  We study higher-order elliptic operators on one-dimensional ramified
structures (networks). We introduce a general variational framework for
fourth-order operators that allows us to study features of both hyperbolic and
parabolic equations driven by this class of operators. We observe that they
extend to the higher-order case and discuss well-posedness and conservation of
energy of beam equations, along with regularizing properties of polyharmonic
heat kernels. A noteworthy finding is the discovery of a new class of
well-posed evolution equations with Wentzell-type boundary conditions.
