Why is it easier to predict the epidemic curve than to reconstruct the underlying contact network?

We study the deterministic Susceptible-Infected-Susceptible (SIS) epidemic model on weighted graphs. van Mieghem et al. have shown that it is possible to learn an estimated network from a finite time sample of the trajectories of the dynamics that in turn can give an accurate prediction beyond the sample time range, even though the estimated network might be qualitatively far from the ground truth. We give a mathematically rigorous derivation for this phenomenon, notably that for large networks, prediction of the epidemic curves is robust, while reconstructing the underlying network is ill-conditioned. Furthermore, we also provide an explicit formula for the underlying network when reconstruction is possible. At the heart of the explanation, we rely on Szemerédi's weak regularity lemma.