and $j$ is roughly $\frac{d_id_j}{\sum_\ell{d_\ell}}$. Many
network data sets are neither large, sparse, nor simple, and in these cases the
standard approximation no longer applies. We derive a novel estimator using a
dynamical approach: the estimator emerges from the stationarity conditions of a
class of Markov Chain Monte Carlo algorithms for graph sampling. We derive
error bounds for this estimator, and provide an efficient scheme with which to
compute it. We test the estimator on synthetic and empirical degree sequences,
finding that it enjoys relative error against ground truth a full order of
magnitude smaller than the standard approximation. We then compare modularity
maximization techniques using both the standard and novel estimator, finding
that the qualitative structure of the optimization landscape depends
significantly on the estimator choice. Our results emphasize the importance of
using carefully specified random graph models in data scientific applications.
