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A two-dimensional spin-directed $\mathbb{Z}^{\,}_{2}$ network model isconstructed that describes the combined effects of dimerization and disorderfor the surface states of a weak three-dimensional $\mathbb{Z}^{\,}_{2}$topological insulator. The network model consists of helical edge states oftwo-dimensional layers of $\mathbb{Z}^{\,}_{2}$ topological insulators whichare coupled by time-reversal symmetric interlayer tunneling. It is argued that,without dimerization of interlayer couplings, the network model has noinsulating phase for any disorder strength. However, a sufficiently strongdimerization induces a transition from a metallic phase to an insulating phase.The critical exponent $\nu$ for the diverging localization length atmetal-insulator transition points is obtained by finite-size scaling analysisof numerical data from simulations of this network model. It is shown that thephase transition belongs to the two-dimensional symplectic universality classof Anderson transition.

Spin-directed network model for the surface states of weak three-dimensional $\mathbb{Z}^{\,}_{2}$ topological insulators