Generic Local Identifiability Does Not Imply Generic Global Identifiability in Linear Networks

Abstract

We give a counterexample to the implication from generic local to generic global identifiability for linear networks with a fixed topology, independently unknown edge coefficients, and selected excited and measured nodes. A six-node network with nine unknown coefficients has exactly two compatible parameter vectors generically. Adding six excited and measured nodes preserves this ambiguity while making the added coefficients directly recoverable. The resulting twelve-node structure has forty-two independent unknown coefficients and zero diagonal. For every positive bound on the largest eigenvalue magnitude, a nonempty open set of parameters has two distinct compatible networks satisfying that bound. This holds for both real and complex coefficients. Other open regions have exactly one admissible solution. The proof uses a rational transformation exchanging the two solutions and a common limiting matrix whose fourth power vanishes. Exact rational-number certificates verify an explicit pair. The construction also extends algebraically when the unknown coordinates are whole rational transfer functions. Genericity in prescribed numerator and denominator coefficient families, and the requirement that every transfer function vanish at infinite frequency, remain separate questions.