Directed graph learning naturally leads to trainable nonsymmetric propagation operators with distinct right and left spectral structures. Building on the two-sided cone Rayleigh framework for generalized pencils \[ B_θ-λG, \] we develop a learning-oriented methodology for spectral certification, sensitivity analysis, and control without requiring symmetry, nonnegativity, or cone preservation. In the positive-orthant setting, computable lower and upper cone bounds provide an a posteriori enclosure of a distinguished cone level, while smooth soft-min/max surrogates preserve rigorous one-sided bounds with explicit approximation errors and remain differentiable with respect to the trainable parameters. For a simple interior level, the right and left modes satisfy \[ Dλ_C(B)[H]=v_C^T H u_C, \] yielding first-order optimal graph-supported interventions under prescribed perturbation budgets and motivating adaptive spectral control. Numerical experiments demonstrate the applicability of the approach beyond cone-preserving operators and in directed learning settings. Signed nonsymmetric perturbations reveal a transition from interior eigenpairs to boundary complementary quasi-pairs, including non-spectral cone levels, while controlled experiments show that symmetrization can remove predictive information carried solely by edge direction. On the directed Cora citation network, adaptive recomputation of the right--left sensitivity reduces the distinguished spectral level by approximately $21.5\%$ under a cumulative edge-weight reduction budget of $0.5\%$, with no observed change in test accuracy for the trained model and data split considered.
While signed-directed graphs have been studied using linear Laplacians in the design of graph neural networks, relatively little research has focused on developing non-linear Laplacian operators for such networks. We introduce a non-linear Laplacian operator specific to signed and directed networks (NLSD). This non-linear operator extends the concepts of the signed Laplacian for signed graphs and the Laplacian for directed graphs. The NLSD calculates node-specific potentials based on features More precisely, if the potential discrepancy is not aligned with the edge direction, we ignore it (and vice versa) leveraging message-passing techniques only across edges where potential discrepancies align with the edge's direction. Utilizing this novel operator, we propose an efficient spectral GNN framework (NLSD-GNN). We conducted comprehensive evaluations focusing on node classification and link prediction, examining scenarios involving signed, directional, or both types of information. Our findings reveal that this spectral GNN framework not only integrates signed and directional data effectively but also achieves superior performance across diverse datasets.
Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_θ(A_q)\,R$, where $A_q$ is a normalized magnetic operator, $h_θ$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \emph{matrix function} of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that $k = O(\log(1/\varepsilon))$ block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with $1/\sqrt{s}$ Monte-Carlo error, and the undirected $q{=}0$ case improves heterophilous benchmarks over no-PE and polynomial baselines.
Many representation learning problems involve directed relations, such as lexical entailment, sentence entailment, ontology hierarchy, and citation links. Standard Euclidean, cosine, and Mahalanobis heads are symmetric, while generic neural scorers can model directionality but provide limited geometric structure. This paper proposes a role-aware neural convex divergence head for asymmetric representation learning. The head applies source- and target-role projections before evaluating an input-convex neural Bregman divergence, yielding a nonnegative structured score in the role-projected space. We characterize its projected-space identity, source-role convexity, directional-gap decomposition, and Hessian-based local curvature. Experiments on lexical, sentence, ontology, and directed graph benchmarks compare symmetric distances, unstructured asymmetric scorers, order/hyperbolic baselines, plain ICNN-Bregman heads, and the proposed role-aware variant. Across ten random seeds on the main semantic and ontology benchmarks, role-aware projections consistently improve directional accuracy over plain ICNN-Bregman heads while preserving zero observed negative divergence rate. The results also identify a boundary case: on large fixed-feature citation prediction, specialized symmetric or hyperbolic baselines remain stronger in ranking accuracy. Overall, the proposed head is best understood as a structured and interpretable plug-in distance module for tasks where directional relations matter.