  We study the voter model and related random-copying processes on arbitrarily
complex network structures. Through a representation of the dynamics as a
particle reaction process, we show that a quantity measuring the degree of
order in a finite system is, under certain conditions, exactly governed by a
universal diffusion equation. Whenever this reduction occurs, the details of
the network structure and random-copying process affect only a single parameter
in the diffusion equation. The validity of the reduction can be established
with considerably less information than one might expect: it suffices to know
just two characteristic timescales within the dynamics of a single pair of
reacting particles. We develop methods to identify these timescales, and apply
them to deterministic and random network structures. We focus in particular on
how the ordering time is affected by degree correlations, since such effects
are hard to access by existing theoretical approaches.
