Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation

Implicit Graph Neural Networks (IGNNs) define node representations as fixed points of graph neural operators, enabling effectively infinite-depth propagation, iteration-independent parameterization, and flexible test-time computation. Yet these benefits depend on the equilibrium being unique and reached by fixed-point iteration. Existing constructions often impose constraints on recurrent updates to obtain these guarantees, limiting the transformations available at equilibrium. This raises a central question: can IGNNs gain expressiveness through richer, edge-dependent transformations while retaining the inherent strengths of their equilibrium formulation? We introduce SheafDEQ, a subhomogeneous deep-equilibrium architecture with adaptive neural-sheaf propagation. Its learned, matrix-valued sheaf restriction maps can align, mix, or reverse neighbouring representations. Under mild regularity conditions, we prove that SheafDEQ admits a unique equilibrium reached globally by fixed-point iteration from any positive initial state. Contractivity further guarantees convergence under bounded communication staleness. We evaluate SheafDEQ on distributed-inference tasks requiring repeated nonlocal aggregation and on community detection whose rewiring increasingly favours cross-community interactions. SheafDEQ improves over implicit baselines on Sums, MNIST Terrain, and Coordinates, and on community detection as connectivity becomes increasingly heterophilic. SheafDEQ remains competitive with recurrent models without equilibrium guarantees. Continued-iteration diagnostics show decreasing residuals and retained test accuracy through 100 iterations, unlike the finite-horizon control, while delayed-update experiments show low sensitivity to bounded communication staleness.