Consistent Fixed Points and Negative Gain

We discuss the stabilization properties of networks that are composed of ¿displacement elements¿. Each displacement element is defined by an integer K, called the displacement of the element, an input variable x, and an output variable y, where the values of x and y are non-negative integers. An execution step of this element assigns to y the maximum of 0 and K + x. The objective of our discussion is to demonstrate that two principles play an important role in ensuring that a network N is stabilizing, i. e. starting from any global state, network N is guaranteed to reach a global fixed point. Specifically, the principle of consistent fixed points is analogous to the requirement that a control system be free from self-oscillations. And the principle of negative gain is analogous to the requirement that the feedback loop of a sum of displacements along every directed loop in network N is negative.