It is not weakness of nonlinearity but sparseness of nonlinear couplings that plays a key role in the sparse direct-interaction perturbation (SDIP), which is a secound-order moment closure theory for nonlinear dynamical systems. The SDIP is similar but different in procedure from Kraichnan's direct-interaction approximation (DIA). Homogeneous Navier-Stokes turbulence is an example of dynamical systems in which nonlinearity is strong in magnitude but sparse in coupling. In order to clarify the importance of sparseness of coupling, we formulate SDIP for a model equation which has three parameters: coupling density, strength of nonlinearity and the number of degrees of freedom. By the help of numerical simulations, it is shown that SDIP is applicable when the coupling density is sufficiently smaller than the square root of the number of degrees of freedom, even if the strength of nonlinearity is infinitely large. This implies that the applicability of SDIP has nothing to do with the Gaussianity of a dynamical variable, whereas DIA is often explained as a theory based on it. The SDIP is also applied to a shell model of turbulence with sparse coupling which exhibits the Kolmogorov similarity. The solution to SDIP equations is consistent with inertial range properties such as the -5/3 power law of the energy spectrum.