arXiv:2608.23655 2026 Architecture 2 ideas extracted · analyzed Aug 29, 2026
The paper provides a principled way to attach unit-modulus complex phases to directed graph edges while preserving a useful gauge symmetry: node-wise phase reparameterizations change individual edge gains but leave cycle products invariant. This suggests graph-neural layers whose messages are complex and gauge-equivariant, allowing models to represent directional frustration, circulation, and phase consistency rather than only edge magnitudes. The spectral comparison theorem also gives a safety guarantee: phase-weighted propagation cannot have larger spectral radius than the nonnegative underlying adjacency, which can be used to build stable graph filters or constrain learned propagation operators.
Replace real nonnegative graph-edge weights in a directed GNN with unit-modulus complex gains and complex node states. Enforce node-wise U(1) gauge equivariance, so arbitrary phase choices at individual nodes cannot change predictions while gauge-invariant cycle phases remain available to encode directed relational structure.
Use unit-gain directed propagation as a drop-in graph filter whose spectral radius is provably bounded by that of the underlying nonnegative adjacency. Add an inexpensive power-iteration monitor or rescaling rule so learned graph propagation remains no more expansive than the corresponding magnitude-only operator.