Stability-optimized shift/operator selection for directed graphs

Identify, for a fixed directed graph topology G and a specified class of admissible shift operators S (including adjacency variants, directed Laplacians, and transition operators), an operator A_opt in S that minimizes the eigenbasis condition number kappa(V_A), thereby yielding the most numerically robust Biorthogonal Graph Fourier Transform (BGFT) for G.

Background

The paper develops a BGFT framework for directed graphs using biorthogonal left/right eigenvectors of the adjacency operator, emphasizing the impact of non-normality and eigenbasis conditioning on stability. Different operator choices (adjacency, directed Laplacian, transition matrices) can lead to markedly different conditioning and numerical behavior.

This open problem asks for a principled operator selection within an admissible class that optimizes the conditioning of the spectral basis, directly targeting robustness of BGFT-based analysis, filtering, and sampling on a given directed topology.