Anisotropic fibrous networks, especially transverse isotropic fibrous networks, are widely used to model the microstructures of biological tissues, polymer gels, fibrous thermal insulations, and other fibrous materials. In this letter, we build a three-dimensional transverse isotropic fibrous network model and study its mechanical properties along the through-thickness direction. We propose a measurement of anisotropy for transverse isotropic fibrous networks and then study the influence of anisotropy on the networks' mechanical properties, including its elastic modulus, maximum elongation, and stress-strain curve, by means of finite-element simulation. We also study theoretically the influence of anisotropy on maximum elongation. We find that as the anisotropy of the networks becomes stronger, the elastic modulus decreases and the maximum elongation increases, indicating a transition in mechanical properties from brittle to ductile. We identify this transition as the "ductile-brittle transition." This transition can help guide the design and regulate the mechanical properties of a transverse isotropic fibrous network.