  We study the voter model, under node and link update, and the related
invasion process on a single strongly connected component of a directed
network. We implement an analytical treatment in the thermodynamic limit using
the heterogeneous mean field assumption. From the dynamical rules at the
microscopic level, we find the equations for the evolution of the relative
densities of nodes in a given state on heterogeneous networks with arbitrary
degree distribution and degree-degree correlations. We prove that conserved
quantities as weighted linear superpositions of spin states exist for all three
processes and, for uncorrelated directed networks, we derive their specific
expressions. We also discuss the time evolution of the relative densities that
decay exponentially to a homogeneous stationary value given by the conserved
quantity. The conservation laws obtained in the thermodynamic limit for a
system that does not order in that limit determine the probabilities of
reaching the absorbing state for a finite system. The contribution of each
degree class to the conserved quantity is determined by a local property.
Depending on the dynamics, the highest contribution is associated to
influential nodes reaching a large number of outgoing neighbors, not too
influenceable ones with a low number of incoming connections, or both at the
same time.
