Actin filaments assemble into network-like structures and play an important role in various cellular mechanical processes. It is known that the response of actin networks cross-linked by stiff proteins is characterized by two distinct regimes: (i) a linear stress strain response for small deformations and (ii) a power law relation between elastic shear modulus and stress with an exponent 3/2 for large deformations. If the response is interpreted by the entropic single filament model, this hardening is attributed to the reduction of the thermal filament fluctuations as the filaments undergo affine stretching. By contrast, here we numerically study the elastic properties of a discrete, fully three-dimensional model for an isotropic filamentous networks, where athermal filaments of finite length are interconnected by rigid cross-links. By analyzing the network microstructure, the network connectivity is quantified in terms of the ratio of the mean filament length and the mean cross-linking distance along the filament. We derive a scaling relation for the initial network shear modulus-i.e., regime i-as a function of the network connectivity. When sheared to large strains, the simulated networks exhibit nonlinear strain hardening characterized by a 3/2 power law-regime ii. The origin of the strain hardening is shown to be associated now with the stretching of percolating and nearly fully extended stress paths in the network. In addition, we show that the strain at the onset of the strain hardening regime only depends on the network connectivity.