The analysis of equilibria of complex systems is challenging due to the high dimensionality and non-linear interactions. There are different kinds of complex systems such as cooperative systems in which all interactions help growth and mixed-weight systems, in which some interactions help growth, while others hinder it. In this thesis, we focus on the correlation between the equilibria of cooperative and mixed-weight systems. We simplified the equilibrium equations by combining the attributes of a system into a one-dimensional equation. This reduced equation is easy to compute and provides an upper bound on the equilibrium value of each node. Although this bound may exceed many actual equilibrium values, it still defines the subspace in which all equilibria must lie. For cooperative systems, we presented a theorem that provides constraints on two vectors. If these vectors satisfy the given conditions, then there exists an equilibrium between the components. We also discussed methods to find such vector pairs. We applied this theorem to relate the equilibria of mixed-weight and cooperative systems. The equilibria of the mixed-weight system are always less than or equal to some equilibrium in the cooperative system. We introduced a framework for classifying cooperative equilibria. On any subset of nodes, an equilibrium may have entries that are maximal compared to all other equilibria on that subset. This leads to a single equilibrium that is the largest at every entry, called the principal equilibrium, which is component-wise maximal. The principal equilibrium upper bounds all equilibria of the mixed-weight system. Finally, we discussed the inherent difficulty of translating cooperative equilibria into the mixed-weight system, which stems from high dimensionality and non-linearity. We stated the conditions that mixedweight equilibria must satisfy and provided constraints determining if a cooperative-system equilibrium remains valid when competitive interactions are added. This concludes the comparison by showing that the principal equilibrium provides a component-wise upper bound for all equilibria of the mixed-weight system.