Relaxed coherence sufficiency in matrix-weighted networks

Determine whether a relaxed coherence condition for matrix-weighted networks—where all edge-weight matrices along any directed cycle share a common eigenvector associated with eigenvalue 1—is sufficient to guarantee non-trivial long-time behavior (e.g., steady-state subspace) when 1 is the largest eigenvalue of the governing operator such as the random-walk transition matrix; and investigate alternative formulations in which the shared eigenvector along a cycle may have eigenvalues whose product equals 1.

Background

The paper defines coherence for matrix-weighted networks (MWNs) as the requirement that the product of the edge transformations along every directed cycle equals the identity matrix. This strict condition ensures the existence of non-trivial steady states for dynamics such as consensus and random walks.

In the Supplemental Material, the authors propose relaxing coherence by requiring that the edge-weight matrices along a directed cycle share a common eigenvector associated with eigenvalue 1, so that coherence holds within the subspace spanned by this eigenvector. They suggest that this relaxed condition might be sufficient in the long-time limit when 1 is the largest eigenvalue (as for random-walk transition matrices), and they also point to an alternative formulation where the product of eigenvalues along a cycle equals 1 even if the shared eigenvector is not always associated with eigenvalue 1. They explicitly leave these questions for future research.