The large-scale structure of the Universe, as traced by the distribution of galaxies, is now being revealed by large-volume cosmological surveys. The structure is characterized by galaxies distributed along filaments, the filaments connecting in turn to form a percolating network. Our objective here is to quantitatively specify the underlying mechanisms that drive the formation of the cosmic network: By combining percolation-based analyses with N-body simulations of gravitational structure formation, we elucidate how the network has its origin in the properties of the initial density field (nature) and how its contrast is then amplified by the nonlinear mapping induced by the gravitational instability (nurture).