  Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian
${\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$
by $\dot x=-{\cal L}x$ for consensus and $\dot p=-p{\cal L}$ for diffusion. We
consider both these processes as well their discrete time analogues. We define
a basis of row vectors $\{\bar \gamma_i\}_{i=1}^k$ of the left null-space of
${\cal L}$ and a basis of column vectors $\{\gamma_i\}_{i=1}^k$ of the right
null-space of ${\cal L}$ in terms of the partition of $G$ into strongly
connected components. This allows for complete characterization of the
asymptotic behavior of both diffusion and consensus --- discrete and continuous
--- in terms of these eigenvectors.
  As an application of these ideas, we present a treatment of the pagerank
algorithm that is dual to the usual one. We further show that the teleporting
feature usually included in the algorithm is not strictly necessary.
  This is a complete and self-contained treatment of the asymptotics of
consensus and diffusion on digraphs. Many of the ideas presented here can be
found scattered in the literature, though mostly outside mainstream mathematics
and not always with complete proofs. This paper seeks to remedy this by
providing a compact and accessible survey.
