Determine stability conditions linking CNBT eigenvectors to belief propagation fixed points

Determine conditions under which eigenvectors of the complex non-backtracking matrix B_alpha correspond to stable solutions (fixed points) of belief propagation on directed graphs, and characterize the stability criteria explicitly in terms of the eigenstructure of B_alpha.

Background

In deriving a connection between CNBT and BP, the authors arrive at an eigenvalue problem involving B_alpha. However, it is not yet established when such eigenvectors correspond to BP fixed points that are stable, which is crucial for theoretical guarantees and practical algorithms.

Clarifying stability in terms of CNBT eigenstructure would bridge spectral and message-passing viewpoints for directed community detection.