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Relax or generalize the uniform-prior and degree-homogeneity assumptions in the CNBT–BP link

Ascertain to what extent the uniform prior over cluster labels and the degree homogeneity assumption in the belief propagation derivation can be generalized or relaxed while preserving the theoretical connection to the complex non-backtracking matrix B_alpha for directed graphs.

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Background

The derivation connecting BP to CNBT relies on simplifying assumptions, specifically a uniform prior over cluster memberships and approximate degree homogeneity across clusters. These assumptions may be restrictive in real networks with heterogeneous degrees and priors.

Understanding how far these assumptions can be relaxed without breaking the CNBT–BP connection is essential for extending theory and improving the robustness of algorithms.

References

Finally, an important concern is the extent to which the assumptions introduced in the previous section (namely, a uniform prior shared by all vertices and the degree homogeneity) can be generalized or relaxed. Addressing these open questions is critical for advancing our understanding of the properties of complex-valued matrix representations.

— Complex non-backtracking matrix for directed graphs (2507.12503 - Sando et al., 16 Jul 2025) in Subsubsection 'Discussion', within Subsection 'Relationship with Belief Propagation' (Section 4)