  We consider an $\varepsilon$-periodic ($\varepsilon\to 0$) tubular structure,
modelled as a magnetic Laplacian on a metric graph, which is periodic along a
single axis. We show that the corresponding Hamiltonian admits norm-resolvent
convergence to an ODE on $\mathbb{R}$ which is fourth order at a discrete set
of values of the magnetic potential (\emph{critical points}) and second-order
generically. In a vicinity of critical points we establish a mixed-order
asymptotics. The rate of convergence is also estimated. This represents a
physically viable model of a phase transition as the strength of the (constant)
magnetic field increases.
