Eigenvalue Pairing in the Response Matrix for a Class of Network Models with Circular Symmetry

Abstract

We consider the response matrices in certain weighted networks that display a circular symmetry. It had been observed empirically that these exhibit several paired (multiplicity two) eigenvalues. Here, this pairing is explained analytically for a version of the model more general than the original. The exact number of necessarily paired eigenvalues is given in terms of the structure of the model, and the special structure of the eigenvectors is also described. Examples are provided.