Scalable BGFT without explicit Jordan chains

Develop scalable computational methods for BGFT on large sparse directed graphs that avoid explicit Jordan decomposition, including (i) Krylov or other non-Hermitian subspace techniques to approximate biorthogonal spectral coordinates and (ii) stable polynomial or rational approximations for spectral filters h(A) that bypass explicit spectral factorization.

Background

When the adjacency operator is non-diagonalizable, Jordan chains are required for a complete spectral representation, but computing Jordan forms is impractical for large sparse graphs. The paper highlights the need for scalable alternatives to enable BGFT in realistic directed-network settings.

This open problem seeks algorithmic substitutes that retain the benefits of BGFT—such as spectral-domain filtering and analysis—without relying on brittle or infeasible Jordan computations.