Modular Equilibrium: Structure-Aware Solving for Graph Neural Networks

Iteration-based implicit graph solvers—deep equilibrium models and their graph variants—claim to compute an equilibrium, but the equilibrium is nominal: the forward pass stops far from its tolerance, yet forcing full convergence changes test performance not at all. We show that the "infinite depth" of these models is carried entirely by cycles, and propose modular equilibrium, a fixed-point graph layer whose solver is organized around exactly that structure: decompose the graph into cyclic modules—strongly connected components for directed learned graphs, and bridge (2-edge-connected) decomposition for undirected ones, which we show is the correct analogue since SCCs degenerate on bidirected graphs—solve the fixed point within each module with a bounded budget, and propagate between modules by bounded feed-forward sweeps along the module graph, with gradients exact for the unrolled computation: bounded where computation is bounded in practice, iterative exactly where feedback lives. We prove that modular propagation is exact in the converged limit—it reaches the global fixed point itself (block back-substitution), not an approximation of it—and that its budget follows structural complexity: iterations are spent inside dense cyclic modules, whose truncation tail scales as 1/(1-r), and saved on the structurally simple inter-module links. On complex superpixel graphs, a depth-tuned, parameter-matched local baseline only ties global equilibrium solving, while modular solving exceeds both; on small molecular graphs, where a few iterations nearly suffice, it matches global solving at the default coupling and degrades more gracefully as coupling strengthens, while enabling what global solvers cannot express: causal module-importance analysis, which identifies the molecules' macrocyclic rings as the load-bearing units. The advantage tracks the predicted boundary conditions—it vanishes when structural encodings already supply the cyclic structure—and random partitions of the same size distribution destroy the benefit.