Characterization of normal asymmetric matrices relevant to directed networks

Develop a complete characterization of normal asymmetric matrices (normal but non-symmetric weight matrices) that can arise as weighted adjacency matrices of directed networks, clarifying their structural properties and implications for dynamical behavior.

Background

The paper contrasts trophic directedness and non-normality as distinct notions of directedness. It points out that while all symmetric matrices are normal and undirected networks correspond to normal matrices, there exist asymmetric matrices that are normal, and their full characterization is lacking.

This gap matters for understanding when directed networks can be normal (hence potentially exhibit particular dynamical properties) and for differentiating hierarchical asymmetry from other forms of directedness. The paper provides examples (e.g., circulant matrices) but notes a comprehensive classification is missing.