  We study the differential operator $A=\frac{d^4}{dx^4}$ acting on a connected
network $\mathcal{G}$ along with $\mathcal L^2$, the square of the discrete
Laplacian acting on a connected discrete graph $\mathsf{G}$. For both operators
we discuss well-posedness of the associated {linear} parabolic problems \[
  \frac{\partial u}{\partial t}=-Au,\qquad\frac{df}{dt}=-\mathcal L^2 f, \] on
$L^p(\mathcal{G})$ or $\ell^p(\mathsf{V})$, respectively, for $1\leq
p\leq\infty$. In view of the well-known lack of parabolic maximum principle for
all elliptic differential operators of order $2N$ for $N>1$, our most
surprising finding is that, after some transient time, the parabolic equations
driven by $-A$ may display Markovian features, depending on the imposed
transmission conditions in the vertices. Analogous results seem to be unknown
in the case of general domains and even bounded intervals. Our analysis is
based on a detailed study of bi-harmonic functions complemented by simple
combinatorial arguments. We elaborate on analogous issues for the discrete
bi-Laplacian; a characterization of complete graphs in terms of the Markovian
property of the semigroup generated by $-\mathcal L^2$ is also presented.
