  The networks of this -- primarily (but not exclusively) expository --
compendium are strongly connected, finite directed graphs $X$, where each
oriented edge $(x,y)$ is equipped with a positive weight (conductance)
$a(x,y)$. We are not assuming symmetry of this function, and in general we do
not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give
rise to a difference operator, the normalised version of which we consider as
our Laplace operator. It is associated with a Markov chain with state space
$X$. A non-empty subset of $X$ is designated as the boundary. We provide a
systematic exposition of the different types of Laplace equations, starting
with the Poisson equation, Dirichlet problem and Neumann problem. For the
latter, we discuss the definition of outer normal derivatives. We then pass to
Laplace equations involving potentials, thereby also addressing the Robin
boundary problem. Next, we study the bi-Laplacian and associated equations: the
iterated Poisson equation, the bi-Laplace Neumann and Dirichlet problems, and
the "plate equation". It turns out that the bi-Laplace Dirichlet to Neumann map
is of non-trivial interest. The exposition concludes with two detailed
examples.
