Social influence in networks is commonly modeled by linear threshold and independent cascade dynamics and analyzed via iterative simulation, which can be costly and can obscure the structure of steady-state outcomes. This research proposes a complementary formulation that recasts influence propagation as a system of nonlinear equations whose fixed points are the diffusion equilibria. A connection between system of nonlinear equation equilibria and classical simulation outcomes is established. Numerical and global optimization techniques are leveraged in the solution of the system of nonlinear equations to locate multiple solutions and examine local stability. In computational experiments, the system of nonlinear equation approach reproduces benchmark cascade sizes while efficiently exploring many seed sets and parameter regimes. It also finds alternative equilibria and provides sensitivity and stability diagnostics that are cumbersome to obtain via simulation alone. Overall, framing diffusion as a system of nonlinear equations yields a practical and informative complement to simulation for analysis and decision-making in influence propagation.