Multivaluedness in Networks: Theory

Abstract

<div><div>This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded1 systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufficient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. </div><div><br></div><div>The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems.</div><div><br></div><div>The significance of the work presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.</div></div>

Multivaluedness in Networks: Shannon’s Noisy-Channel Coding Theorem

“…Currently, work is in progress to extend the results of [2] to scenarios where both measurement noise and exogenous noise propagating through the system will be accounted for. Once this undertaking has been completed, this would enable one to also consider the individual encoders E n as well.…”

“…Figure 1 depicts what [2] refers to as the counter-cascaded configuration of the systems M and N • T . The symbols M , T and N serve the dual purpose of representing the nonlinear systems shown as well as their respective mathematical descriptions as nonlinear operators [6] M : U → W , T : U → V and N : V → X that map from and into the signal spaces indicated.…”

“…Surprisingly, multivaluedness appears to have been confronted for the first time only recently in [1], where inverse modeling in microwave filters using neural networks was studied. Subsequently, [2] proposed a counter-cascaded systems analysis framework, which led to a necessary and sufficient condition for the manifestation of multivaluedness. Further analytical results were also presented there for the characterization of multivaluedness.…”

Multivaluedness in Networks: Exemplars

“…Multivaluedness appears to have been confronted for the first time in [1], where inverse modeling in microwave filters using neural networks was studied. Subsequently, van Wyk [2] proposed a counter-cascaded systems analysis framework, which led to a necessary and sufficient condition for the manifestation of multivaluedness. Furthermore, its application to the structural reduction of complex networks [2] demonstrated the fundamental nature of this framework in the context of signals-andsystems theory.…”

“…II. COUNTER-CASCADED SYSTEMS AND MULTIVALUEDNESS Figure 1 depicts the counter-cascaded configuration of the systems M and N • T introduced in [2]. The symbols M , T and N serve the dual purpose of representing the systems in Fig.…”