  A commonly used approach to study stability in a complex system is by
analyzing the Jacobian matrix at an equilibrium point of a dynamical system.
The equilibrium point is stable if all eigenvalues have negative real parts.
Here, by obtaining eigenvalue bounds of the Jacobian, we show that stable
complex systems will favor mutualistic and competitive relationships that are
asymmetrical (non-reciprocative) and trophic relationships that are symmetrical
(reciprocative). Additionally, we define a measure called the interdependence
diversity that quantifies how distributed the dependencies are between the
dynamical variables in the system. We find that increasing interdependence
diversity has a destabilizing effect on the equilibrium point, and the effect
is greater for trophic relationships than for mutualistic and competitive
relationships. These predictions are consistent with empirical observations in
ecology. More importantly, our findings suggest stabilization algorithms that
can apply very generally to a variety of complex systems.
