Which Communication Routes Help a GNN? A Graphon Theory of Task-Conditioned Edge Sensitivity

Which communication routes in a graph neural network actually help its task, and does the answer remain meaningful when the same population is represented by graphs of different sizes or by different random samples? On a single finite graph, this question can be answered locally by differentiating the task loss with respect to edge weights. Across graphs, however, those derivatives live in different coordinates and need not be directly comparable. We use graphons to make this question tractable at the population level. A graphon provides common coordinates for communication in large dense graphs, allowing the local value of changing a route to be represented by a signed task-conditioned sensitivity field. For nonlinear residual message passing, we derive this field exactly: each route is valued through the interaction between the representation difference available to cross it and a backward signal describing what the task needs. We prove that the complete field is stable under graphon cut convergence. In particular, dense Bernoulli graphs may remain noisy edge by edge in raw $L^2$, while their sensitivities converge in expected $L^1$ at rate $O(n^{-1/2})$ to those of the corresponding mean communication kernel. Under stronger aligned $L^2$ assumptions, finite communication updates also track a common population trajectory over fixed horizons. Experiments separate structural stability from operational usefulness. Synthetic and CiteSeer-calibrated studies show stable sensitivities despite persistent edge noise, and a fixed-forward diagnostic shows that changing the task alone can change communication preferences. A controlled routing problem shows that route-specific forward–backward alignment can matter for intervention, whereas on METR-LA much of the transferred intervention benefit is captured by a simpler signed magnitude score. Together, these results show when task-conditioned communication has a coherent population meaning while separating that guarantee from how much task-specific detail is useful in a particular model.