Identification of the dark complement block under higher-order gain variation

Determine whether the complement block of the resolvent interaction matrix is identified through higher-order joint variation of the varying gains in the exact nonlinear fixed-point model, and characterize the resulting identified set when all gain derivatives expose only products of cross-pattern entries.

Background

The cross-identification theorem establishes exact point identification of a cross-pattern subset of the resolvent interaction matrix: varying gains at nodes with outgoing feedback identify the corresponding rows and columns. At first order, however, the block indexed by nodes whose gains do not vary remains unidentified.

The paper notes that higher-order nonlinear dependence might transmit information about this dark block through the identified cross-pattern entries, but it does not establish whether that dependence is sufficient for identification. Resolving this question would determine the full identified set under partial gain variation and clarify the limits of exact nonlinear identification.