Explain why the community-structure eigenvalue is real-dominant with near-zero imaginary part

Characterize and prove the theoretical reasons under which, for directed graphs, the eigenvalue associated with the eigenvector encoding cluster structure for the complex non-backtracking matrix B_alpha has the largest real part and an almost zero imaginary part.

Background

Empirically, the authors observe that the eigenvalue tied to community-structure eigenvectors tends to have the largest real part and nearly zero imaginary part, which underlies the effectiveness of their spectral method.

A theoretical explanation would clarify the spectral signatures of community structure in CNBT, potentially yielding diagnostic metrics for cluster strength and stability.