Clarify the theoretical relationship between the CNBT matrix and belief propagation

Establish a rigorous theoretical characterization of the relationship between the complex non-backtracking matrix B_alpha for directed graphs and belief propagation on directed graphs, by determining how the spectrum and eigenvectors of B_alpha correspond to message dynamics, linearizations, and fixed points of belief propagation.

Background

The paper introduces the complex non-backtracking (CNBT) matrix for directed graphs and derives a link to belief propagation (BP) via a linearization that yields a structure resembling B_alpha. While this suggests a fundamental connection, the authors note that the relationship is not yet fully understood.

A precise theoretical mapping between CNBT spectral features and BP behavior would help explain why CNBT-based spectral clustering performs well in sparse directed settings and could guide algorithm design and analysis.