We consider an \varepsilon-periodic (\varepsilon \rightarrow 0) tubular structure, modeled 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 \BbbR which is fourth order at a discrete set of values of the magnetic potential (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.