  We explore the voter model dynamics on a directed random graph model ensemble
(digraphs), given by the Directed Configuration Model. The voter model captures
the evolution of opinions over time on a graph where each vertex represents an
individual holding a binary opinion. Our primary interest lies in the density
of discordant edges, defined as the fraction of edges connecting vertices with
different opinions, and its asymptotic behavior as the graph size grows to
infinity. This analysis provides valuable insights, not only into the consensus
time behavior but also into how the process approaches this absorption time on
shorter time scales. Our analysis is based on the study of certain annealed
random walk processes evolving on out-directed, marked Galton-Watson trees,
which describe the locally tree-like nature of the considered random graph
model. Additionally, we employ innovative coupling techniques that exploit the
classical stochastic dual process of coalescing random walks. We extend
existing results on random regular graphs to the more general setting of
heterogeneous and directed configurations, highlighting the role of graph
topology in the opinion dynamics.
