Matrix Criteria for Arbitrary Reliability in Iterated Neural Nets

In a previous paper by this author, the problem of achieving arbitrary reliability for combinatorial nets from arbitrarily unreliable elements was reduced to the study of the convergence properties of an associated polynomial system. In this paper simple criteria which specify the convergence of such a system to a nodal fixed point are obtained from known results in matrix theory. (Convergence to a nodal fixed point implies that the corresponding net approaches reliability arbitrarily near 1 for a particular function.) Theorems are also given which show that it is possible to obtain, from a single system converging to a nodal fixed point, many systems having this property. In a recent paper by this author [1] the organization and reliability of large combinatorial nets were investigated. In that paper several theorems were proved whose ultimate purpose was to delineate conditions under which arbitrarily reliable homogeneous nets could be obtained from arbitrarily unreliable elements. It is the main purpose of the present note to amplify these results as well as prove certain new theorems from which arbitrary reliability for a particular net may be tested by making use of easily applied theorems and associated algorithms. We begin by reviewing certain necessary concepts from the aforementioned paper. This paper should be consulted for a more detailed treatment of these ideas.