Brief Announcement: 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. The first principle, named consistent fixed points, states that if a variable is written by two subnetworks of N , then the values of this variable, when these two subnetworks reach fixed points, are equal. The second principle, named negative gain, states that the sum of displacements along every directed loop in network N is negative.